Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.7% → 84.1%

Time: 11.9s

Alternatives: 27

Speedup: 1.9×

Specification

Math

\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (*
 (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
 (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 65.7% accurate, 1.0× speedup

Math

\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (*
 (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
 (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)\\ \mathbf{if}\;h \leq -1.62 \cdot 10^{+156}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{h}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 1.2 \cdot 10^{-304}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell} \cdot \sqrt{h}} \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (fma
         (/ (* M D) (* -2.0 d))
         (/ (* (* 0.25 (/ (* M D) d)) h) l)
         1.0)))
  (if (<= h -1.62e+156)
    (*
     (* (sqrt (/ d l)) (* (sqrt (- d)) (sqrt (/ -1.0 h))))
     (-
      1.0
      (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
    (if (<= h 1.2e-304)
      (* (/ (fabs d) (sqrt (* h l))) t_0)
      (* (/ (fabs d) (* (sqrt l) (sqrt h))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fma(((M * D) / (-2.0 * d)), (((0.25 * ((M * D) / d)) * h) / l), 1.0);
	double tmp;
	if (h <= -1.62e+156) {
		tmp = (sqrt((d / l)) * (sqrt(-d) * sqrt((-1.0 / h)))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else if (h <= 1.2e-304) {
		tmp = (fabs(d) / sqrt((h * l))) * t_0;
	} else {
		tmp = (fabs(d) / (sqrt(l) * sqrt(h))) * t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fma(Float64(Float64(M * D) / Float64(-2.0 * d)), Float64(Float64(Float64(0.25 * Float64(Float64(M * D) / d)) * h) / l), 1.0)
	tmp = 0.0
	if (h <= -1.62e+156)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / h)))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	elseif (h <= 1.2e-304)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * t_0);
	else
		tmp = Float64(Float64(abs(d) / Float64(sqrt(l) * sqrt(h))) * t_0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[h, -1.62e+156], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.2e-304], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.6200000000000001e156

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. mult-flip N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{1}{\mathsf{neg}\left(h\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. sqrt-prod N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(h\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-unsound-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(h\right)}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-unsound-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(h\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\color{blue}{-d}} \cdot \sqrt{\frac{1}{\mathsf{neg}\left(h\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-unsound-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{-d} \cdot \color{blue}{\sqrt{\frac{1}{\mathsf{neg}\left(h\right)}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(h\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)}{\mathsf{neg}\left(h\right)}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. frac-2neg-rev N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{h}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{-d} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{h}}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. metadata-eval 37.2%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{\color{blue}{-1}}{h}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 37.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{\frac{-1}{h}}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -1.6200000000000001e156 < h < 1.2e-304

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 77.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)} \]

   if 1.2e-304 < h

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 77.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      2. pow1/2 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      6. pow1/2 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      9. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      10. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      11. lower-unsound-sqrt.f64 44.4%

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

   9. Applied rewrites 44.4%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 84.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)\\ \mathbf{if}\;h \leq -1.62 \cdot 10^{+156}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 1.2 \cdot 10^{-304}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell} \cdot \sqrt{h}} \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (fma
         (/ (* M D) (* -2.0 d))
         (/ (* (* 0.25 (/ (* M D) d)) h) l)
         1.0)))
  (if (<= h -1.62e+156)
    (*
     (* (sqrt (/ d l)) (/ (sqrt (- d)) (sqrt (- h))))
     (-
      1.0
      (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
    (if (<= h 1.2e-304)
      (* (/ (fabs d) (sqrt (* h l))) t_0)
      (* (/ (fabs d) (* (sqrt l) (sqrt h))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fma(((M * D) / (-2.0 * d)), (((0.25 * ((M * D) / d)) * h) / l), 1.0);
	double tmp;
	if (h <= -1.62e+156) {
		tmp = (sqrt((d / l)) * (sqrt(-d) / sqrt(-h))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else if (h <= 1.2e-304) {
		tmp = (fabs(d) / sqrt((h * l))) * t_0;
	} else {
		tmp = (fabs(d) / (sqrt(l) * sqrt(h))) * t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fma(Float64(Float64(M * D) / Float64(-2.0 * d)), Float64(Float64(Float64(0.25 * Float64(Float64(M * D) / d)) * h) / l), 1.0)
	tmp = 0.0
	if (h <= -1.62e+156)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	elseif (h <= 1.2e-304)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * t_0);
	else
		tmp = Float64(Float64(abs(d) / Float64(sqrt(l) * sqrt(h))) * t_0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[h, -1.62e+156], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 1.2e-304], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.6200000000000001e156

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-unsound-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-unsound-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-neg.f64 37.2%

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 37.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -1.6200000000000001e156 < h < 1.2e-304

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 77.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)} \]

   if 1.2e-304 < h

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 77.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      2. pow1/2 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      6. pow1/2 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      9. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      10. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      11. lower-unsound-sqrt.f64 44.4%

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

   9. Applied rewrites 44.4%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_1 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_3 := t\_1 \cdot t\_2\\ t_4 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_3}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_5 := \frac{t\_3}{-2 \cdot d}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(t\_5, \frac{\left(0.25 \cdot \left(t\_2 \cdot \frac{t\_1}{d}\right)\right) \cdot h}{\ell}, 1\right)\\ \mathbf{elif}\;t\_4 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(t\_5, \frac{\left(0.25 \cdot \frac{t\_3}{d}\right) \cdot h}{\ell}, 1\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (/ (fabs d) (sqrt (* h l))))
       (t_1 (fmin M (fabs D)))
       (t_2 (fmax M (fabs D)))
       (t_3 (* t_1 t_2))
       (t_4
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ t_3 (* 2.0 d)) 2.0)) (/ h l)))))
       (t_5 (/ t_3 (* -2.0 d))))
  (if (<= t_4 0.0)
    (* t_0 (fma t_5 (/ (* (* 0.25 (* t_2 (/ t_1 d))) h) l) 1.0))
    (if (<= t_4 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (* t_0 (fma t_5 (/ (* (* 0.25 (/ t_3 d)) h) l) 1.0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(d) / sqrt((h * l));
	double t_1 = fmin(M, fabs(D));
	double t_2 = fmax(M, fabs(D));
	double t_3 = t_1 * t_2;
	double t_4 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_3 / (2.0 * d)), 2.0)) * (h / l)));
	double t_5 = t_3 / (-2.0 * d);
	double tmp;
	if (t_4 <= 0.0) {
		tmp = t_0 * fma(t_5, (((0.25 * (t_2 * (t_1 / d))) * h) / l), 1.0);
	} else if (t_4 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_0 * fma(t_5, (((0.25 * (t_3 / d)) * h) / l), 1.0);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(d) / sqrt(Float64(h * l)))
	t_1 = fmin(M, abs(D))
	t_2 = fmax(M, abs(D))
	t_3 = Float64(t_1 * t_2)
	t_4 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(t_3 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_5 = Float64(t_3 / Float64(-2.0 * d))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(t_0 * fma(t_5, Float64(Float64(Float64(0.25 * Float64(t_2 * Float64(t_1 / d))) * h) / l), 1.0));
	elseif (t_4 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(t_0 * fma(t_5, Float64(Float64(Float64(0.25 * Float64(t_3 / d)) * h) / l), 1.0));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(t$95$3 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(t$95$0 * N[(t$95$5 * N[(N[(N[(0.25 * N[(t$95$2 * N[(t$95$1 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(t$95$0 * N[(t$95$5 * N[(N[(N[(0.25 * N[(t$95$3 / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 77.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \color{blue}{\frac{M \cdot D}{d}}\right) \cdot h}{\ell}, 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{\color{blue}{M \cdot D}}{d}\right) \cdot h}{\ell}, 1\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{\color{blue}{D \cdot M}}{d}\right) \cdot h}{\ell}, 1\right) \]

      4. associate-/l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot h}{\ell}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot h}{\ell}, 1\right) \]

      6. lower-/.f64 75.8%

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \left(D \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot h}{\ell}, 1\right) \]

   9. Applied rewrites 75.8%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot h}{\ell}, 1\right) \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 77.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_1 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_2 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_3 := t\_1 \cdot t\_2\\ t_4 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_3}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\frac{t\_3}{-2 \cdot d}, \frac{\left(0.25 \cdot \left(t\_2 \cdot \frac{t\_1}{d}\right)\right) \cdot h}{\ell}, 1\right)\\ \mathbf{elif}\;t\_4 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \frac{\left(0.25 \cdot t\_3\right) \cdot \frac{t\_3 \cdot h}{\left(d + d\right) \cdot \ell}}{d}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (/ (fabs d) (sqrt (* h l))))
       (t_1 (fmin M (fabs D)))
       (t_2 (fmax M (fabs D)))
       (t_3 (* t_1 t_2))
       (t_4
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ t_3 (* 2.0 d)) 2.0)) (/ h l))))))
  (if (<= t_4 0.0)
    (*
     t_0
     (fma
      (/ t_3 (* -2.0 d))
      (/ (* (* 0.25 (* t_2 (/ t_1 d))) h) l)
      1.0))
    (if (<= t_4 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (*
       t_0
       (- 1.0 (/ (* (* 0.25 t_3) (/ (* t_3 h) (* (+ d d) l))) d)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs(d) / sqrt((h * l));
	double t_1 = fmin(M, fabs(D));
	double t_2 = fmax(M, fabs(D));
	double t_3 = t_1 * t_2;
	double t_4 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow((t_3 / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = t_0 * fma((t_3 / (-2.0 * d)), (((0.25 * (t_2 * (t_1 / d))) * h) / l), 1.0);
	} else if (t_4 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_0 * (1.0 - (((0.25 * t_3) * ((t_3 * h) / ((d + d) * l))) / d));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(abs(d) / sqrt(Float64(h * l)))
	t_1 = fmin(M, abs(D))
	t_2 = fmax(M, abs(D))
	t_3 = Float64(t_1 * t_2)
	t_4 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(t_3 / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(t_0 * fma(Float64(t_3 / Float64(-2.0 * d)), Float64(Float64(Float64(0.25 * Float64(t_2 * Float64(t_1 / d))) * h) / l), 1.0));
	elseif (t_4 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(0.25 * t_3) * Float64(Float64(t_3 * h) / Float64(Float64(d + d) * l))) / d)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(t$95$3 / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(t$95$0 * N[(N[(t$95$3 / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.25 * N[(t$95$2 * N[(t$95$1 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(N[(0.25 * t$95$3), $MachinePrecision] * N[(N[(t$95$3 * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 77.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \color{blue}{\frac{M \cdot D}{d}}\right) \cdot h}{\ell}, 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{\color{blue}{M \cdot D}}{d}\right) \cdot h}{\ell}, 1\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{\color{blue}{D \cdot M}}{d}\right) \cdot h}{\ell}, 1\right) \]

