Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.0% → 74.9%

Time: 8.2s

Alternatives: 14

Speedup: 1.4×

Specification

Math

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

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: 67.0% accurate, 1.0× speedup

Math

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

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: 74.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-231}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{-1}}\right)\right) \cdot \left(1 - \frac{\left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right) \cdot -0.125}{h}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2e-231)
   (*
    (* d (- (sqrt (pow (* h l) -1.0))))
    (- 1.0 (/ (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) 0.5) h) l)))
   (if (<= d 3e-162)
     (/ (* (* (pow (/ h l) 1.5) (/ (pow (* M D) 2.0) d)) -0.125) h)
     (if (<= d 5e+115)
       (*
        (* (/ (sqrt d) (sqrt l)) (sqrt (/ d h)))
        (/ (fma (/ (* (pow (* D M) 2.0) h) (* d d)) -0.125 l) l))
       (*
        (* (sqrt (pow (* l h) -1.0)) d)
        (-
         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) {
	double tmp;
	if (d <= -2e-231) {
		tmp = (d * -sqrt(pow((h * l), -1.0))) * (1.0 - (((pow(((D * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else if (d <= 3e-162) {
		tmp = ((pow((h / l), 1.5) * (pow((M * D), 2.0) / d)) * -0.125) / h;
	} else if (d <= 5e+115) {
		tmp = ((sqrt(d) / sqrt(l)) * sqrt((d / h))) * (fma(((pow((D * M), 2.0) * h) / (d * d)), -0.125, l) / l);
	} else {
		tmp = (sqrt(pow((l * h), -1.0)) * d) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2e-231)
		tmp = Float64(Float64(d * Float64(-sqrt((Float64(h * l) ^ -1.0)))) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * 0.5) * h) / l)));
	elseif (d <= 3e-162)
		tmp = Float64(Float64(Float64((Float64(h / l) ^ 1.5) * Float64((Float64(M * D) ^ 2.0) / d)) * -0.125) / h);
	elseif (d <= 5e+115)
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(l)) * sqrt(Float64(d / h))) * Float64(fma(Float64(Float64((Float64(D * M) ^ 2.0) * h) / Float64(d * d)), -0.125, l) / l));
	else
		tmp = Float64(Float64(sqrt((Float64(l * h) ^ -1.0)) * d) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2e-231], N[(N[(d * (-N[Sqrt[N[Power[N[(h * l), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3e-162], N[(N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 5e+115], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.125 + l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $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]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2e-231

   1. Initial program 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 73.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 73.4

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 73.6

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left({\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   8. Applied rewrites 73.6%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   9. Taylor expanded in h around -inf

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

       1. sqrt-div N/A

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

       2. sqrt-pow2 N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. inv-pow N/A

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

       11. lower-pow.f64 N/A

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

       12. lower-*.f64 78.0

           \[\leadsto \left(d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{-1}}\right)\right) \cdot \left(1 - \frac{\left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   11. Applied rewrites 78.0%

       \[\leadsto \color{blue}{\left(d \cdot \left(-\sqrt{{\left(h \cdot \ell\right)}^{-1}}\right)\right)} \cdot \left(1 - \frac{\left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if -2e-231 < d < 2.99999999999999999e-162

   1. Initial program 43.2%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right)}{h} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}}{h} \]

   8. Applied rewrites 59.0%

      \[\leadsto \frac{\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right) \cdot -0.125}{h} \]

   if 2.99999999999999999e-162 < d < 5.00000000000000008e115

   1. Initial program 73.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

   5. Applied rewrites 76.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d}}{\ell}} \]

   7. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      3. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      6. lower-sqrt.f64 80.4

         \[\leadsto \left(\frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell} \]

