Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.4% → 84.2%

Time: 10.5s

Alternatives: 14

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_1 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\\ t_2 := \left(t\_1 \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_0\\ t_3 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(h \cdot \left(\frac{D \cdot M}{d} \cdot -0.125\right), \frac{\frac{D \cdot M}{\ell}}{d}, 1\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\left(t\_1 \cdot {\left(\frac{1}{\ell} \cdot d\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
        (t_1 (pow (/ d h) (/ 1.0 2.0)))
        (t_2 (* (* t_1 (pow (/ d l) (/ 1.0 2.0))) t_0))
        (t_3
         (*
          (/ (fabs d) (sqrt (* l h)))
          (fma (* h (* (/ (* D M) d) -0.125)) (/ (/ (* D M) l) d) 1.0))))
   (if (<= t_2 0.0)
     t_3
     (if (<= t_2 2e+238)
       (* (* t_1 (pow (* (/ 1.0 l) d) (/ 1.0 2.0))) t_0)
       t_3))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_1 = pow((d / h), (1.0 / 2.0));
	double t_2 = (t_1 * pow((d / l), (1.0 / 2.0))) * t_0;
	double t_3 = (fabs(d) / sqrt((l * h))) * fma((h * (((D * M) / d) * -0.125)), (((D * M) / l) / d), 1.0);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 2e+238) {
		tmp = (t_1 * pow(((1.0 / l) * d), (1.0 / 2.0))) * t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	t_1 = Float64(d / h) ^ Float64(1.0 / 2.0)
	t_2 = Float64(Float64(t_1 * (Float64(d / l) ^ Float64(1.0 / 2.0))) * t_0)
	t_3 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(h * Float64(Float64(Float64(D * M) / d) * -0.125)), Float64(Float64(Float64(D * M) / l) / d), 1.0))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 2e+238)
		tmp = Float64(Float64(t_1 * (Float64(Float64(1.0 / l) * d) ^ Float64(1.0 / 2.0))) * t_0);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * M), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 2e+238], N[(N[(t$95$1 * N[Power[N[(N[(1.0 / l), $MachinePrecision] * d), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 2.0000000000000001e238 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 52.9%

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. rem-sqrt-square N/A

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

      20. lower-fabs.f64 75.5%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 75.5%

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

   5. Applied rewrites 75.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   7. Applied rewrites 76.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 77.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e238

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 66.4%

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

   3. Applied rewrites 66.4%

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

Alternative 2: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(h \cdot \left(\frac{D \cdot M}{d} \cdot -0.125\right), \frac{\frac{D \cdot M}{\ell}}{d}, 1\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \mathsf{fma}\left(\frac{\left(0.25 \cdot \frac{M \cdot D}{d}\right) \cdot h}{-2 \cdot d}, \frac{M \cdot D}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1
         (*
          (/ (fabs d) (sqrt (* l h)))
          (fma (* h (* (/ (* D M) d) -0.125)) (/ (/ (* D M) l) d) 1.0))))
   (if (<= t_0 0.0)
     t_1
     (if (<= t_0 2e+238)
       (*
        (* (sqrt (/ d l)) (sqrt (/ d h)))
        (fma (/ (* (* 0.25 (/ (* M D) d)) h) (* -2.0 d)) (/ (* M D) l) 1.0))
       t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fabs(d) / sqrt((l * h))) * fma((h * (((D * M) / d) * -0.125)), (((D * M) / l) / d), 1.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+238) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * fma((((0.25 * ((M * D) / d)) * h) / (-2.0 * d)), ((M * D) / l), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(h * Float64(Float64(Float64(D * M) / d) * -0.125)), Float64(Float64(Float64(D * M) / l) / d), 1.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+238)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * fma(Float64(Float64(Float64(0.25 * Float64(Float64(M * D) / d)) * h) / Float64(-2.0 * d)), Float64(Float64(M * D) / l), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * M), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+238], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / N[(-2.0 * d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 2.0000000000000001e238 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 52.9%

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. rem-sqrt-square N/A

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

      20. lower-fabs.f64 75.5%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 75.5%

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

   5. Applied rewrites 75.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   7. Applied rewrites 76.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 77.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e238

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 52.9%

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. rem-sqrt-square N/A

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

      20. lower-fabs.f64 75.5%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 75.5%

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

   5. Applied rewrites 75.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   7. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 76.1%

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

      4. lower-/.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-undiv N/A

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

      11. pow1/2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. pow-prod-down N/A