      4. associate-/l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot h}{\ell}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot h}{\ell}, 1\right) \]

      6. lower-/.f64 75.8%

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \left(D \cdot \color{blue}{\frac{M}{d}}\right)\right) \cdot h}{\ell}, 1\right) \]

   9. Applied rewrites 75.8%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right) \cdot h}{\ell}, 1\right) \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right)} \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)} \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\color{blue}{\frac{D \cdot M}{d}} \cdot \frac{1}{4}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(D \cdot M\right) \cdot \frac{1}{4}}{d}} \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(\left(D \cdot M\right) \cdot \frac{1}{4}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)}{d}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(\left(D \cdot M\right) \cdot \frac{1}{4}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)}{d}}\right) \]

   7. Applied rewrites 74.9%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(0.25 \cdot \left(M \cdot D\right)\right) \cdot \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell}}{d}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)\\ \mathbf{if}\;h \leq 1.2 \cdot 10^{-304}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell} \cdot \sqrt{h}} \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (fma
         (/ (* M D) (* -2.0 d))
         (/ (* (* 0.25 (/ (* M D) d)) h) l)
         1.0)))
  (if (<= h 1.2e-304)
    (* (/ (fabs d) (sqrt (* h l))) t_0)
    (* (/ (fabs d) (* (sqrt l) (sqrt h))) t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fma(((M * D) / (-2.0 * d)), (((0.25 * ((M * D) / d)) * h) / l), 1.0);
	double tmp;
	if (h <= 1.2e-304) {
		tmp = (fabs(d) / sqrt((h * l))) * t_0;
	} else {
		tmp = (fabs(d) / (sqrt(l) * sqrt(h))) * t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fma(Float64(Float64(M * D) / Float64(-2.0 * d)), Float64(Float64(Float64(0.25 * Float64(Float64(M * D) / d)) * h) / l), 1.0)
	tmp = 0.0
	if (h <= 1.2e-304)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * t_0);
	else
		tmp = Float64(Float64(abs(d) / Float64(sqrt(l) * sqrt(h))) * t_0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[h, 1.2e-304], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 1.2e-304

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 77.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)} \]

   if 1.2e-304 < h

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 77.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      2. pow1/2 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{{\left(h \cdot \ell\right)}^{\frac{1}{2}}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      6. pow1/2 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      9. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      10. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell}} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

      11. lower-unsound-sqrt.f64 44.4%

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]

   9. Applied rewrites 44.4%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2 \cdot \left(1 - t\_0 \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(1 - \frac{\left(0.25 \cdot \left(M \cdot D\right)\right) \cdot t\_0}{d}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (/ (* (* M D) h) (* (+ d d) l)))
       (t_1
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_2 (/ (fabs d) (sqrt (* h l)))))
  (if (<= t_1 0.0)
    (* t_2 (- 1.0 (* t_0 (* 0.25 (/ (* M D) d)))))
    (if (<= t_1 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (* t_2 (- 1.0 (/ (* (* 0.25 (* M D)) t_0) d)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((M * D) * h) / ((d + d) * l);
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = fabs(d) / sqrt((h * l));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2 * (1.0 - (t_0 * (0.25 * ((M * D) / d))));
	} else if (t_1 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_2 * (1.0 - (((0.25 * (M * D)) * t_0) / d));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((m * d_1) * h) / ((d + d) * l)
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_2 = abs(d) / sqrt((h * l))
    if (t_1 <= 0.0d0) then
        tmp = t_2 * (1.0d0 - (t_0 * (0.25d0 * ((m * d_1) / d))))
    else if (t_1 <= 1d+159) then
        tmp = (sqrt(((1.0d0 / l) * d)) * sqrt((d / h))) * 1.0d0
    else
        tmp = t_2 * (1.0d0 - (((0.25d0 * (m * d_1)) * t_0) / d))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = ((M * D) * h) / ((d + d) * l);
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.abs(d) / Math.sqrt((h * l));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2 * (1.0 - (t_0 * (0.25 * ((M * D) / d))));
	} else if (t_1 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_2 * (1.0 - (((0.25 * (M * D)) * t_0) / d));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = ((M * D) * h) / ((d + d) * l)
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.fabs(d) / math.sqrt((h * l))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_2 * (1.0 - (t_0 * (0.25 * ((M * D) / d))))
	elif t_1 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_2 * (1.0 - (((0.25 * (M * D)) * t_0) / d))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(M * D) * h) / Float64(Float64(d + d) * l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(abs(d) / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(t_2 * Float64(1.0 - Float64(t_0 * Float64(0.25 * Float64(Float64(M * D) / d)))));
	elseif (t_1 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(0.25 * Float64(M * D)) * t_0) / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = ((M * D) * h) / ((d + d) * l);
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = abs(d) / sqrt((h * l));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_2 * (1.0 - (t_0 * (0.25 * ((M * D) / d))));
	elseif (t_1 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_2 * (1.0 - (((0.25 * (M * D)) * t_0) / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(t$95$2 * N[(1.0 - N[(t$95$0 * N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(t$95$2 * N[(1.0 - N[(N[(N[(0.25 * N[(M * D), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right)} \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(D \cdot M\right) \cdot \color{blue}{\frac{h}{\left(d + d\right) \cdot \ell}}\right) \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}} \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}} \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)\right) \]

      9. lower-*.f64 74.7%

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{\left(D \cdot M\right) \cdot h}}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{D \cdot M}{d} \cdot 0.25\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{\left(D \cdot M\right)} \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{\left(M \cdot D\right)} \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)\right) \]

      12. lift-*.f64 74.7%

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{\left(M \cdot D\right)} \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{D \cdot M}{d} \cdot 0.25\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \]

      15. lower-*.f64 74.7%

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \color{blue}{\left(0.25 \cdot \frac{D \cdot M}{d}\right)}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{1}{4} \cdot \frac{\color{blue}{D \cdot M}}{d}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{1}{4} \cdot \frac{\color{blue}{M \cdot D}}{d}\right)\right) \]

      18. lift-*.f64 74.7%

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{\color{blue}{M \cdot D}}{d}\right)\right) \]

   7. Applied rewrites 74.7%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)}\right) \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right)} \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)} \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\color{blue}{\frac{D \cdot M}{d}} \cdot \frac{1}{4}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(D \cdot M\right) \cdot \frac{1}{4}}{d}} \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(\left(D \cdot M\right) \cdot \frac{1}{4}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)}{d}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(\left(D \cdot M\right) \cdot \frac{1}{4}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)}{d}}\right) \]

   7. Applied rewrites 74.9%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(0.25 \cdot \left(M \cdot D\right)\right) \cdot \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell}}{d}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_1
        (*
         (/ (fabs d) (sqrt (* h l)))
         (-
          1.0
          (*
           (/ (* (* M D) h) (* (+ d d) l))
           (* 0.25 (/ (* M D) d)))))))
  (if (<= t_0 0.0)
    t_1
    (if (<= t_0 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fabs(d) / sqrt((h * l))) * (1.0 - ((((M * D) * h) / ((d + d) * l)) * (0.25 * ((M * D) / d))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = (abs(d) / sqrt((h * l))) * (1.0d0 - ((((m * d_1) * h) / ((d + d) * l)) * (0.25d0 * ((m * d_1) / d))))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 1d+159) then
        tmp = (sqrt(((1.0d0 / l) * d)) * sqrt((d / h))) * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - ((((M * D) * h) / ((d + d) * l)) * (0.25 * ((M * D) / d))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - ((((M * D) * h) / ((d + d) * l)) * (0.25 * ((M * D) / d))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(Float64(M * D) * h) / Float64(Float64(d + d) * l)) * Float64(0.25 * Float64(Float64(M * D) / d)))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = (abs(d) / sqrt((h * l))) * (1.0 - ((((M * D) * h) / ((d + d) * l)) * (0.25 * ((M * D) / d))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision] / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right)} \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \]

      3. associate-*l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(D \cdot M\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(D \cdot M\right) \cdot \color{blue}{\frac{h}{\left(d + d\right) \cdot \ell}}\right) \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}} \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}} \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)\right) \]

      9. lower-*.f64 74.7%

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{\left(D \cdot M\right) \cdot h}}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{D \cdot M}{d} \cdot 0.25\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{\left(D \cdot M\right)} \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{\left(M \cdot D\right)} \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)\right) \]

      12. lift-*.f64 74.7%

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\color{blue}{\left(M \cdot D\right)} \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{D \cdot M}{d} \cdot 0.25\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{D \cdot M}{d}\right)}\right) \]

      15. lower-*.f64 74.7%

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \color{blue}{\left(0.25 \cdot \frac{D \cdot M}{d}\right)}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{1}{4} \cdot \frac{\color{blue}{D \cdot M}}{d}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(\frac{1}{4} \cdot \frac{\color{blue}{M \cdot D}}{d}\right)\right) \]

      18. lift-*.f64 74.7%

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{\color{blue}{M \cdot D}}{d}\right)\right) \]

   7. Applied rewrites 74.7%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\frac{\left(M \cdot D\right) \cdot h}{\left(d + d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{M \cdot D}{d}\right)}\right) \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{\left(0.25 \cdot \left(D \cdot M\right)\right) \cdot h}{\ell \cdot d} \cdot \left(D \cdot M\right), \frac{-0.5}{d}, 1\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_1
        (*
         (/ (fabs d) (sqrt (* h l)))
         (fma
          (* (/ (* (* 0.25 (* D M)) h) (* l d)) (* D M))
          (/ -0.5 d)
          1.0))))
  (if (<= t_0 0.0)
    t_1
    (if (<= t_0 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fabs(d) / sqrt((h * l))) * fma(((((0.25 * (D * M)) * h) / (l * d)) * (D * M)), (-0.5 / d), 1.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * fma(Float64(Float64(Float64(Float64(0.25 * Float64(D * M)) * h) / Float64(l * d)) * Float64(D * M)), Float64(-0.5 / d), 1.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.25 * N[(D * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 77.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{M \cdot D}{-2 \cdot d} \cdot \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell} + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\color{blue}{\frac{M \cdot D}{-2 \cdot d}} \cdot \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell} + 1\right) \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}}{-2 \cdot d}} + 1\right) \]

      4. mult-flip N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}\right) \cdot \frac{1}{-2 \cdot d}} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\left(M \cdot D\right) \cdot \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, \frac{1}{-2 \cdot d}, 1\right)} \]