   9. Applied rewrites 80.4%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell} \]

   if 5.00000000000000008e115 < d

   1. Initial program 72.7%

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

   3. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

Alternative 2: 59.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-157}:\\ \;\;\;\;t\_2 \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), -0.125, \ell\right)}{\ell}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;t\_2 \cdot 1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_0\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h 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 (* (sqrt (/ d l)) (sqrt (/ d h)))))
   (if (<= t_1 -5e-157)
     (* t_2 (/ (fma (* (* D D) (* (* M M) (/ h (* d d)))) -0.125 l) l))
     (if (<= t_1 5e+188)
       (* t_2 1.0)
       (if (<= t_1 INFINITY) (/ (* t_0 d) h) (/ (* d (- t_0)) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / 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 = sqrt((d / l)) * sqrt((d / h));
	double tmp;
	if (t_1 <= -5e-157) {
		tmp = t_2 * (fma(((D * D) * ((M * M) * (h / (d * d)))), -0.125, l) / l);
	} else if (t_1 <= 5e+188) {
		tmp = t_2 * 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / 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(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (t_1 <= -5e-157)
		tmp = Float64(t_2 * Float64(fma(Float64(Float64(D * D) * Float64(Float64(M * M) * Float64(h / Float64(d * d)))), -0.125, l) / l));
	elseif (t_1 <= 5e+188)
		tmp = Float64(t_2 * 1.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_0)) / h);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / 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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-157], N[(t$95$2 * N[(N[(N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+188], N[(t$95$2 * 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $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)))) < -5.0000000000000002e-157

   1. Initial program 86.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

   5. Applied rewrites 64.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d}}{\ell}} \]

   7. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      6. unpow-prod-down N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      7. associate-*r* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      8. pow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}, \frac{-1}{8}, \ell\right)}{\ell} \]

      9. associate-/l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{8}, \ell\right)}{\ell} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{8}, \ell\right)}{\ell} \]

      11. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{8}, \ell\right)}{\ell} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}, \frac{-1}{8}, \ell\right)}{\ell} \]

      13. associate-/l* N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{8}, \ell\right)}{\ell} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{8}, \ell\right)}{\ell} \]

      15. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{8}, \ell\right)}{\ell} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{8}, \ell\right)}{\ell} \]

      17. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{{d}^{2}}\right), \frac{-1}{8}, \ell\right)}{\ell} \]

      18. pow2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), \frac{-1}{8}, \ell\right)}{\ell} \]

      19. lift-*.f64 50.9

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), -0.125, \ell\right)}{\ell} \]

   9. Applied rewrites 50.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right), -0.125, \ell\right)}{\ell} \]

   if -5.0000000000000002e-157 < (*.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)))) < 5.0000000000000001e188

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

   5. Applied rewrites 69.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d}}{\ell}} \]

   7. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell}} \]

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 86.2%

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

   if 5.0000000000000001e188 < (*.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 61.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-*.f64 76.4

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   8. Applied rewrites 76.4%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   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 0.0%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{h}{\ell}}\right)\right)}{h} \]

      7. lower-neg.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 15.8

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

   8. Applied rewrites 15.8%

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

Alternative 3: 52.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-78}:\\ \;\;\;\;\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_0\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h 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))))))
   (if (<= t_1 -4e-78)
     (* (* -0.125 (* (* D D) (/ (* M M) d))) (sqrt (/ h (* (* l l) l))))
     (if (<= t_1 5e+188)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (if (<= t_1 INFINITY) (/ (* t_0 d) h) (/ (* d (- t_0)) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / 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 tmp;
	if (t_1 <= -4e-78) {
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * sqrt((h / ((l * l) * l)));
	} else if (t_1 <= 5e+188) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / 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 tmp;
	if (t_1 <= -4e-78) {
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * Math.sqrt((h / ((l * l) * l)));
	} else if (t_1 <= 5e+188) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / 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)))
	tmp = 0
	if t_1 <= -4e-78:
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * math.sqrt((h / ((l * l) * l)))
	elif t_1 <= 5e+188:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = (d * -t_0) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / 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))))
	tmp = 0.0
	if (t_1 <= -4e-78)
		tmp = Float64(Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(M * M) / d))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (t_1 <= 5e+188)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_0)) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / 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)));
	tmp = 0.0;
	if (t_1 <= -4e-78)
		tmp = (-0.125 * ((D * D) * ((M * M) / d))) * sqrt((h / ((l * l) * l)));
	elseif (t_1 <= 5e+188)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = (d * -t_0) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / 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]}, If[LessEqual[t$95$1, -4e-78], N[(N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+188], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $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)))) < -4e-78