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

      18. metadata-eval N/A

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

      19. pow1/2 N/A

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

      20. lift-sqrt.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval N/A

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

      23. unpow1/2 N/A

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

      24. lower-sqrt.f64 N/A

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

      25. lower-/.f64 69.2%

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

   9. Applied rewrites 69.2%

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

Alternative 3: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_0\\ t_2 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(h \cdot \left(\frac{D \cdot M}{d} \cdot -0.125\right), \frac{\frac{D \cdot M}{\ell}}{d}, 1\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
        (t_1 (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) t_0))
        (t_2
         (*
          (/ (fabs d) (sqrt (* l h)))
          (fma (* h (* (/ (* D M) d) -0.125)) (/ (/ (* D M) l) d) 1.0))))
   (if (<= t_1 0.0)
     t_2
     (if (<= t_1 2e+238) (* (* (sqrt (/ d l)) (sqrt (/ d h))) t_0) t_2))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * t_0;
	double t_2 = (fabs(d) / sqrt((l * h))) * fma((h * (((D * M) / d) * -0.125)), (((D * M) / l) / d), 1.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+238) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * t_0)
	t_2 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(h * Float64(Float64(Float64(D * M) / d) * -0.125)), Float64(Float64(Float64(D * M) / l) / d), 1.0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2e+238)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, 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] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * M), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 2e+238], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $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)))) < 0.0 or 2.0000000000000001e238 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 52.9%

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. rem-sqrt-square N/A

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

      20. lower-fabs.f64 75.5%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 75.5%

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

   5. Applied rewrites 75.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   7. Applied rewrites 76.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 77.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e238

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.4%

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 66.4%

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 66.4%

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

   3. Applied rewrites 66.4%

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

Alternative 4: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(h \cdot \left(\frac{D \cdot M}{d} \cdot -0.125\right), \frac{\frac{D \cdot M}{\ell}}{d}, 1\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1
         (*
          (/ (fabs d) (sqrt (* l h)))
          (fma (* h (* (/ (* D M) d) -0.125)) (/ (/ (* D M) l) d) 1.0))))
   (if (<= t_0 0.0)
     t_1
     (if (<= t_0 2e+238) (* (sqrt (/ d h)) (sqrt (/ d l))) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fabs(d) / sqrt((l * h))) * fma((h * (((D * M) / d) * -0.125)), (((D * M) / l) / d), 1.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+238) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(h * Float64(Float64(Float64(D * M) / d) * -0.125)), Float64(Float64(Float64(D * M) / l) / d), 1.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+238)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D * M), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+238], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 2.0000000000000001e238 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 52.9%

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. rem-sqrt-square N/A

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

      20. lower-fabs.f64 75.5%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 75.5%

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

   5. Applied rewrites 75.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   7. Applied rewrites 76.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 77.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e238

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.3%

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

   4. Applied rewrites 23.3%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 31.9%

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

   7. Applied rewrites 31.9%

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

   8. Taylor expanded in h around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 39.2%

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

   10. Applied rewrites 39.2%

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

Alternative 5: 82.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{D \cdot M}{\ell} \cdot \left(\frac{D \cdot M}{d} \cdot -0.125\right), \frac{h}{d}, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1
         (/
          (*
           (fma (* (/ (* D M) l) (* (/ (* D M) d) -0.125)) (/ h d) 1.0)
           (fabs d))
          (sqrt (* h l)))))
   (if (<= t_0 -2e-107)
     t_1
     (if (<= t_0 0.0)
       (* (/ (fabs d) (sqrt (* l h))) 1.0)
       (if (<= t_0 2e+238) (* (sqrt (/ d h)) (sqrt (/ d l))) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fma((((D * M) / l) * (((D * M) / d) * -0.125)), (h / d), 1.0) * fabs(d)) / sqrt((h * l));
	double tmp;
	if (t_0 <= -2e-107) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (fabs(d) / sqrt((l * h))) * 1.0;
	} else if (t_0 <= 2e+238) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(fma(Float64(Float64(Float64(D * M) / l) * Float64(Float64(Float64(D * M) / d) * -0.125)), Float64(h / d), 1.0) * abs(d)) / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= -2e-107)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0);
	elseif (t_0 <= 2e+238)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(D * M), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-107], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+238], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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)))) < -2e-107 or 2.0000000000000001e238 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 52.9%

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. rem-sqrt-square N/A

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

      20. lower-fabs.f64 75.5%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 75.5%