   9. Applied rewrites 74.9%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(0.25 \cdot \left(D \cdot M\right)\right) \cdot h}{\ell \cdot d} \cdot \left(D \cdot M\right), \frac{-0.5}{d}, 1\right)} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(D \cdot M, \frac{\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_1
        (*
         (/ (fabs d) (sqrt (* h l)))
         (fma
          (* D M)
          (/ (* (* (/ (* D M) d) 0.25) h) (* (* -2.0 d) l))
          1.0))))
  (if (<= t_0 0.0)
    t_1
    (if (<= t_0 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fabs(d) / sqrt((h * l))) * fma((D * M), (((((D * M) / d) * 0.25) * h) / ((-2.0 * d) * l)), 1.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * fma(Float64(D * M), Float64(Float64(Float64(Float64(Float64(D * M) / d) * 0.25) * h) / Float64(Float64(-2.0 * d) * l)), 1.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] * N[(N[(N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * 0.25), $MachinePrecision] * h), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

   7. Applied rewrites 77.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{M \cdot D}{-2 \cdot d}, \frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}, 1\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{M \cdot D}{-2 \cdot d} \cdot \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell} + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\color{blue}{\frac{M \cdot D}{-2 \cdot d}} \cdot \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell} + 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\frac{M \cdot D}{-2 \cdot d} \cdot \color{blue}{\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\ell}} + 1\right) \]

      4. frac-times N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\color{blue}{\frac{\left(M \cdot D\right) \cdot \left(\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h\right)}{\left(-2 \cdot d\right) \cdot \ell}} + 1\right) \]

      5. associate-/l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\color{blue}{\left(M \cdot D\right) \cdot \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell}} + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(M \cdot D, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\color{blue}{M \cdot D}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\color{blue}{D \cdot M}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\color{blue}{D \cdot M}, \frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(D \cdot M, \color{blue}{\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell}}, 1\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(D \cdot M, \frac{\color{blue}{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right)} \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(D \cdot M, \frac{\color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)} \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(D \cdot M, \frac{\color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{1}{4}\right)} \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(D \cdot M, \frac{\left(\frac{\color{blue}{M \cdot D}}{d} \cdot \frac{1}{4}\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right) \]

      15. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(D \cdot M, \frac{\left(\frac{\color{blue}{D \cdot M}}{d} \cdot \frac{1}{4}\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(D \cdot M, \frac{\left(\frac{\color{blue}{D \cdot M}}{d} \cdot \frac{1}{4}\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right) \]

      17. lower-*.f64 74.7%

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(D \cdot M, \frac{\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot h}{\color{blue}{\left(-2 \cdot d\right) \cdot \ell}}, 1\right) \]

   9. Applied rewrites 74.7%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(D \cdot M, \frac{\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell}, 1\right)} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 79.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\frac{\left(M \cdot D\right) \cdot \frac{M \cdot D}{d}}{d} \cdot 0.125\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\frac{\left(M \cdot h\right) \cdot \left(D \cdot \left(\frac{D \cdot M}{d} \cdot -0.125\right)\right)}{\ell \cdot d} - -1\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_0 0.0)
    (*
     (/ (fabs d) (sqrt (* l h)))
     (- 1.0 (* (* (/ (* (* M D) (/ (* M D) d)) d) 0.125) (/ h l))))
    (if (<= t_0 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (*
       (/ (fabs d) (sqrt (* h l)))
       (-
        (/ (* (* M h) (* D (* (/ (* D M) d) -0.125))) (* l d))
        -1.0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (fabs(d) / sqrt((l * h))) * (1.0 - (((((M * D) * ((M * D) / d)) / d) * 0.125) * (h / l)));
	} else if (t_0 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= 0.0d0) then
        tmp = (abs(d) / sqrt((l * h))) * (1.0d0 - (((((m * d_1) * ((m * d_1) / d)) / d) * 0.125d0) * (h / l)))
    else if (t_0 <= 1d+159) then
        tmp = (sqrt(((1.0d0 / l) * d)) * sqrt((d / h))) * 1.0d0
    else
        tmp = (abs(d) / sqrt((h * l))) * ((((m * h) * (d_1 * (((d_1 * m) / d) * (-0.125d0)))) / (l * d)) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * (1.0 - (((((M * D) * ((M * D) / d)) / d) * 0.125) * (h / l)));
	} else if (t_0 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * (1.0 - (((((M * D) * ((M * D) / d)) / d) * 0.125) * (h / l)))
	elif t_0 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M * D) * Float64(Float64(M * D) / d)) / d) * 0.125) * Float64(h / l))));
	elseif (t_0 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(Float64(Float64(Float64(M * h) * Float64(D * Float64(Float64(Float64(D * M) / d) * -0.125))) / Float64(l * d)) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (abs(d) / sqrt((l * h))) * (1.0 - (((((M * D) * ((M * D) / d)) / d) * 0.125) * (h / l)));
	elseif (t_0 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	else
		tmp = (abs(d) / sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * 0.125), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M * h), $MachinePrecision] * N[(D * N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lower-*.f64 70.0%

         \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \color{blue}{\frac{h}{\left(d + d\right) \cdot \ell}}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\color{blue}{\left(d + d\right) \cdot \ell}}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\color{blue}{\left(d + d\right)} \cdot \ell}\right) \]

      9. count-2-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\color{blue}{\left(2 \cdot d\right)} \cdot \ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\color{blue}{\left(2 \cdot d\right)} \cdot \ell}\right) \]

      11. times-frac N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)}{2 \cdot d} \cdot \frac{h}{\ell}}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)}{2 \cdot d} \cdot \color{blue}{\frac{h}{\ell}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \color{blue}{\frac{\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)}{2 \cdot d} \cdot \frac{h}{\ell}}\right) \]

   7. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\frac{\left(M \cdot D\right) \cdot \frac{M \cdot D}{d}}{d} \cdot 0.125\right) \cdot \frac{h}{\ell}\right)} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell} + 1\right)} \]

   7. Applied rewrites 69.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d}, \frac{M \cdot h}{\ell}, 1\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} + 1\right)} \]

      2. add-flip N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} - \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} - \color{blue}{-1}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} - -1\right)} \]

   9. Applied rewrites 71.0%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(M \cdot h\right) \cdot \left(D \cdot \left(\frac{D \cdot M}{d} \cdot -0.125\right)\right)}{\ell \cdot d} - -1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left|d\right| \cdot \frac{1 - \left(\frac{\left(M \cdot D\right) \cdot \frac{M \cdot D}{d}}{d} \cdot 0.125\right) \cdot \frac{h}{\ell}}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;t\_0 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\frac{\left(M \cdot h\right) \cdot \left(D \cdot \left(\frac{D \cdot M}{d} \cdot -0.125\right)\right)}{\ell \cdot d} - -1\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_0 0.0)
    (*
     (fabs d)
     (/
      (- 1.0 (* (* (/ (* (* M D) (/ (* M D) d)) d) 0.125) (/ h l)))
      (sqrt (* l h))))
    (if (<= t_0 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (*
       (/ (fabs d) (sqrt (* h l)))
       (-
        (/ (* (* M h) (* D (* (/ (* D M) d) -0.125))) (* l d))
        -1.0))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fabs(d) * ((1.0 - (((((M * D) * ((M * D) / d)) / d) * 0.125) * (h / l))) / sqrt((l * h)));
	} else if (t_0 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= 0.0d0) then
        tmp = abs(d) * ((1.0d0 - (((((m * d_1) * ((m * d_1) / d)) / d) * 0.125d0) * (h / l))) / sqrt((l * h)))
    else if (t_0 <= 1d+159) then
        tmp = (sqrt(((1.0d0 / l) * d)) * sqrt((d / h))) * 1.0d0
    else
        tmp = (abs(d) / sqrt((h * l))) * ((((m * h) * (d_1 * (((d_1 * m) / d) * (-0.125d0)))) / (l * d)) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.abs(d) * ((1.0 - (((((M * D) * ((M * D) / d)) / d) * 0.125) * (h / l))) / Math.sqrt((l * h)));
	} else if (t_0 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.fabs(d) * ((1.0 - (((((M * D) * ((M * D) / d)) / d) * 0.125) * (h / l))) / math.sqrt((l * h)))
	elif t_0 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(abs(d) * Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M * D) * Float64(Float64(M * D) / d)) / d) * 0.125) * Float64(h / l))) / sqrt(Float64(l * h))));
	elseif (t_0 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(Float64(Float64(Float64(M * h) * Float64(D * Float64(Float64(Float64(D * M) / d) * -0.125))) / Float64(l * d)) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = abs(d) * ((1.0 - (((((M * D) * ((M * D) / d)) / d) * 0.125) * (h / l))) / sqrt((l * h)));
	elseif (t_0 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	else
		tmp = (abs(d) / sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Abs[d], $MachinePrecision] * N[(N[(1.0 - N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * 0.125), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M * h), $MachinePrecision] * N[(D * N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left|d\right| \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)}{\sqrt{h \cdot \ell}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left|d\right| \cdot \frac{1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}{\sqrt{h \cdot \ell}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left|d\right| \cdot \frac{1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}{\sqrt{h \cdot \ell}}} \]

      6. lower-/.f64 71.5%

         \[\leadsto \left|d\right| \cdot \color{blue}{\frac{1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}{\sqrt{h \cdot \ell}}} \]

   7. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\left|d\right| \cdot \frac{1 - \left(\frac{\left(M \cdot D\right) \cdot \frac{M \cdot D}{d}}{d} \cdot 0.125\right) \cdot \frac{h}{\ell}}{\sqrt{\ell \cdot h}}} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell} + 1\right)} \]

   7. Applied rewrites 69.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d}, \frac{M \cdot h}{\ell}, 1\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} + 1\right)} \]

      2. add-flip N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} - \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} - \color{blue}{-1}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} - -1\right)} \]

   9. Applied rewrites 71.0%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(M \cdot h\right) \cdot \left(D \cdot \left(\frac{D \cdot M}{d} \cdot -0.125\right)\right)}{\ell \cdot d} - -1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 77.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\frac{\left(M \cdot h\right) \cdot \left(D \cdot \left(\frac{D \cdot M}{d} \cdot -0.125\right)\right)}{\ell \cdot d} - -1\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_1
        (*
         (/ (fabs d) (sqrt (* h l)))
         (-
          (/ (* (* M h) (* D (* (/ (* D M) d) -0.125))) (* l d))
          -1.0))))
  (if (<= t_0 0.0)
    t_1
    (if (<= t_0 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fabs(d) / sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = (abs(d) / sqrt((h * l))) * ((((m * h) * (d_1 * (((d_1 * m) / d) * (-0.125d0)))) / (l * d)) - (-1.0d0))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 1d+159) then
        tmp = (sqrt(((1.0d0 / l) * d)) * sqrt((d / h))) * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (Math.abs(d) / Math.sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = (math.fabs(d) / math.sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(Float64(Float64(Float64(M * h) * Float64(D * Float64(Float64(Float64(D * M) / d) * -0.125))) / Float64(l * d)) - -1.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = (abs(d) / sqrt((h * l))) * ((((M * h) * (D * (((D * M) / d) * -0.125))) / (l * d)) - -1.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M * h), $MachinePrecision] * N[(D * N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot 0.25\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell}\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{D \cdot M}{d} \cdot \frac{1}{4}\right) \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{h}{\left(d + d\right) \cdot \ell} + 1\right)} \]