   1. Initial program 86.1%

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

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      5. pow-prod-down N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      7. lower-pow.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      12. lower-pow.f64 35.0

          \[\leadsto \left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

   5. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      4. unpow-prod-down N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      5. associate-/l* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{\color{blue}{3}}}} \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. lower-*.f64 28.8

          \[\leadsto \left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

   7. Applied rewrites 28.8%

      \[\leadsto \left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      2. unpow3 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{2} \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{2} \cdot \ell}} \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      6. lower-*.f64 28.8

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

   9. Applied rewrites 28.8%

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

   if -4e-78 < (*.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)))) < 5.0000000000000001e188

   1. Initial program 87.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

   5. Applied rewrites 68.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d}}{\ell}} \]

   7. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell}} \]

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 84.6%

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

   if 5.0000000000000001e188 < (*.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 61.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-*.f64 76.4

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   8. Applied rewrites 76.4%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   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 0.0%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{h}{\ell}}\right)\right)}{h} \]

      7. lower-neg.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 15.8

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

   8. Applied rewrites 15.8%

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

Alternative 4: 50.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{d \cdot \left(-t\_0\right)}{h}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-157}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h 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 (/ (* d (- t_0)) h)))
   (if (<= t_1 -5e-157)
     t_2
     (if (<= t_1 5e+188)
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
       (if (<= t_1 INFINITY) (/ (* t_0 d) h) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / 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 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= -5e-157) {
		tmp = t_2;
	} else if (t_1 <= 5e+188) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} 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.sqrt((h / 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 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= -5e-157) {
		tmp = t_2;
	} else if (t_1 <= 5e+188) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / 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 = (d * -t_0) / h
	tmp = 0
	if t_1 <= -5e-157:
		tmp = t_2
	elif t_1 <= 5e+188:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / 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(Float64(d * Float64(-t_0)) / h)
	tmp = 0.0
	if (t_1 <= -5e-157)
		tmp = t_2;
	elseif (t_1 <= 5e+188)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / 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 = (d * -t_0) / h;
	tmp = 0.0;
	if (t_1 <= -5e-157)
		tmp = t_2;
	elseif (t_1 <= 5e+188)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / 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[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-157], t$95$2, If[LessEqual[t$95$1, 5e+188], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $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)))) < -5.0000000000000002e-157 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 56.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{h}{\ell}}\right)\right)}{h} \]

      7. lower-neg.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 20.2

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

   8. Applied rewrites 20.2%

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

   if -5.0000000000000002e-157 < (*.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)))) < 5.0000000000000001e188

   1. Initial program 87.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

   5. Applied rewrites 69.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d}}{\ell}} \]

   7. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell}} \]

   8. Taylor expanded in d around inf

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

   10. Applied rewrites 86.2%

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

   if 5.0000000000000001e188 < (*.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 61.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-*.f64 76.4

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   8. Applied rewrites 76.4%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right) \cdot -0.125}{h}\\ \end{array} \end{array} \]

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 5e+188)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) 0.5) h) l)))
     (if (<= t_0 INFINITY)
       (/ (* (sqrt (/ h l)) d) h)
       (/ (* (* (pow (/ h l) 1.5) (/ (pow (* M D) 2.0) d)) -0.125) h)))))

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 <= 5e+188) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((pow(((D * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = ((pow((h / l), 1.5) * (pow((M * D), 2.0) / d)) * -0.125) / h;
	}
	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 tmp;
	if (t_0 <= 5e+188) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((Math.pow(((D * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = ((Math.pow((h / l), 1.5) * (Math.pow((M * D), 2.0) / d)) * -0.125) / h;
	}
	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 <= 5e+188:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((math.pow(((D * M) / (2.0 * d)), 2.0) * 0.5) * h) / l))
	elif t_0 <= math.inf:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = ((math.pow((h / l), 1.5) * (math.pow((M * D), 2.0) / d)) * -0.125) / h
	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 <= 5e+188)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * 0.5) * h) / l)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(Float64(Float64((Float64(h / l) ^ 1.5) * Float64((Float64(M * D) ^ 2.0) / d)) * -0.125) / h);
	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 <= 5e+188)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((D * M) / (2.0 * d)) ^ 2.0) * 0.5) * h) / l));
	elseif (t_0 <= Inf)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = ((((h / l) ^ 1.5) * (((M * D) ^ 2.0) / d)) * -0.125) / h;
	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, 5e+188], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / h), $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)))) < 5.0000000000000001e188