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

   5. Applied rewrites 75.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   7. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 74.8%

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

   if -2e-107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 52.9%

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. rem-sqrt-square N/A

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

      20. lower-fabs.f64 75.5%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 75.5%

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 42.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e238

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.3%

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

   4. Applied rewrites 23.3%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 31.9%

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

   7. Applied rewrites 31.9%

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

   8. Taylor expanded in h around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 39.2%

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

   10. Applied rewrites 39.2%

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

Alternative 6: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{D \cdot M}{d}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(-0.125 \cdot \frac{t\_1 \cdot h}{d}\right) \cdot M, \frac{D}{\ell}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{D \cdot M}{\ell} \cdot \left(t\_1 \cdot -0.125\right), \frac{h}{d}, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}\\ \end{array} \]

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 (/ (* D M) d)))
   (if (<= t_0 0.0)
     (*
      (/ (fabs d) (sqrt (* l h)))
      (fma (* (* -0.125 (/ (* t_1 h) d)) M) (/ D l) 1.0))
     (if (<= t_0 2e+238)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (/
        (* (fma (* (/ (* D M) l) (* t_1 -0.125)) (/ h d) 1.0) (fabs d))
        (sqrt (* h l)))))))

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 = (D * M) / d;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (fabs(d) / sqrt((l * h))) * fma(((-0.125 * ((t_1 * h) / d)) * M), (D / l), 1.0);
	} else if (t_0 <= 2e+238) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = (fma((((D * M) / l) * (t_1 * -0.125)), (h / d), 1.0) * fabs(d)) / sqrt((h * l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(D * M) / d)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(-0.125 * Float64(Float64(t_1 * h) / d)) * M), Float64(D / l), 1.0));
	elseif (t_0 <= 2e+238)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(D * M) / l) * Float64(t_1 * -0.125)), Float64(h / d), 1.0) * abs(d)) / sqrt(Float64(h * l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.125 * N[(N[(t$95$1 * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * N[(D / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+238], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(D * M), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 * -0.125), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 52.9%

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. rem-sqrt-square N/A

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

      20. lower-fabs.f64 75.5%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 75.5%

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

   5. Applied rewrites 75.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   7. Applied rewrites 76.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 73.0%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e238

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.3%

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

   4. Applied rewrites 23.3%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 31.9%

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

   7. Applied rewrites 31.9%

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

   8. Taylor expanded in h around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 39.2%

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

   10. Applied rewrites 39.2%

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

   if 2.0000000000000001e238 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 52.9%

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. rem-sqrt-square N/A

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

      20. lower-fabs.f64 75.5%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 75.5%

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

   5. Applied rewrites 75.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

   7. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 74.8%

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

Alternative 7: 57.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{\ell \cdot d}} \cdot \left(-d\right)\right) \cdot \sqrt{h \cdot d}}{h}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (/ (fabs d) (sqrt (* l h))) 1.0)))
   (if (<= t_0 -2e-107)
     (/ (* (* (sqrt (/ 1.0 (* l d))) (- d)) (sqrt (* h d))) h)
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 2e+238) (* (sqrt (/ d h)) (sqrt (/ d l))) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fabs(d) / sqrt((l * h))) * 1.0;
	double tmp;
	if (t_0 <= -2e-107) {
		tmp = ((sqrt((1.0 / (l * d))) * -d) * sqrt((h * d))) / h;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+238) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = (abs(d) / sqrt((l * h))) * 1.0d0
    if (t_0 <= (-2d-107)) then
        tmp = ((sqrt((1.0d0 / (l * d))) * -d) * sqrt((h * d))) / h
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+238) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	double tmp;
	if (t_0 <= -2e-107) {
		tmp = ((Math.sqrt((1.0 / (l * d))) * -d) * Math.sqrt((h * d))) / h;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+238) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	tmp = 0
	if t_0 <= -2e-107:
		tmp = ((math.sqrt((1.0 / (l * d))) * -d) * math.sqrt((h * d))) / h
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+238:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0)
	tmp = 0.0
	if (t_0 <= -2e-107)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / Float64(l * d))) * Float64(-d)) * sqrt(Float64(h * d))) / h);
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+238)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = (abs(d) / sqrt((l * h))) * 1.0;
	tmp = 0.0;
	if (t_0 <= -2e-107)
		tmp = ((sqrt((1.0 / (l * d))) * -d) * sqrt((h * d))) / h;
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+238)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-107], N[(N[(N[(N[Sqrt[N[(1.0 / N[(l * d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision] * N[Sqrt[N[(h * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+238], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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)))) < -2e-107