   7. Applied rewrites 69.4%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d}, \frac{M \cdot h}{\ell}, 1\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} + 1\right)} \]

      2. add-flip N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} - \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} - \color{blue}{-1}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(\frac{1}{4} \cdot \frac{M \cdot D}{d}\right) \cdot D}{-2 \cdot d} \cdot \frac{M \cdot h}{\ell} - -1\right)} \]

   9. Applied rewrites 71.0%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{\left(M \cdot h\right) \cdot \left(D \cdot \left(\frac{D \cdot M}{d} \cdot -0.125\right)\right)}{\ell \cdot d} - -1\right)} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0 \cdot t\_1}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := t\_1 \cdot t\_1\\ t_4 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;-0.125 \cdot \left(\frac{\left(t\_3 \cdot t\_0\right) \cdot t\_0}{\ell \cdot d} \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h}{d}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\left|d\right|}{t\_4} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{t\_3}{\ell \cdot d} \cdot \left(t\_0 \cdot \left(\left(t\_0 \cdot h\right) \cdot \frac{\left|d\right|}{t\_4 \cdot d}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin M (fabs D)))
       (t_1 (fmax M (fabs D)))
       (t_2
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* t_0 t_1) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_3 (* t_1 t_1))
       (t_4 (sqrt (* l h))))
  (if (<= t_2 0.0)
    (*
     -0.125
     (*
      (/ (* (* t_3 t_0) t_0) (* l d))
      (/ (* (/ (fabs d) (sqrt (* h l))) h) d)))
    (if (<= t_2 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (if (<= t_2 INFINITY)
        (* (/ (fabs d) t_4) 1.0)
        (*
         -0.125
         (*
          (/ t_3 (* l d))
          (* t_0 (* (* t_0 h) (/ (fabs d) (* t_4 d)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(M, fabs(D));
	double t_1 = fmax(M, fabs(D));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = t_1 * t_1;
	double t_4 = sqrt((l * h));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -0.125 * ((((t_3 * t_0) * t_0) / (l * d)) * (((fabs(d) / sqrt((h * l))) * h) / d));
	} else if (t_2 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (fabs(d) / t_4) * 1.0;
	} else {
		tmp = -0.125 * ((t_3 / (l * d)) * (t_0 * ((t_0 * h) * (fabs(d) / (t_4 * d)))));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(M, Math.abs(D));
	double t_1 = fmax(M, Math.abs(D));
	double t_2 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = t_1 * t_1;
	double t_4 = Math.sqrt((l * h));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -0.125 * ((((t_3 * t_0) * t_0) / (l * d)) * (((Math.abs(d) / Math.sqrt((h * l))) * h) / d));
	} else if (t_2 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.abs(d) / t_4) * 1.0;
	} else {
		tmp = -0.125 * ((t_3 / (l * d)) * (t_0 * ((t_0 * h) * (Math.abs(d) / (t_4 * d)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(M, math.fabs(D))
	t_1 = fmax(M, math.fabs(D))
	t_2 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)))
	t_3 = t_1 * t_1
	t_4 = math.sqrt((l * h))
	tmp = 0
	if t_2 <= 0.0:
		tmp = -0.125 * ((((t_3 * t_0) * t_0) / (l * d)) * (((math.fabs(d) / math.sqrt((h * l))) * h) / d))
	elif t_2 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	elif t_2 <= math.inf:
		tmp = (math.fabs(d) / t_4) * 1.0
	else:
		tmp = -0.125 * ((t_3 / (l * d)) * (t_0 * ((t_0 * h) * (math.fabs(d) / (t_4 * d)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(M, abs(D))
	t_1 = fmax(M, abs(D))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_0 * t_1) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(t_1 * t_1)
	t_4 = sqrt(Float64(l * h))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(t_3 * t_0) * t_0) / Float64(l * d)) * Float64(Float64(Float64(abs(d) / sqrt(Float64(h * l))) * h) / d)));
	elseif (t_2 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(abs(d) / t_4) * 1.0);
	else
		tmp = Float64(-0.125 * Float64(Float64(t_3 / Float64(l * d)) * Float64(t_0 * Float64(Float64(t_0 * h) * Float64(abs(d) / Float64(t_4 * d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(M, abs(D));
	t_1 = max(M, abs(D));
	t_2 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((t_0 * t_1) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_3 = t_1 * t_1;
	t_4 = sqrt((l * h));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = -0.125 * ((((t_3 * t_0) * t_0) / (l * d)) * (((abs(d) / sqrt((h * l))) * h) / d));
	elseif (t_2 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	elseif (t_2 <= Inf)
		tmp = (abs(d) / t_4) * 1.0;
	else
		tmp = -0.125 * ((t_3 / (l * d)) * (t_0 * ((t_0 * h) * (abs(d) / (t_4 * d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(-0.125 * N[(N[(N[(N[(t$95$3 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Abs[d], $MachinePrecision] / t$95$4), $MachinePrecision] * 1.0), $MachinePrecision], N[(-0.125 * N[(N[(t$95$3 / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(t$95$0 * h), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[(t$95$4 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

   6. Applied rewrites 29.8%

      \[\leadsto -0.125 \cdot \left(\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{\ell \cdot d} \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h}{d}}\right) \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2}} \cdot \ell} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \color{blue}{\ell}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} \]

      8. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

      9. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \left(\ell \cdot \color{blue}{d}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(\ell \cdot d\right) \cdot \color{blue}{d}} \]

      12. times-frac N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \color{blue}{\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      16. lower-/.f64 24.2%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{\color{blue}{d}}\right) \]

   6. Applied rewrites 27.7%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{\color{blue}{d}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}}}{d}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right)\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      14. lower-*.f64 31.8%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot \color{blue}{d}}\right)\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot d}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

      17. lower-*.f64 31.8%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

   8. Applied rewrites 31.8%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)}\right)\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 72.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{\left(-0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot D\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d}\\ \mathbf{elif}\;t\_0 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\left|d\right|}{t\_1} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{t\_1 \cdot d}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_1 (sqrt (* l h))))
  (if (<= t_0 -1e-80)
    (/
     (/
      (*
       (* -0.125 (* (* (* D M) D) M))
       (* (/ (fabs d) (sqrt (* h l))) h))
      (* l d))
     d)
    (if (<= t_0 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (if (<= t_0 INFINITY)
        (* (/ (fabs d) t_1) 1.0)
        (*
         -0.125
         (*
          (/ (* D D) (* l d))
          (* M (* (* M h) (/ (fabs d) (* t_1 d)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((l * h));
	double tmp;
	if (t_0 <= -1e-80) {
		tmp = (((-0.125 * (((D * M) * D) * M)) * ((fabs(d) / sqrt((h * l))) * h)) / (l * d)) / d;
	} else if (t_0 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (fabs(d) / t_1) * 1.0;
	} else {
		tmp = -0.125 * (((D * D) / (l * d)) * (M * ((M * h) * (fabs(d) / (t_1 * d)))));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((l * h));
	double tmp;
	if (t_0 <= -1e-80) {
		tmp = (((-0.125 * (((D * M) * D) * M)) * ((Math.abs(d) / Math.sqrt((h * l))) * h)) / (l * d)) / d;
	} else if (t_0 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.abs(d) / t_1) * 1.0;
	} else {
		tmp = -0.125 * (((D * D) / (l * d)) * (M * ((M * h) * (Math.abs(d) / (t_1 * d)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((l * h))
	tmp = 0
	if t_0 <= -1e-80:
		tmp = (((-0.125 * (((D * M) * D) * M)) * ((math.fabs(d) / math.sqrt((h * l))) * h)) / (l * d)) / d
	elif t_0 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	elif t_0 <= math.inf:
		tmp = (math.fabs(d) / t_1) * 1.0
	else:
		tmp = -0.125 * (((D * D) / (l * d)) * (M * ((M * h) * (math.fabs(d) / (t_1 * d)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(l * h))
	tmp = 0.0
	if (t_0 <= -1e-80)
		tmp = Float64(Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(D * M) * D) * M)) * Float64(Float64(abs(d) / sqrt(Float64(h * l))) * h)) / Float64(l * d)) / d);
	elseif (t_0 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(abs(d) / t_1) * 1.0);
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(l * d)) * Float64(M * Float64(Float64(M * h) * Float64(abs(d) / Float64(t_1 * d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((l * h));
	tmp = 0.0;
	if (t_0 <= -1e-80)
		tmp = (((-0.125 * (((D * M) * D) * M)) * ((abs(d) / sqrt((h * l))) * h)) / (l * d)) / d;
	elseif (t_0 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	elseif (t_0 <= Inf)
		tmp = (abs(d) / t_1) * 1.0;
	else
		tmp = -0.125 * (((D * D) / (l * d)) * (M * ((M * h) * (abs(d) / (t_1 * d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-80], N[(N[(N[(N[(-0.125 * N[(N[(N[(D * M), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[t$95$0, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Abs[d], $MachinePrecision] / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(N[(M * h), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[(t$95$1 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999996e-81

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

   6. Applied rewrites 29.2%

      \[\leadsto \frac{\frac{\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{\color{blue}{d}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(D \cdot D\right)\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{-1}{8} \cdot \left(\left(M \cdot \left(D \cdot D\right)\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\frac{\left(\frac{-1}{8} \cdot \left(\left(\left(M \cdot D\right) \cdot D\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{-1}{8} \cdot \left(\left(\left(M \cdot D\right) \cdot D\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d} \]

      6. lower-*.f64 31.6%

         \[\leadsto \frac{\frac{\left(-0.125 \cdot \left(\left(\left(M \cdot D\right) \cdot D\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{-1}{8} \cdot \left(\left(\left(M \cdot D\right) \cdot D\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot M\right) \cdot D\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d} \]

      9. lower-*.f64 31.6%

         \[\leadsto \frac{\frac{\left(-0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot D\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d} \]

   8. Applied rewrites 31.6%

      \[\leadsto \frac{\frac{\left(-0.125 \cdot \left(\left(\left(D \cdot M\right) \cdot D\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d} \]

   if -9.9999999999999996e-81 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2}} \cdot \ell} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \color{blue}{\ell}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} \]

      8. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

      9. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \left(\ell \cdot \color{blue}{d}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(\ell \cdot d\right) \cdot \color{blue}{d}} \]