   1. Initial program 86.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 85.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 85.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 86.5

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left({\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if 5.0000000000000001e188 < (*.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 61.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-*.f64 76.4

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   8. Applied rewrites 76.4%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   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 0.0%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right)}{h} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}}{h} \]

   8. Applied rewrites 38.4%

      \[\leadsto \frac{\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{{\left(M \cdot D\right)}^{2}}{d}\right) \cdot -0.125}{h} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 72.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_1 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_1\right)}{h}\\ \end{array} \end{array} \]

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 (/ h l))))
   (if (<= t_0 5e+188)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* (* (pow (/ (* D M) (* 2.0 d)) 2.0) 0.5) h) l)))
     (if (<= t_0 INFINITY) (/ (* t_1 d) h) (/ (* d (- t_1)) h)))))

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((h / l));
	double tmp;
	if (t_0 <= 5e+188) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((pow(((D * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (t_1 * d) / h;
	} else {
		tmp = (d * -t_1) / h;
	}
	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((h / l));
	double tmp;
	if (t_0 <= 5e+188) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - (((Math.pow(((D * M) / (2.0 * d)), 2.0) * 0.5) * h) / l));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (t_1 * d) / h;
	} else {
		tmp = (d * -t_1) / h;
	}
	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((h / l))
	tmp = 0
	if t_0 <= 5e+188:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - (((math.pow(((D * M) / (2.0 * d)), 2.0) * 0.5) * h) / l))
	elif t_0 <= math.inf:
		tmp = (t_1 * d) / h
	else:
		tmp = (d * -t_1) / h
	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(h / l))
	tmp = 0.0
	if (t_0 <= 5e+188)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * 0.5) * h) / l)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(t_1 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_1)) / h);
	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((h / l));
	tmp = 0.0;
	if (t_0 <= 5e+188)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((((D * M) / (2.0 * d)) ^ 2.0) * 0.5) * h) / l));
	elseif (t_0 <= Inf)
		tmp = (t_1 * d) / h;
	else
		tmp = (d * -t_1) / h;
	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[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 5e+188], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(t$95$1 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$1)), $MachinePrecision] / h), $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)))) < 5.0000000000000001e188

   1. Initial program 86.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 85.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 85.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 86.5

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left({\left(\frac{D \cdot M}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   8. Applied rewrites 86.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left({\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if 5.0000000000000001e188 < (*.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 61.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-*.f64 76.4

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   8. Applied rewrites 76.4%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   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 0.0%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{h}{\ell}}\right)\right)}{h} \]

      7. lower-neg.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 15.8

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

   8. Applied rewrites 15.8%

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

Alternative 7: 71.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{D}{d} \cdot \frac{M}{2}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(t\_2 \cdot t\_2\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_0\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h 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 (* (/ D d) (/ M 2.0))))
   (if (<= t_1 5e+188)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* (* (* t_2 t_2) 0.5) h) l)))
     (if (<= t_1 INFINITY) (/ (* t_0 d) h) (/ (* d (- t_0)) h)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / 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 = (D / d) * (M / 2.0);
	double tmp;
	if (t_1 <= 5e+188) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((t_2 * t_2) * 0.5) * h) / l));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / 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 = (D / d) * (M / 2.0);
	double tmp;
	if (t_1 <= 5e+188) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((((t_2 * t_2) * 0.5) * h) / l));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / 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 = (D / d) * (M / 2.0)
	tmp = 0
	if t_1 <= 5e+188:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((((t_2 * t_2) * 0.5) * h) / l))
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = (d * -t_0) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / 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(Float64(D / d) * Float64(M / 2.0))
	tmp = 0.0
	if (t_1 <= 5e+188)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(t_2 * t_2) * 0.5) * h) / l)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_0)) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / 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 = (D / d) * (M / 2.0);
	tmp = 0.0;
	if (t_1 <= 5e+188)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((t_2 * t_2) * 0.5) * h) / l));
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = (d * -t_0) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / 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[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+188], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $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)))) < 5.0000000000000001e188

   1. Initial program 86.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 85.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 85.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\color{blue}{\frac{d}{h}}}\right) \cdot \left(1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   6. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.7