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.3%

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

   4. Applied rewrites 23.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

      5. pow-to-exp N/A

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

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

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

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

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

      8. lower-unsound-log.f64 22.4%

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 22.4%

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

      12. lift-/.f64 N/A

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

      13. metadata-eval 22.4%

          \[\leadsto \frac{e^{\log \left(h \cdot d\right) \cdot 0.5} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   6. Applied rewrites 22.4%

      \[\leadsto \frac{e^{\log \left(h \cdot d\right) \cdot 0.5} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   7. Taylor expanded in d around -inf

      \[\leadsto \frac{e^{\log \left(h \cdot d\right) \cdot 0.5} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)}{h} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 14.4%

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

   9. Applied rewrites 14.4%

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

   11. Applied rewrites 14.4%

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

   if -2e-107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 2.0000000000000001e238 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 52.9%

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. rem-sqrt-square N/A

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

      20. lower-fabs.f64 75.5%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 75.5%

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 42.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e238

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 23.3%

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

   4. Applied rewrites 23.3%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 31.9%

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

   7. Applied rewrites 31.9%

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

   8. Taylor expanded in h around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 39.2%

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

   10. Applied rewrites 39.2%

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

Alternative 8: 55.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;-1 \cdot \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (/ (fabs d) (sqrt (* l h))) 1.0)))
   (if (<= t_0 -2e-107)
     (* -1.0 (/ d (* h (sqrt (/ l h)))))
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 2e+238) (* (sqrt (/ d h)) (sqrt (/ d l))) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fabs(d) / sqrt((l * h))) * 1.0;
	double tmp;
	if (t_0 <= -2e-107) {
		tmp = -1.0 * (d / (h * sqrt((l / h))));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+238) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = (abs(d) / sqrt((l * h))) * 1.0d0
    if (t_0 <= (-2d-107)) then
        tmp = (-1.0d0) * (d / (h * sqrt((l / h))))
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+238) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	double tmp;
	if (t_0 <= -2e-107) {
		tmp = -1.0 * (d / (h * Math.sqrt((l / h))));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+238) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	tmp = 0
	if t_0 <= -2e-107:
		tmp = -1.0 * (d / (h * math.sqrt((l / h))))
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+238:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0)
	tmp = 0.0
	if (t_0 <= -2e-107)
		tmp = Float64(-1.0 * Float64(d / Float64(h * sqrt(Float64(l / h)))));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+238)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = (abs(d) / sqrt((l * h))) * 1.0;
	tmp = 0.0;
	if (t_0 <= -2e-107)
		tmp = -1.0 * (d / (h * sqrt((l / h))));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+238)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-107], N[(-1.0 * N[(d / N[(h * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+238], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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)))) < -2e-107

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. sqrt-div N/A

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

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

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

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

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

      14. lower-*.f64 N/A

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

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

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

      16. lower-*.f64 48.6%

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

   3. Applied rewrites 48.6%

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

   4. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 26.3%

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

   6. Applied rewrites 26.3%

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

   7. Taylor expanded in h around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 12.3%

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

   9. Applied rewrites 12.3%

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

   if -2e-107 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 2.0000000000000001e238 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

   3. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-undiv N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-/.f64 52.9%

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

      17. lift-sqrt.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. rem-sqrt-square N/A

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

      20. lower-fabs.f64 75.5%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 75.5%

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 42.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e238

   1. Initial program 66.4%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.9%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.9%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 0.0)
     (/ (- d) (sqrt (* h l)))
     (if (<= t_0 2e+238)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (* (/ (fabs d) (sqrt (* l h))) 1.0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -d / sqrt((h * l));
	} else if (t_0 <= 2e+238) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= 0.0d0) then
        tmp = -d / sqrt((h * l))
    else if (t_0 <= 2d+238) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else
        tmp = (abs(d) / sqrt((l * h))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -d / Math.sqrt((h * l));
	} else if (t_0 <= 2e+238) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = -d / math.sqrt((h * l))
	elif t_0 <= 2e+238:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	else:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(-d) / sqrt(Float64(h * l)));
	elseif (t_0 <= 2e+238)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = -d / sqrt((h * l));
	elseif (t_0 <= 2e+238)
		tmp = sqrt((d / h)) * sqrt((d / l));
	else
		tmp = (abs(d) / sqrt((l * h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[((-d) / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+238], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.6%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e238