      12. times-frac N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \color{blue}{\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      16. lower-/.f64 24.2%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{\color{blue}{d}}\right) \]

   6. Applied rewrites 27.7%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{\color{blue}{d}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}}}{d}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right)\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      14. lower-*.f64 31.8%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot \color{blue}{d}}\right)\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot d}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

      17. lower-*.f64 31.8%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

   8. Applied rewrites 31.8%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)}\right)\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 71.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(M, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(M, \left|D\right|\right)\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0 \cdot t\_1}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := t\_1 \cdot t\_1\\ t_4 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{\left(-0.125 \cdot \left(\left(t\_3 \cdot t\_0\right) \cdot t\_0\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{d}\\ \mathbf{elif}\;t\_2 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\left|d\right|}{t\_4} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{t\_3}{\ell \cdot d} \cdot \left(t\_0 \cdot \left(\left(t\_0 \cdot h\right) \cdot \frac{\left|d\right|}{t\_4 \cdot d}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin M (fabs D)))
       (t_1 (fmax M (fabs D)))
       (t_2
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* t_0 t_1) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_3 (* t_1 t_1))
       (t_4 (sqrt (* l h))))
  (if (<= t_2 -5e+28)
    (/
     (/
      (*
       (* -0.125 (* (* t_3 t_0) t_0))
       (* (/ (fabs d) (sqrt (* h l))) h))
      (* l d))
     d)
    (if (<= t_2 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (if (<= t_2 INFINITY)
        (* (/ (fabs d) t_4) 1.0)
        (*
         -0.125
         (*
          (/ t_3 (* l d))
          (* t_0 (* (* t_0 h) (/ (fabs d) (* t_4 d)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(M, fabs(D));
	double t_1 = fmax(M, fabs(D));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = t_1 * t_1;
	double t_4 = sqrt((l * h));
	double tmp;
	if (t_2 <= -5e+28) {
		tmp = (((-0.125 * ((t_3 * t_0) * t_0)) * ((fabs(d) / sqrt((h * l))) * h)) / (l * d)) / d;
	} else if (t_2 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (fabs(d) / t_4) * 1.0;
	} else {
		tmp = -0.125 * ((t_3 / (l * d)) * (t_0 * ((t_0 * h) * (fabs(d) / (t_4 * d)))));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(M, Math.abs(D));
	double t_1 = fmax(M, Math.abs(D));
	double t_2 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = t_1 * t_1;
	double t_4 = Math.sqrt((l * h));
	double tmp;
	if (t_2 <= -5e+28) {
		tmp = (((-0.125 * ((t_3 * t_0) * t_0)) * ((Math.abs(d) / Math.sqrt((h * l))) * h)) / (l * d)) / d;
	} else if (t_2 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.abs(d) / t_4) * 1.0;
	} else {
		tmp = -0.125 * ((t_3 / (l * d)) * (t_0 * ((t_0 * h) * (Math.abs(d) / (t_4 * d)))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(M, math.fabs(D))
	t_1 = fmax(M, math.fabs(D))
	t_2 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)))
	t_3 = t_1 * t_1
	t_4 = math.sqrt((l * h))
	tmp = 0
	if t_2 <= -5e+28:
		tmp = (((-0.125 * ((t_3 * t_0) * t_0)) * ((math.fabs(d) / math.sqrt((h * l))) * h)) / (l * d)) / d
	elif t_2 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	elif t_2 <= math.inf:
		tmp = (math.fabs(d) / t_4) * 1.0
	else:
		tmp = -0.125 * ((t_3 / (l * d)) * (t_0 * ((t_0 * h) * (math.fabs(d) / (t_4 * d)))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(M, abs(D))
	t_1 = fmax(M, abs(D))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_0 * t_1) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(t_1 * t_1)
	t_4 = sqrt(Float64(l * h))
	tmp = 0.0
	if (t_2 <= -5e+28)
		tmp = Float64(Float64(Float64(Float64(-0.125 * Float64(Float64(t_3 * t_0) * t_0)) * Float64(Float64(abs(d) / sqrt(Float64(h * l))) * h)) / Float64(l * d)) / d);
	elseif (t_2 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(abs(d) / t_4) * 1.0);
	else
		tmp = Float64(-0.125 * Float64(Float64(t_3 / Float64(l * d)) * Float64(t_0 * Float64(Float64(t_0 * h) * Float64(abs(d) / Float64(t_4 * d))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(M, abs(D));
	t_1 = max(M, abs(D));
	t_2 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((t_0 * t_1) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_3 = t_1 * t_1;
	t_4 = sqrt((l * h));
	tmp = 0.0;
	if (t_2 <= -5e+28)
		tmp = (((-0.125 * ((t_3 * t_0) * t_0)) * ((abs(d) / sqrt((h * l))) * h)) / (l * d)) / d;
	elseif (t_2 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	elseif (t_2 <= Inf)
		tmp = (abs(d) / t_4) * 1.0;
	else
		tmp = -0.125 * ((t_3 / (l * d)) * (t_0 * ((t_0 * h) * (abs(d) / (t_4 * d)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[M, N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -5e+28], N[(N[(N[(N[(-0.125 * N[(N[(t$95$3 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[t$95$2, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Abs[d], $MachinePrecision] / t$95$4), $MachinePrecision] * 1.0), $MachinePrecision], N[(-0.125 * N[(N[(t$95$3 / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(t$95$0 * h), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[(t$95$4 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999996e28

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

   6. Applied rewrites 29.2%

      \[\leadsto \frac{\frac{\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{\color{blue}{d}} \]

   if -4.9999999999999996e28 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2}} \cdot \ell} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \color{blue}{\ell}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} \]

      8. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

      9. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \left(\ell \cdot \color{blue}{d}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(\ell \cdot d\right) \cdot \color{blue}{d}} \]

      12. times-frac N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \color{blue}{\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      16. lower-/.f64 24.2%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{\color{blue}{d}}\right) \]

   6. Applied rewrites 27.7%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{\color{blue}{d}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}}}{d}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right)\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      14. lower-*.f64 31.8%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot \color{blue}{d}}\right)\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot d}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

      17. lower-*.f64 31.8%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

   8. Applied rewrites 31.8%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)}\right)\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 16: 70.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{D \cdot D}{\ell \cdot d}\\ t_1 := \sqrt{\ell \cdot h}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \frac{\left|d\right|}{t\_1 \cdot d}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;-0.125 \cdot \left(t\_0 \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot t\_3\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\left|d\right|}{t\_1} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(t\_0 \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot t\_3\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (/ (* D D) (* l d)))
       (t_1 (sqrt (* l h)))
       (t_2
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_3 (/ (fabs d) (* t_1 d))))
  (if (<= t_2 0.0)
    (* -0.125 (* t_0 (* (* M M) (* h t_3))))
    (if (<= t_2 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (if (<= t_2 INFINITY)
        (* (/ (fabs d) t_1) 1.0)
        (* -0.125 (* t_0 (* M (* (* M h) t_3)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * D) / (l * d);
	double t_1 = sqrt((l * h));
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = fabs(d) / (t_1 * d);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -0.125 * (t_0 * ((M * M) * (h * t_3)));
	} else if (t_2 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (fabs(d) / t_1) * 1.0;
	} else {
		tmp = -0.125 * (t_0 * (M * ((M * h) * t_3)));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D * D) / (l * d);
	double t_1 = Math.sqrt((l * h));
	double t_2 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = Math.abs(d) / (t_1 * d);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -0.125 * (t_0 * ((M * M) * (h * t_3)));
	} else if (t_2 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.abs(d) / t_1) * 1.0;
	} else {
		tmp = -0.125 * (t_0 * (M * ((M * h) * t_3)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D * D) / (l * d)
	t_1 = math.sqrt((l * h))
	t_2 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_3 = math.fabs(d) / (t_1 * d)
	tmp = 0
	if t_2 <= 0.0:
		tmp = -0.125 * (t_0 * ((M * M) * (h * t_3)))
	elif t_2 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	elif t_2 <= math.inf:
		tmp = (math.fabs(d) / t_1) * 1.0
	else:
		tmp = -0.125 * (t_0 * (M * ((M * h) * t_3)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D * D) / Float64(l * d))
	t_1 = sqrt(Float64(l * h))
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(abs(d) / Float64(t_1 * d))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(-0.125 * Float64(t_0 * Float64(Float64(M * M) * Float64(h * t_3))));
	elseif (t_2 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(abs(d) / t_1) * 1.0);
	else
		tmp = Float64(-0.125 * Float64(t_0 * Float64(M * Float64(Float64(M * h) * t_3))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D * D) / (l * d);
	t_1 = sqrt((l * h));
	t_2 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_3 = abs(d) / (t_1 * d);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = -0.125 * (t_0 * ((M * M) * (h * t_3)));
	elseif (t_2 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	elseif (t_2 <= Inf)
		tmp = (abs(d) / t_1) * 1.0;
	else
		tmp = -0.125 * (t_0 * (M * ((M * h) * t_3)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D * D), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[d], $MachinePrecision] / N[(t$95$1 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(-0.125 * N[(t$95$0 * N[(N[(M * M), $MachinePrecision] * N[(h * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Abs[d], $MachinePrecision] / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], N[(-0.125 * N[(t$95$0 * N[(M * N[(N[(M * h), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2}} \cdot \ell} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \color{blue}{\ell}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} \]

      8. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

      9. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \left(\ell \cdot \color{blue}{d}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(\ell \cdot d\right) \cdot \color{blue}{d}} \]

      12. times-frac N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \color{blue}{\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      16. lower-/.f64 24.2%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{\color{blue}{d}}\right) \]

   6. Applied rewrites 27.7%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{\color{blue}{d}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(h \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(h \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right)\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)\right)\right) \]

      9. associate-/l/ N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      11. lower-*.f64 30.7%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot \color{blue}{d}}\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot d}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

      14. lower-*.f64 30.7%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

   8. Applied rewrites 30.7%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\left(h \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)}\right)\right) \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2}} \cdot \ell} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \color{blue}{\ell}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} \]

      8. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

      9. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \left(\ell \cdot \color{blue}{d}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(\ell \cdot d\right) \cdot \color{blue}{d}} \]

      12. times-frac N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \color{blue}{\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      16. lower-/.f64 24.2%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{\color{blue}{d}}\right) \]

   6. Applied rewrites 27.7%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{\color{blue}{d}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}}}{d}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right)\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      14. lower-*.f64 31.8%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot \color{blue}{d}}\right)\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot d}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