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\color{blue}{\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right)} \cdot 0.5\right) \cdot h}{\ell}\right) \]

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 85.7

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(\left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 85.7

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\color{blue}{\frac{D}{d}} \cdot \frac{M}{2}\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right) \]

   8. Applied rewrites 85.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\color{blue}{\left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)\right)} \cdot 0.5\right) \cdot h}{\ell}\right) \]

   if 5.0000000000000001e188 < (*.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 61.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-*.f64 76.4

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   8. Applied rewrites 76.4%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   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 0.0%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{h}{\ell}}\right)\right)}{h} \]

      7. lower-neg.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 15.8

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

   8. Applied rewrites 15.8%

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

Alternative 8: 47.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{d \cdot \left(-t\_0\right)}{h}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-157}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h 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 (/ (* d (- t_0)) h)))
   (if (<= t_1 -5e-157) t_2 (if (<= t_1 INFINITY) (/ (* t_0 d) h) t_2))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / 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 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= -5e-157) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} 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.sqrt((h / 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 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= -5e-157) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / 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 = (d * -t_0) / h
	tmp = 0
	if t_1 <= -5e-157:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / 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(Float64(d * Float64(-t_0)) / h)
	tmp = 0.0
	if (t_1 <= -5e-157)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / 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 = (d * -t_0) / h;
	tmp = 0.0;
	if (t_1 <= -5e-157)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / 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[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-157], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], t$95$2]]]]]

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)))) < -5.0000000000000002e-157 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 56.1%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 40.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in l around -inf

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

      1. sqrt-pow2 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{h}{\ell}}\right)\right)}{h} \]

      7. lower-neg.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 20.2

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

   8. Applied rewrites 20.2%

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

   if -5.0000000000000002e-157 < (*.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 78.7%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-*.f64 77.0

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   8. Applied rewrites 77.0%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 61.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+88}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.3e+154)
   (* (- d) (pow (* l h) -0.5))
   (if (<= d 7.2e+88)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (/ (fma (/ (* (* (* D M) (* D M)) h) (* d d)) -0.125 l) l))
     (* (/ 1.0 (* (sqrt l) (sqrt h))) d))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.3e+154) {
		tmp = -d * pow((l * h), -0.5);
	} else if (d <= 7.2e+88) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (fma(((((D * M) * (D * M)) * h) / (d * d)), -0.125, l) / l);
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.3e+154)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	elseif (d <= 7.2e+88)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(fma(Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) * h) / Float64(d * d)), -0.125, l) / l));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.3e+154], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.2e+88], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.125 + l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.29999999999999994e154

   1. Initial program 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 77.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right) \cdot h}{\ell}}\right) \]

   5. Taylor expanded in l around -inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

   7. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}} \]

   if -1.29999999999999994e154 < d < 7.2000000000000004e88

   1. Initial program 64.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

   5. Applied rewrites 57.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d}}{\ell}} \]

   7. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      3. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      6. lift-*.f64 57.7

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell} \]

   9. Applied rewrites 57.7%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell} \]

   if 7.2000000000000004e88 < d

   1. Initial program 73.0%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 63.8

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   5. Applied rewrites 63.8%

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

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      4. *-commutative N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 63.9

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   7. Applied rewrites 63.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 73.3

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   9. Applied rewrites 73.3%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 7.2 \cdot 10^{+88}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 7.2e+88)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (/ (fma (/ (* (* (* D M) (* D M)) h) (* d d)) -0.125 l) l))
   (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 7.2e+88) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (fma(((((D * M) * (D * M)) * h) / (d * d)), -0.125, l) / l);
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 7.2e+88)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(fma(Float64(Float64(Float64(Float64(D * M) * Float64(D * M)) * h) / Float64(d * d)), -0.125, l) / l));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 7.2e+88], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.125 + l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 7.2000000000000004e88

   1. Initial program 65.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\ell}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\ell - \frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

   5. Applied rewrites 56.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\frac{\ell - \frac{0.125 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d}}{\ell}} \]

   7. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      3. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, \frac{-1}{8}, \ell\right)}{\ell} \]

      6. lift-*.f64 56.2

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell} \]