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 23.3%

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.9%

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.9%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      5. lower-/.f64 39.2%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   10. Applied rewrites 39.2%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   if 2.0000000000000001e238 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 66.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. frac-times N/A

         \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. lift-/.f64 52.9%

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      20. lower-fabs.f64 75.5%

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      22. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      23. lower-*.f64 75.5%

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.8%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 48.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -2e-107)
     (/ d (sqrt (* h l)))
     (if (<= t_0 2e+238)
       (/ d (* h (sqrt (/ l h))))
       (* (/ (fabs d) (sqrt (* l h))) 1.0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-107) {
		tmp = d / sqrt((h * l));
	} else if (t_0 <= 2e+238) {
		tmp = d / (h * sqrt((l / h)));
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-2d-107)) then
        tmp = d / sqrt((h * l))
    else if (t_0 <= 2d+238) then
        tmp = d / (h * sqrt((l / h)))
    else
        tmp = (abs(d) / sqrt((l * h))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-107) {
		tmp = d / Math.sqrt((h * l));
	} else if (t_0 <= 2e+238) {
		tmp = d / (h * Math.sqrt((l / h)));
	} else {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -2e-107:
		tmp = d / math.sqrt((h * l))
	elif t_0 <= 2e+238:
		tmp = d / (h * math.sqrt((l / h)))
	else:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -2e-107)
		tmp = Float64(d / sqrt(Float64(h * l)));
	elseif (t_0 <= 2e+238)
		tmp = Float64(d / Float64(h * sqrt(Float64(l / h))));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -2e-107)
		tmp = d / sqrt((h * l));
	elseif (t_0 <= 2e+238)
		tmp = d / (h * sqrt((l / h)));
	else
		tmp = (abs(d) / sqrt((l * h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-107], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+238], N[(d / N[(h * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-107

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.6%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lower-*.f64 26.4%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

   11. Applied rewrites 26.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   if -2e-107 < (*.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)))) < 2.0000000000000001e238

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.6%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \frac{d}{\color{blue}{h \cdot \sqrt{\frac{\ell}{h}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{d}{h \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

       4. lower-/.f64 39.3%

          \[\leadsto \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

   11. Applied rewrites 39.3%

       \[\leadsto \frac{d}{\color{blue}{h \cdot \sqrt{\frac{\ell}{h}}}} \]

   if 2.0000000000000001e238 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. 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-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \color{blue}{\left(D \cdot M\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      18. count-2-rev N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{\color{blue}{d + d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

   3. Applied rewrites 66.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      9. frac-times N/A

         \[\leadsto {\color{blue}{\left(\frac{d \cdot d}{h \cdot \ell}\right)}}^{\frac{1}{2}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto {\left(\frac{\color{blue}{d \cdot d}}{h \cdot \ell}\right)}^{\frac{1}{2}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto {\left(\frac{d \cdot d}{\color{blue}{h \cdot \ell}}\right)}^{\frac{1}{2}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      12. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      13. sqrt-undiv N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      16. lift-/.f64 52.9%

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      17. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      19. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      20. lower-fabs.f64 75.5%

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      22. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

      23. lower-*.f64 75.5%

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{\ell \cdot h}}} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot \left(1 - \frac{0.5 \cdot \left(D \cdot M\right)}{d + d} \cdot \frac{\left(D \cdot M\right) \cdot h}{\left(d + d\right) \cdot \ell}\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.8%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 44.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d}{\ell \cdot \sqrt{\frac{h}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ d (sqrt (* h l)))))
   (if (<= t_0 -2e-96)
     t_1
     (if (<= t_0 INFINITY) (/ d (* l (sqrt (/ h l)))) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = d / sqrt((h * l));
	double tmp;
	if (t_0 <= -2e-96) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = d / (l * sqrt((h / l)));
	} else {
		tmp = t_1;
	}
	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 = d / Math.sqrt((h * l));
	double tmp;
	if (t_0 <= -2e-96) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = d / (l * Math.sqrt((h / l)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = d / math.sqrt((h * l))
	tmp = 0
	if t_0 <= -2e-96:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = d / (l * math.sqrt((h / l)))
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= -2e-96)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(d / Float64(l * sqrt(Float64(h / l))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = d / sqrt((h * l));
	tmp = 0.0;
	if (t_0 <= -2e-96)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = d / (l * sqrt((h / l)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-96], t$95$1, If[LessEqual[t$95$0, Infinity], N[(d / N[(l * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999998e-96 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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.6%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lower-*.f64 26.4%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