      17. lower-*.f64 31.8%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

   8. Applied rewrites 31.8%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)}\right)\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 70.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\ell \cdot h}\\ t_2 := -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{t\_1 \cdot d}\right)\right)\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\left|d\right|}{t\_1} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_1 (sqrt (* l h)))
       (t_2
        (*
         -0.125
         (*
          (/ (* D D) (* l d))
          (* M (* (* M h) (/ (fabs d) (* t_1 d))))))))
  (if (<= t_0 0.0)
    t_2
    (if (<= t_0 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (if (<= t_0 INFINITY) (* (/ (fabs d) t_1) 1.0) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((l * h));
	double t_2 = -0.125 * (((D * D) / (l * d)) * (M * ((M * h) * (fabs(d) / (t_1 * d)))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (fabs(d) / t_1) * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((l * h));
	double t_2 = -0.125 * (((D * D) / (l * d)) * (M * ((M * h) * (Math.abs(d) / (t_1 * d)))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.abs(d) / t_1) * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((l * h))
	t_2 = -0.125 * (((D * D) / (l * d)) * (M * ((M * h) * (math.fabs(d) / (t_1 * d)))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_2
	elif t_0 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	elif t_0 <= math.inf:
		tmp = (math.fabs(d) / t_1) * 1.0
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(l * h))
	t_2 = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(l * d)) * Float64(M * Float64(Float64(M * h) * Float64(abs(d) / Float64(t_1 * d))))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(abs(d) / t_1) * 1.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((l * h));
	t_2 = -0.125 * (((D * D) / (l * d)) * (M * ((M * h) * (abs(d) / (t_1 * d)))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	elseif (t_0 <= Inf)
		tmp = (abs(d) / t_1) * 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(N[(M * h), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[(t$95$1 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Abs[d], $MachinePrecision] / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2}} \cdot \ell} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \color{blue}{\ell}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} \]

      8. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

      9. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \left(\ell \cdot \color{blue}{d}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(\ell \cdot d\right) \cdot \color{blue}{d}} \]

      12. times-frac N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \color{blue}{\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      16. lower-/.f64 24.2%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{\color{blue}{d}}\right) \]

   6. Applied rewrites 27.7%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{\color{blue}{d}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \]

      3. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}}}{d}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}}{d}\right)\right)\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell} \cdot d}}\right)\right)\right) \]

      14. lower-*.f64 31.8%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot \color{blue}{d}}\right)\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell} \cdot d}\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

      17. lower-*.f64 31.8%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)\right)\right) \]

   8. Applied rewrites 31.8%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \left(M \cdot \color{blue}{\left(\left(M \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right)}\right)\right) \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 69.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\mathsf{min}\left(M, D\right) \cdot \mathsf{min}\left(M, D\right)\right) \cdot h\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\ell \cdot h}\\ t_3 := t\_2 \cdot d\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\left(\left(\left(\mathsf{max}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\right) \cdot t\_0\right) \cdot \frac{\left|d\right|}{t\_3}\right) \cdot -0.125}{\ell \cdot d}\\ \mathbf{elif}\;t\_1 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\left|d\right|}{t\_2} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{t\_0 \cdot \left|d\right|}{t\_3} \cdot \mathsf{max}\left(M, D\right)\right) \cdot \frac{\mathsf{max}\left(M, D\right)}{\ell \cdot d}\right)\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (* (* (fmin M D) (fmin M D)) h))
       (t_1
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (*
            (/ 1.0 2.0)
            (pow (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_2 (sqrt (* l h)))
       (t_3 (* t_2 d)))
  (if (<= t_1 0.0)
    (/
     (* (* (* (* (fmax M D) (fmax M D)) t_0) (/ (fabs d) t_3)) -0.125)
     (* l d))
    (if (<= t_1 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (if (<= t_1 INFINITY)
        (* (/ (fabs d) t_2) 1.0)
        (*
         -0.125
         (*
          (* (/ (* t_0 (fabs d)) t_3) (fmax M D))
          (/ (fmax M D) (* l d)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (fmin(M, D) * fmin(M, D)) * h;
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((l * h));
	double t_3 = t_2 * d;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((((fmax(M, D) * fmax(M, D)) * t_0) * (fabs(d) / t_3)) * -0.125) / (l * d);
	} else if (t_1 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (fabs(d) / t_2) * 1.0;
	} else {
		tmp = -0.125 * ((((t_0 * fabs(d)) / t_3) * fmax(M, D)) * (fmax(M, D) / (l * d)));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (fmin(M, D) * fmin(M, D)) * h;
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.sqrt((l * h));
	double t_3 = t_2 * d;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((((fmax(M, D) * fmax(M, D)) * t_0) * (Math.abs(d) / t_3)) * -0.125) / (l * d);
	} else if (t_1 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.abs(d) / t_2) * 1.0;
	} else {
		tmp = -0.125 * ((((t_0 * Math.abs(d)) / t_3) * fmax(M, D)) * (fmax(M, D) / (l * d)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (fmin(M, D) * fmin(M, D)) * h
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.sqrt((l * h))
	t_3 = t_2 * d
	tmp = 0
	if t_1 <= 0.0:
		tmp = ((((fmax(M, D) * fmax(M, D)) * t_0) * (math.fabs(d) / t_3)) * -0.125) / (l * d)
	elif t_1 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	elif t_1 <= math.inf:
		tmp = (math.fabs(d) / t_2) * 1.0
	else:
		tmp = -0.125 * ((((t_0 * math.fabs(d)) / t_3) * fmax(M, D)) * (fmax(M, D) / (l * d)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(fmin(M, D) * fmin(M, D)) * h)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(l * h))
	t_3 = Float64(t_2 * d)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(fmax(M, D) * fmax(M, D)) * t_0) * Float64(abs(d) / t_3)) * -0.125) / Float64(l * d));
	elseif (t_1 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(abs(d) / t_2) * 1.0);
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(t_0 * abs(d)) / t_3) * fmax(M, D)) * Float64(fmax(M, D) / Float64(l * d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (min(M, D) * min(M, D)) * h;
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((min(M, D) * max(M, D)) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = sqrt((l * h));
	t_3 = t_2 * d;
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = ((((max(M, D) * max(M, D)) * t_0) * (abs(d) / t_3)) * -0.125) / (l * d);
	elseif (t_1 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	elseif (t_1 <= Inf)
		tmp = (abs(d) / t_2) * 1.0;
	else
		tmp = -0.125 * ((((t_0 * abs(d)) / t_3) * max(M, D)) * (max(M, D) / (l * d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Min[M, D], $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * d), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(N[(N[Max[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Abs[d], $MachinePrecision] / t$95$2), $MachinePrecision] * 1.0), $MachinePrecision], N[(-0.125 * N[(N[(N[(N[(t$95$0 * N[Abs[d], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] * N[(N[Max[M, D], $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2}} \cdot \ell} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \color{blue}{\ell}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} \]

      8. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

      9. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \left(\ell \cdot \color{blue}{d}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(\ell \cdot d\right) \cdot \color{blue}{d}} \]

      12. times-frac N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \color{blue}{\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      16. lower-/.f64 24.2%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{\color{blue}{d}}\right) \]

   6. Applied rewrites 27.7%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \cdot \color{blue}{\frac{-1}{8}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \cdot \frac{-1}{8} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \cdot \frac{-1}{8} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}{\ell \cdot d} \cdot \frac{-1}{8} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \cdot \frac{-1}{8}}{\color{blue}{\ell \cdot d}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \cdot \frac{-1}{8}}{\color{blue}{\ell \cdot d}} \]

   8. Applied rewrites 29.7%

      \[\leadsto \frac{\left(\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right) \cdot -0.125}{\color{blue}{\ell \cdot d}} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2}} \cdot \ell} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \color{blue}{\ell}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} \]

      8. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

      9. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \left(\ell \cdot \color{blue}{d}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(\ell \cdot d\right) \cdot \color{blue}{d}} \]

      12. times-frac N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \color{blue}{\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      16. lower-/.f64 24.2%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{\color{blue}{d}}\right) \]

   6. Applied rewrites 27.7%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d} \cdot \color{blue}{\frac{D \cdot D}{\ell \cdot d}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d} \cdot \frac{D \cdot D}{\color{blue}{\ell \cdot d}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d} \cdot \frac{D \cdot D}{\color{blue}{\ell} \cdot d}\right) \]

      5. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d} \cdot \left(D \cdot \color{blue}{\frac{D}{\ell \cdot d}}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d} \cdot D\right) \cdot \color{blue}{\frac{D}{\ell \cdot d}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d} \cdot D\right) \cdot \color{blue}{\frac{D}{\ell \cdot d}}\right) \]

   8. Applied rewrites 31.3%

      \[\leadsto -0.125 \cdot \left(\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left|d\right|}{\sqrt{\ell \cdot h} \cdot d} \cdot D\right) \cdot \color{blue}{\frac{D}{\ell \cdot d}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 19: 68.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\ell \cdot h}\\ t_2 := \frac{\left(\left(\left(\mathsf{max}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\right) \cdot \left(\left(\mathsf{min}\left(M, D\right) \cdot \mathsf{min}\left(M, D\right)\right) \cdot h\right)\right) \cdot \frac{\left|d\right|}{t\_1 \cdot d}\right) \cdot -0.125}{\ell \cdot d}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\left|d\right|}{t\_1} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (*
            (/ 1.0 2.0)
            (pow (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
           (/ h l)))))
       (t_1 (sqrt (* l h)))
       (t_2
        (/
         (*
          (*
           (*
            (* (fmax M D) (fmax M D))
            (* (* (fmin M D) (fmin M D)) h))
           (/ (fabs d) (* t_1 d)))
          -0.125)
         (* l d))))
  (if (<= t_0 0.0)
    t_2
    (if (<= t_0 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (if (<= t_0 INFINITY) (* (/ (fabs d) t_1) 1.0) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((l * h));
	double t_2 = ((((fmax(M, D) * fmax(M, D)) * ((fmin(M, D) * fmin(M, D)) * h)) * (fabs(d) / (t_1 * d))) * -0.125) / (l * d);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (fabs(d) / t_1) * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((l * h));
	double t_2 = ((((fmax(M, D) * fmax(M, D)) * ((fmin(M, D) * fmin(M, D)) * h)) * (Math.abs(d) / (t_1 * d))) * -0.125) / (l * d);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.abs(d) / t_1) * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((l * h))
	t_2 = ((((fmax(M, D) * fmax(M, D)) * ((fmin(M, D) * fmin(M, D)) * h)) * (math.fabs(d) / (t_1 * d))) * -0.125) / (l * d)
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_2
	elif t_0 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	elif t_0 <= math.inf:
		tmp = (math.fabs(d) / t_1) * 1.0
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(l * h))
	t_2 = Float64(Float64(Float64(Float64(Float64(fmax(M, D) * fmax(M, D)) * Float64(Float64(fmin(M, D) * fmin(M, D)) * h)) * Float64(abs(d) / Float64(t_1 * d))) * -0.125) / Float64(l * d))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(abs(d) / t_1) * 1.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((min(M, D) * max(M, D)) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((l * h));
	t_2 = ((((max(M, D) * max(M, D)) * ((min(M, D) * min(M, D)) * h)) * (abs(d) / (t_1 * d))) * -0.125) / (l * d);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	elseif (t_0 <= Inf)
		tmp = (abs(d) / t_1) * 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[Max[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Min[M, D], $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[(t$95$1 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Abs[d], $MachinePrecision] / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2}} \cdot \ell} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{\color{blue}{d}}^{2} \cdot \ell} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \color{blue}{\ell}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} \]