   9. Applied rewrites 56.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\mathsf{fma}\left(\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{d \cdot d}, -0.125, \ell\right)}{\ell} \]

   if 7.2000000000000004e88 < d

   1. Initial program 73.0%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 63.8

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   5. Applied rewrites 63.8%

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

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      4. *-commutative N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 63.9

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   7. Applied rewrites 63.9%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lower-sqrt.f64 73.3

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   9. Applied rewrites 73.3%

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

Alternative 11: 39.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -2.1 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -2.1e-196) (/ (* (sqrt (/ h l)) d) h) (/ (* 1.0 d) (sqrt (* h l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -2.1e-196) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = (1.0 * d) / sqrt((h * l));
	}
	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) :: tmp
    if (h <= (-2.1d-196)) then
        tmp = (sqrt((h / l)) * d) / h
    else
        tmp = (1.0d0 * d) / sqrt((h * l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -2.1e-196) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = (1.0 * d) / Math.sqrt((h * l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -2.1e-196:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = (1.0 * d) / math.sqrt((h * l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -2.1e-196)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(Float64(1.0 * d) / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -2.1e-196)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = (1.0 * d) / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -2.1e-196], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(1.0 * d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -2.09999999999999988e-196

   1. Initial program 67.2%

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

   3. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{h}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{{h}^{3}}{{\ell}^{3}}}\right) + d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   5. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}, \sqrt{{\left(\frac{h}{\ell}\right)}^{3}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}} \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-*.f64 38.9

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   8. Applied rewrites 38.9%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   if -2.09999999999999988e-196 < h

   1. Initial program 66.8%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. inv-pow N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      7. lower-*.f64 40.1

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   5. Applied rewrites 40.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

      4. *-commutative N/A

         \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

      5. inv-pow N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      6. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      11. lift-*.f64 40.2

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   7. Applied rewrites 40.2%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{d} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      5. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

      10. lower-*.f64 40.3

          \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   9. Applied rewrites 40.3%

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

Alternative 12: 27.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ (* 1.0 d) (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return (1.0 * d) / sqrt((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 = (1.0d0 * d) / sqrt((h * l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(1.0 * d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.0%

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

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. inv-pow N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   5. lower-pow.f64 N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   6. *-commutative N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   7. lower-*.f64 27.3

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

5. Applied rewrites 27.3%

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

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   2. lift-pow.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. *-commutative N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   5. inv-pow N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   6. sqrt-div N/A

      \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   11. lift-*.f64 27.1

       \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

7. Applied rewrites 27.1%

   \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot \color{blue}{d} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   5. associate-*l/ N/A

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1 \cdot d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

   10. lower-*.f64 27.2

       \[\leadsto \frac{1 \cdot d}{\sqrt{h \cdot \ell}} \]

9. Applied rewrites 27.2%

   \[\leadsto \frac{1 \cdot d}{\color{blue}{\sqrt{h \cdot \ell}}} \]
10. Add Preprocessing

Alternative 13: 27.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\ell \cdot h}} \cdot d \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* (/ 1.0 (sqrt (* l h))) d))

C

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

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 = (1.0d0 / sqrt((l * h))) * d
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(1.0 / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 67.0%

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

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. inv-pow N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   5. lower-pow.f64 N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   6. *-commutative N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   7. lower-*.f64 27.3

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

5. Applied rewrites 27.3%

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

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   2. lift-pow.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   4. *-commutative N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   5. inv-pow N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   6. sqrt-div N/A

      \[\leadsto \frac{\sqrt{1}}{\sqrt{h \cdot \ell}} \cdot d \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   11. lift-*.f64 27.1

       \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

7. Applied rewrites 27.1%

   \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
8. Add Preprocessing

Alternative 14: 27.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* (sqrt (/ 1.0 (* l h))) d))

C

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

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 = sqrt((1.0d0 / (l * h))) * d
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]

Derivation

1. Initial program 67.0%

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

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. inv-pow N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   5. lower-pow.f64 N/A

      \[\leadsto \sqrt{{\left(h \cdot \ell\right)}^{-1}} \cdot d \]

   6. *-commutative N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   7. lower-*.f64 27.3

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

5. Applied rewrites 27.3%

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

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   2. lift-pow.f64 N/A

      \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]

   3. unpow-1 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   5. lift-*.f64 27.3

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

7. Applied rewrites 27.3%

   \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(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)))))