   11. Applied rewrites 26.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   if -1.9999999999999998e-96 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.6%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

   9. Taylor expanded in l around -inf

      \[\leadsto \frac{d}{\color{blue}{\ell \cdot \sqrt{\frac{h}{\ell}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{d}{\ell \cdot \color{blue}{\sqrt{\frac{h}{\ell}}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{d}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

       4. lower-/.f64 39.6%

          \[\leadsto \frac{d}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

   11. Applied rewrites 39.6%

       \[\leadsto \frac{d}{\color{blue}{\ell \cdot \sqrt{\frac{h}{\ell}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 43.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{d}{h \cdot \sqrt{\frac{\ell}{h}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ d (sqrt (* h l)))))
   (if (<= t_0 -2e-107)
     t_1
     (if (<= t_0 INFINITY) (/ d (* h (sqrt (/ l h)))) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = d / sqrt((h * l));
	double tmp;
	if (t_0 <= -2e-107) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = d / (h * sqrt((l / h)));
	} else {
		tmp = t_1;
	}
	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 = d / Math.sqrt((h * l));
	double tmp;
	if (t_0 <= -2e-107) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = d / (h * Math.sqrt((l / h)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = d / math.sqrt((h * l))
	tmp = 0
	if t_0 <= -2e-107:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = d / (h * math.sqrt((l / h)))
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (t_0 <= -2e-107)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(d / Float64(h * sqrt(Float64(l / h))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = d / sqrt((h * l));
	tmp = 0.0;
	if (t_0 <= -2e-107)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = d / (h * sqrt((l / h)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-107], t$95$1, If[LessEqual[t$95$0, Infinity], N[(d / N[(h * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-107 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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.6%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lower-*.f64 26.4%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

   11. Applied rewrites 26.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   if -2e-107 < (*.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 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.6%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \frac{d}{\color{blue}{h \cdot \sqrt{\frac{\ell}{h}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{d}{h \cdot \color{blue}{\sqrt{\frac{\ell}{h}}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

       4. lower-/.f64 39.3%

          \[\leadsto \frac{d}{h \cdot \sqrt{\frac{\ell}{h}}} \]

   11. Applied rewrites 39.3%

       \[\leadsto \frac{d}{\color{blue}{h \cdot \sqrt{\frac{\ell}{h}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 42.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -5.05 \cdot 10^{-231}:\\ \;\;\;\;\frac{-d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* h l)))) (if (<= l -5.05e-231) (/ (- d) t_0) (/ d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (l <= -5.05e-231) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h * l))
    if (l <= (-5.05d-231)) then
        tmp = -d / t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h * l));
	double tmp;
	if (l <= -5.05e-231) {
		tmp = -d / t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h * l))
	tmp = 0
	if l <= -5.05e-231:
		tmp = -d / t_0
	else:
		tmp = d / t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (l <= -5.05e-231)
		tmp = Float64(Float64(-d) / t_0);
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h * l));
	tmp = 0.0;
	if (l <= -5.05e-231)
		tmp = -d / t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.05e-231], N[((-d) / t$95$0), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.0499999999999998e-231

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.6%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

   if -5.0499999999999998e-231 < l

   1. Initial program 66.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-*.f64 48.6%

          \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

      4. lower-*.f64 26.3%

         \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

       3. lower-*.f64 26.4%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

   11. Applied rewrites 26.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 26.4% accurate, 10.2× speedup

Math

\[\frac{d}{\sqrt{h \cdot \ell}} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return 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 = d / sqrt((h * l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. pow-prod-down N/A

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. metadata-eval N/A

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. unpow1/2 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   10. frac-times N/A

       \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   11. sqrt-div N/A

       \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   12. lower-unsound-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   13. lower-unsound-sqrt.f64 N/A

       \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   15. lower-unsound-sqrt.f64 N/A

       \[\leadsto \frac{\sqrt{d \cdot d}}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   16. lower-*.f64 48.6%

       \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Applied rewrites 48.6%

   \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

4. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

   4. lower-*.f64 26.3%

      \[\leadsto -1 \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

6. Applied rewrites 26.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]

8. Applied rewrites 26.3%

   \[\leadsto \color{blue}{\frac{-d}{\sqrt{h \cdot \ell}}} \]

9. Taylor expanded in d around inf

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

    2. lower-sqrt.f64 N/A

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

    3. lower-*.f64 26.4%

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

11. Applied rewrites 26.4%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(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)))))