      8. pow2 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(d \cdot d\right) \cdot \ell} \]

      9. associate-*l* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{d \cdot \left(\ell \cdot \color{blue}{d}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{-1}{8} \cdot \frac{\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\left(\ell \cdot d\right) \cdot \color{blue}{d}} \]

      12. times-frac N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{d}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\color{blue}{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{8} \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \color{blue}{\left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}}{d}\right) \]

      16. lower-/.f64 24.2%

          \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{\color{blue}{d}}\right) \]

   6. Applied rewrites 27.7%

      \[\leadsto -0.125 \cdot \left(\frac{D \cdot D}{\ell \cdot d} \cdot \color{blue}{\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \cdot \color{blue}{\frac{-1}{8}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \cdot \frac{-1}{8} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{D \cdot D}{\ell \cdot d} \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \cdot \frac{-1}{8} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}}{\ell \cdot d} \cdot \frac{-1}{8} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \cdot \frac{-1}{8}}{\color{blue}{\ell \cdot d}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{d}\right) \cdot \frac{-1}{8}}{\color{blue}{\ell \cdot d}} \]

   8. Applied rewrites 29.7%

      \[\leadsto \frac{\left(\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h} \cdot d}\right) \cdot -0.125}{\color{blue}{\ell \cdot d}} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 65.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|M\right|, \left|D\right|\right)\\ t_1 := \mathsf{max}\left(\left|M\right|, \left|D\right|\right)\\ t_2 := \frac{\left(\left(\left(\left(t\_1 \cdot t\_1\right) \cdot t\_0\right) \cdot t\_0\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\\ t_3 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{t\_0 \cdot t\_1}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (fmin (fabs M) (fabs D)))
       (t_1 (fmax (fabs M) (fabs D)))
       (t_2
        (*
         (/
          (*
           (* (* (* (* t_1 t_1) t_0) t_0) h)
           (/ (fabs d) (sqrt (* h l))))
          (* (* d d) l))
         -0.125))
       (t_3
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* t_0 t_1) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_3 -5e+33)
    t_2
    (if (<= t_3 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (if (<= t_3 INFINITY)
        (* (/ (fabs d) (sqrt (* l h))) 1.0)
        t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(fabs(M), fabs(D));
	double t_1 = fmax(fabs(M), fabs(D));
	double t_2 = ((((((t_1 * t_1) * t_0) * t_0) * h) * (fabs(d) / sqrt((h * l)))) / ((d * d) * l)) * -0.125;
	double t_3 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= -5e+33) {
		tmp = t_2;
	} else if (t_3 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (fabs(d) / sqrt((l * h))) * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = fmin(Math.abs(M), Math.abs(D));
	double t_1 = fmax(Math.abs(M), Math.abs(D));
	double t_2 = ((((((t_1 * t_1) * t_0) * t_0) * h) * (Math.abs(d) / Math.sqrt((h * l)))) / ((d * d) * l)) * -0.125;
	double t_3 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_3 <= -5e+33) {
		tmp = t_2;
	} else if (t_3 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = fmin(math.fabs(M), math.fabs(D))
	t_1 = fmax(math.fabs(M), math.fabs(D))
	t_2 = ((((((t_1 * t_1) * t_0) * t_0) * h) * (math.fabs(d) / math.sqrt((h * l)))) / ((d * d) * l)) * -0.125
	t_3 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((t_0 * t_1) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_3 <= -5e+33:
		tmp = t_2
	elif t_3 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	elif t_3 <= math.inf:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = fmin(abs(M), abs(D))
	t_1 = fmax(abs(M), abs(D))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(t_1 * t_1) * t_0) * t_0) * h) * Float64(abs(d) / sqrt(Float64(h * l)))) / Float64(Float64(d * d) * l)) * -0.125)
	t_3 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(t_0 * t_1) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_3 <= -5e+33)
		tmp = t_2;
	elseif (t_3 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = min(abs(M), abs(D));
	t_1 = max(abs(M), abs(D));
	t_2 = ((((((t_1 * t_1) * t_0) * t_0) * h) * (abs(d) / sqrt((h * l)))) / ((d * d) * l)) * -0.125;
	t_3 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((t_0 * t_1) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_3 <= -5e+33)
		tmp = t_2;
	elseif (t_3 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	elseif (t_3 <= Inf)
		tmp = (abs(d) / sqrt((l * h))) * 1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Min[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[M], $MachinePrecision], N[Abs[D], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] * h), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+33], t$95$2, If[LessEqual[t$95$3, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.9999999999999997e33 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

   6. Applied rewrites 24.2%

      \[\leadsto \frac{\left(\left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right) \cdot h\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}}{\left(d \cdot d\right) \cdot \ell} \cdot \color{blue}{-0.125} \]

   if -4.9999999999999997e33 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 57.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\mathsf{min}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{\left(-0.125 \cdot \left(\left(\left(\mathsf{max}\left(M, D\right) \cdot \mathsf{max}\left(M, D\right)\right) \cdot \mathsf{min}\left(M, D\right)\right) \cdot \mathsf{min}\left(M, D\right)\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\frac{\ell}{h}}}}{\ell \cdot d}}{d}\\ \mathbf{elif}\;t\_0 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (*
            (/ 1.0 2.0)
            (pow (/ (* (fmin M D) (fmax M D)) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_0 -1e-80)
    (/
     (/
      (*
       (*
        -0.125
        (* (* (* (fmax M D) (fmax M D)) (fmin M D)) (fmin M D)))
       (/ (fabs d) (sqrt (/ l h))))
      (* l d))
     d)
    (if (<= t_0 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (* (/ (fabs d) (sqrt (* l h))) 1.0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-80) {
		tmp = (((-0.125 * (((fmax(M, D) * fmax(M, D)) * fmin(M, D)) * fmin(M, D))) * (fabs(d) / sqrt((l / h)))) / (l * d)) / d;
	} else if (t_0 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((fmin(m, d_1) * fmax(m, d_1)) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d-80)) then
        tmp = ((((-0.125d0) * (((fmax(m, d_1) * fmax(m, d_1)) * fmin(m, d_1)) * fmin(m, d_1))) * (abs(d) / sqrt((l / h)))) / (l * d)) / d
    else if (t_0 <= 1d+159) then
        tmp = (sqrt(((1.0d0 / l) * d)) * sqrt((d / h))) * 1.0d0
    else
        tmp = (abs(d) / sqrt((l * h))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-80) {
		tmp = (((-0.125 * (((fmax(M, D) * fmax(M, D)) * fmin(M, D)) * fmin(M, D))) * (Math.abs(d) / Math.sqrt((l / h)))) / (l * d)) / d;
	} else if (t_0 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((fmin(M, D) * fmax(M, D)) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e-80:
		tmp = (((-0.125 * (((fmax(M, D) * fmax(M, D)) * fmin(M, D)) * fmin(M, D))) * (math.fabs(d) / math.sqrt((l / h)))) / (l * d)) / d
	elif t_0 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(fmin(M, D) * fmax(M, D)) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-80)
		tmp = Float64(Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(fmax(M, D) * fmax(M, D)) * fmin(M, D)) * fmin(M, D))) * Float64(abs(d) / sqrt(Float64(l / h)))) / Float64(l * d)) / d);
	elseif (t_0 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((min(M, D) * max(M, D)) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -1e-80)
		tmp = (((-0.125 * (((max(M, D) * max(M, D)) * min(M, D)) * min(M, D))) * (abs(d) / sqrt((l / h)))) / (l * d)) / d;
	elseif (t_0 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	else
		tmp = (abs(d) / sqrt((l * h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(N[Min[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-80], N[(N[(N[(N[(-0.125 * N[(N[(N[(N[Max[M, D], $MachinePrecision] * N[Max[M, D], $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision] * N[Min[M, D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[t$95$0, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999996e-81

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

   6. Applied rewrites 29.2%

      \[\leadsto \frac{\frac{\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{\color{blue}{d}} \]

   7. Taylor expanded in h around inf

      \[\leadsto \frac{\frac{\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\frac{\ell}{h}}}}{\ell \cdot d}}{d} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\frac{\ell}{h}}}}{\ell \cdot d}}{d} \]

      2. lower-fabs.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\frac{\ell}{h}}}}{\ell \cdot d}}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\left(\frac{-1}{8} \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\frac{\ell}{h}}}}{\ell \cdot d}}{d} \]

      4. lower-/.f64 19.0%

         \[\leadsto \frac{\frac{\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\frac{\ell}{h}}}}{\ell \cdot d}}{d} \]

   9. Applied rewrites 19.0%

      \[\leadsto \frac{\frac{\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \frac{\left|d\right|}{\sqrt{\frac{\ell}{h}}}}{\ell \cdot d}}{d} \]

   if -9.9999999999999996e-81 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 50.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right)\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_0 0.0)
    (* (* (sqrt (/ d l)) (* -1.0 (* d (sqrt (/ 1.0 (* d h)))))) 1.0)
    (if (<= t_0 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (* (/ (fabs d) (sqrt (* l h))) 1.0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (sqrt((d / l)) * (-1.0 * (d * sqrt((1.0 / (d * h)))))) * 1.0;
	} else if (t_0 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= 0.0d0) then
        tmp = (sqrt((d / l)) * ((-1.0d0) * (d * sqrt((1.0d0 / (d * h)))))) * 1.0d0
    else if (t_0 <= 1d+159) then
        tmp = (sqrt(((1.0d0 / l) * d)) * sqrt((d / h))) * 1.0d0
    else
        tmp = (abs(d) / sqrt((l * h))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (Math.sqrt((d / l)) * (-1.0 * (d * Math.sqrt((1.0 / (d * h)))))) * 1.0;
	} else if (t_0 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (math.sqrt((d / l)) * (-1.0 * (d * math.sqrt((1.0 / (d * h)))))) * 1.0
	elif t_0 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(-1.0 * Float64(d * sqrt(Float64(1.0 / Float64(d * h)))))) * 1.0);
	elseif (t_0 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (sqrt((d / l)) * (-1.0 * (d * sqrt((1.0 / (d * h)))))) * 1.0;
	elseif (t_0 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	else
		tmp = (abs(d) / sqrt((l * h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * N[(d * N[Sqrt[N[(1.0 / N[(d * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right)}\right) \cdot 1 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)}\right)\right) \cdot 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{d \cdot h}}}\right)\right)\right) \cdot 1 \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right)\right) \cdot 1 \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right)\right) \cdot 1 \]

      5. lower-*.f64 22.8%

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right)\right) \cdot 1 \]

   9. Applied rewrites 22.8%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot h}}\right)\right)}\right) \cdot 1 \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 50.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{t\_0}{1 \cdot \left(-d\right)}}\\ \mathbf{elif}\;t\_1 \leq 10^{+159}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (sqrt (* l h)))
       (t_1
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_1 0.0)
    (/ 1.0 (/ t_0 (* 1.0 (- d))))
    (if (<= t_1 1e+159)
      (* (* (sqrt (* (/ 1.0 l) d)) (sqrt (/ d h))) 1.0)
      (* (/ (fabs d) t_0) 1.0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0 / (t_0 / (1.0 * -d));
	} else if (t_1 <= 1e+159) {
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = (fabs(d) / t_0) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((l * h))
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= 0.0d0) then
        tmp = 1.0d0 / (t_0 / (1.0d0 * -d))
    else if (t_1 <= 1d+159) then
        tmp = (sqrt(((1.0d0 / l) * d)) * sqrt((d / h))) * 1.0d0
    else
        tmp = (abs(d) / t_0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((l * h));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0 / (t_0 / (1.0 * -d));
	} else if (t_1 <= 1e+159) {
		tmp = (Math.sqrt(((1.0 / l) * d)) * Math.sqrt((d / h))) * 1.0;
	} else {
		tmp = (Math.abs(d) / t_0) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = 1.0 / (t_0 / (1.0 * -d))
	elif t_1 <= 1e+159:
		tmp = (math.sqrt(((1.0 / l) * d)) * math.sqrt((d / h))) * 1.0
	else:
		tmp = (math.fabs(d) / t_0) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(1.0 / Float64(t_0 / Float64(1.0 * Float64(-d))));
	elseif (t_1 <= 1e+159)
		tmp = Float64(Float64(sqrt(Float64(Float64(1.0 / l) * d)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(Float64(abs(d) / t_0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((l * h));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = 1.0 / (t_0 / (1.0 * -d));
	elseif (t_1 <= 1e+159)
		tmp = (sqrt(((1.0 / l) * d)) * sqrt((d / h))) * 1.0;
	else
		tmp = (abs(d) / t_0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(1.0 / N[(t$95$0 / N[(1.0 * (-d)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+159], N[(N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{\ell \cdot h}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      5. pow1/2 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{{\left(\ell \cdot h\right)}^{\frac{1}{2}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

      9. pow1/2 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      11. div-flip N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{h \cdot \ell}}{\left|d\right| \cdot 1}}} \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{h \cdot \ell}}{\left|d\right| \cdot 1}}} \]

      13. lower-unsound-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{h \cdot \ell}}{\left|d\right| \cdot 1}}} \]

   10. Applied rewrites 25.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell \cdot h}}{1 \cdot \left(-d\right)}}} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{\color{blue}{d \cdot \frac{1}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

      5. lower-/.f64 39.0%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell}} \cdot d} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   8. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\ell} \cdot d}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1 \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 50.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{t\_0}{1 \cdot \left(-d\right)}}\\ \mathbf{elif}\;t\_1 \leq 10^{+159}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (sqrt (* l h)))
       (t_1
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_1 0.0)
    (/ 1.0 (/ t_0 (* 1.0 (- d))))
    (if (<= t_1 1e+159)
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (* (/ (fabs d) t_0) 1.0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0 / (t_0 / (1.0 * -d));
	} else if (t_1 <= 1e+159) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = (fabs(d) / t_0) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((l * h))
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= 0.0d0) then
        tmp = 1.0d0 / (t_0 / (1.0d0 * -d))
    else if (t_1 <= 1d+159) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = (abs(d) / t_0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((l * h));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0 / (t_0 / (1.0 * -d));
	} else if (t_1 <= 1e+159) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = (Math.abs(d) / t_0) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = 1.0 / (t_0 / (1.0 * -d))
	elif t_1 <= 1e+159:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = (math.fabs(d) / t_0) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(1.0 / Float64(t_0 / Float64(1.0 * Float64(-d))));
	elseif (t_1 <= 1e+159)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(abs(d) / t_0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((l * h));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = 1.0 / (t_0 / (1.0 * -d));
	elseif (t_1 <= 1e+159)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = (abs(d) / t_0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(1.0 / N[(t$95$0 / N[(1.0 * (-d)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+159], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{\ell \cdot h}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      5. pow1/2 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{{\left(\ell \cdot h\right)}^{\frac{1}{2}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

      9. pow1/2 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      11. div-flip N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{h \cdot \ell}}{\left|d\right| \cdot 1}}} \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{h \cdot \ell}}{\left|d\right| \cdot 1}}} \]

      13. lower-unsound-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{h \cdot \ell}}{\left|d\right| \cdot 1}}} \]

   10. Applied rewrites 25.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\ell \cdot h}}{1 \cdot \left(-d\right)}}} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

   6. Applied rewrites 29.2%

      \[\leadsto \frac{\frac{\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{\color{blue}{d}} \]

   7. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\color{blue}{d}}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 39.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 49.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1 \cdot \left(-d\right)}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 10^{+159}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (sqrt (* l h)))
       (t_1
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (*
           (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0))
           (/ h l))))))
  (if (<= t_1 0.0)
    (/ (* 1.0 (- d)) t_0)
    (if (<= t_1 1e+159)
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (* (/ (fabs d) t_0) 1.0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (1.0 * -d) / t_0;
	} else if (t_1 <= 1e+159) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = (fabs(d) / t_0) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((l * h))
    t_1 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= 0.0d0) then
        tmp = (1.0d0 * -d) / t_0
    else if (t_1 <= 1d+159) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = (abs(d) / t_0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((l * h));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (1.0 * -d) / t_0;
	} else if (t_1 <= 1e+159) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = (Math.abs(d) / t_0) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = (1.0 * -d) / t_0
	elif t_1 <= 1e+159:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = (math.fabs(d) / t_0) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 * Float64(-d)) / t_0);
	elseif (t_1 <= 1e+159)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(abs(d) / t_0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((l * h));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = (1.0 * -d) / t_0;
	elseif (t_1 <= 1e+159)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = (abs(d) / t_0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 * (-d)), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1e+159], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{\ell \cdot h}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      5. pow1/2 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{{\left(\ell \cdot h\right)}^{\frac{1}{2}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

      9. pow1/2 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{h \cdot \ell}}} \]

   10. Applied rewrites 25.9%

       \[\leadsto \color{blue}{\frac{1 \cdot \left(-d\right)}{\sqrt{\ell \cdot h}}} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999993e158

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}} \]

   4. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]

   6. Applied rewrites 29.2%

      \[\leadsto \frac{\frac{\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot M\right)\right) \cdot \left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right)}{\ell \cdot d}}{\color{blue}{d}} \]

   7. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\color{blue}{d}}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   9. Applied rewrites 39.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   if 9.9999999999999993e158 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 46.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\left(-d\right) \cdot \frac{1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (let* ((t_0 (sqrt (* l h))))
  (if (<=
       (*
        (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
        (-
         1.0
         (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
       -1e-80)
    (* (- d) (/ 1.0 t_0))
    (* (/ (fabs d) t_0) 1.0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-80) {
		tmp = -d * (1.0 / t_0);
	} else {
		tmp = (fabs(d) / t_0) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-1d-80)) then
        tmp = -d * (1.0d0 / t_0)
    else
        tmp = (abs(d) / t_0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-80) {
		tmp = -d * (1.0 / t_0);
	} else {
		tmp = (Math.abs(d) / t_0) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-80:
		tmp = -d * (1.0 / t_0)
	else:
		tmp = (math.fabs(d) / t_0) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -1e-80)
		tmp = Float64(Float64(-d) * Float64(1.0 / t_0));
	else
		tmp = Float64(Float64(abs(d) / t_0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -1e-80)
		tmp = -d * (1.0 / t_0);
	else
		tmp = (abs(d) / t_0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-80], N[((-d) * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999996e-81

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{\ell \cdot h}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      5. pow1/2 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{{\left(\ell \cdot h\right)}^{\frac{1}{2}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

      9. pow1/2 N/A

         \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{\left|d\right| \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left|d\right| \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

   10. Applied rewrites 25.9%

       \[\leadsto \color{blue}{\left(-d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]

   if -9.9999999999999996e-81 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-*.f64 65.7%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 65.7%

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-sqrt.f64 65.7%

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 39.0%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 25.9% accurate, 7.2× speedup

Math

\[\left(-d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}} \]

FPCore

(FPCore (d h l M D)
  :precision binary64
  (* (- d) (/ 1.0 (sqrt (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	return -d * (1.0 / sqrt((l * h)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = -d * (1.0d0 / sqrt((l * h)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return -d * (1.0 / Math.sqrt((l * h)));
}

Python

def code(d, h, l, M, D):
	return -d * (1.0 / math.sqrt((l * h)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64(-d) * Float64(1.0 / sqrt(Float64(l * h))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = -d * (1.0 / sqrt((l * h)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[((-d) * N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. lower-*.f64 65.7%

      \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. metadata-eval N/A

      \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. unpow1/2 N/A

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. lower-sqrt.f64 65.7%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. lift-pow.f64 N/A

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   10. lift-/.f64 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   12. unpow1/2 N/A

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   13. lower-sqrt.f64 65.7%

       \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Applied rewrites 65.7%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

4. Taylor expanded in d around inf

   \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
5. Step-by-step derivation

6. Applied rewrites 39.0%

   \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 43.2%

   \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left|d\right| \cdot 1}{\sqrt{\ell \cdot h}}} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   5. pow1/2 N/A

      \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{{\left(\ell \cdot h\right)}^{\frac{1}{2}}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(\ell \cdot h\right)}}^{\frac{1}{2}}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left|d\right| \cdot 1}{{\color{blue}{\left(h \cdot \ell\right)}}^{\frac{1}{2}}} \]

   9. pow1/2 N/A

      \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \frac{\left|d\right| \cdot 1}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   11. associate-/l* N/A

       \[\leadsto \color{blue}{\left|d\right| \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

   12. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left|d\right| \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

10. Applied rewrites 25.9%

    \[\leadsto \color{blue}{\left(-d\right) \cdot \frac{1}{\sqrt{\ell \cdot h}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025213 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))