Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.0% → 82.0%

Time: 19.6s

Alternatives: 15

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot \frac{M}{2}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt{-d} \cdot \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\right) \cdot \left(1 - \frac{\left({\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) (/ M 2.0))))
   (if (<= h -5e-310)
     (*
      (* (sqrt (- d)) (/ (sqrt (/ d l)) (sqrt (- h))))
      (- 1.0 (/ (* (* (pow (* (/ D 2.0) (/ M d)) 2.0) (/ 1.0 2.0)) h) l)))
     (*
      (/ (* (sqrt d) (sqrt d)) (* (sqrt l) (sqrt h)))
      (- 1.0 (/ (* t_0 (* t_0 (* (/ 1.0 2.0) h))) l))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M / 2.0);
	double tmp;
	if (h <= -5e-310) {
		tmp = (sqrt(-d) * (sqrt((d / l)) / sqrt(-h))) * (1.0 - (((pow(((D / 2.0) * (M / d)), 2.0) * (1.0 / 2.0)) * h) / l));
	} else {
		tmp = ((sqrt(d) * sqrt(d)) / (sqrt(l) * sqrt(h))) * (1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d_1 / d) * (m / 2.0d0)
    if (h <= (-5d-310)) then
        tmp = (sqrt(-d) * (sqrt((d / l)) / sqrt(-h))) * (1.0d0 - ((((((d_1 / 2.0d0) * (m / d)) ** 2.0d0) * (1.0d0 / 2.0d0)) * h) / l))
    else
        tmp = ((sqrt(d) * sqrt(d)) / (sqrt(l) * sqrt(h))) * (1.0d0 - ((t_0 * (t_0 * ((1.0d0 / 2.0d0) * h))) / l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M / 2.0);
	double tmp;
	if (h <= -5e-310) {
		tmp = (Math.sqrt(-d) * (Math.sqrt((d / l)) / Math.sqrt(-h))) * (1.0 - (((Math.pow(((D / 2.0) * (M / d)), 2.0) * (1.0 / 2.0)) * h) / l));
	} else {
		tmp = ((Math.sqrt(d) * Math.sqrt(d)) / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D / d) * (M / 2.0)
	tmp = 0
	if h <= -5e-310:
		tmp = (math.sqrt(-d) * (math.sqrt((d / l)) / math.sqrt(-h))) * (1.0 - (((math.pow(((D / 2.0) * (M / d)), 2.0) * (1.0 / 2.0)) * h) / l))
	else:
		tmp = ((math.sqrt(d) * math.sqrt(d)) / (math.sqrt(l) * math.sqrt(h))) * (1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * Float64(M / 2.0))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) * Float64(sqrt(Float64(d / l)) / sqrt(Float64(-h)))) * Float64(1.0 - Float64(Float64(Float64((Float64(Float64(D / 2.0) * Float64(M / d)) ^ 2.0) * Float64(1.0 / 2.0)) * h) / l)));
	else
		tmp = Float64(Float64(Float64(sqrt(d) * sqrt(d)) / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(Float64(1.0 / 2.0) * h))) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D / d) * (M / 2.0);
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = (sqrt(-d) * (sqrt((d / l)) / sqrt(-h))) * (1.0 - ((((((D / 2.0) * (M / d)) ^ 2.0) * (1.0 / 2.0)) * h) / l));
	else
		tmp = ((sqrt(d) * sqrt(d)) / (sqrt(l) * sqrt(h))) * (1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[Power[N[(N[(D / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(N[(1.0 / 2.0), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 74.0%

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

   4. Applied rewrites 74.1%

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

   6. Applied rewrites 85.4%

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

   if -4.999999999999985e-310 < h

   1. Initial program 67.2%

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

   4. Applied rewrites 71.1%

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

   6. Applied rewrites 74.1%

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

   8. Applied rewrites 74.1%

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

      1. lift-var N/A

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

      2. lift-var N/A

         \[\leadsto \left(\sqrt{\frac{\mathsf{Rewrite<=}\left(lift-var-spec, d\right)}{\color{blue}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-var N/A

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

      6. lift-var N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{\mathsf{Rewrite<=}\left(lift-var-spec, d\right)}{\color{blue}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

   10. Applied rewrites 92.8%

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

Alternative 2: 61.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{D}{d} \cdot \frac{M}{2}\\ \mathbf{if}\;t\_0 \leq 3 \cdot 10^{+160}:\\ \;\;\;\;\left(1 - \frac{\left(\frac{M}{d} \cdot D\right) \cdot \frac{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{D}{2} \cdot M\right)}{d}}{2}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \cdot \left(1 - \frac{t\_1 \cdot \left(t\_1 \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (/ D d) (/ M 2.0))))
   (if (<= t_0 3e+160)
     (*
      (-
       1.0
       (/
        (* (* (/ M d) D) (/ (* (* (/ h l) (/ 1.0 2.0)) (* (/ D 2.0) M)) d))
        2.0))
      (sqrt (* (/ d l) (/ d h))))
     (if (<= t_0 5e+255)
       (/ (* (sqrt d) (sqrt (/ d h))) (sqrt l))
       (*
        (sqrt (* (/ (/ d l) h) d))
        (- 1.0 (/ (* t_1 (* t_1 (* (/ 1.0 2.0) 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 / d) * (M / 2.0);
	double tmp;
	if (t_0 <= 3e+160) {
		tmp = (1.0 - ((((M / d) * D) * ((((h / l) * (1.0 / 2.0)) * ((D / 2.0) * M)) / d)) / 2.0)) * sqrt(((d / l) * (d / h)));
	} else if (t_0 <= 5e+255) {
		tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l);
	} else {
		tmp = sqrt((((d / l) / h) * d)) * (1.0 - ((t_1 * (t_1 * ((1.0 / 2.0) * h))) / l));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: 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 = (d_1 / d) * (m / 2.0d0)
    if (t_0 <= 3d+160) then
        tmp = (1.0d0 - ((((m / d) * d_1) * ((((h / l) * (1.0d0 / 2.0d0)) * ((d_1 / 2.0d0) * m)) / d)) / 2.0d0)) * sqrt(((d / l) * (d / h)))
    else if (t_0 <= 5d+255) then
        tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l)
    else
        tmp = sqrt((((d / l) / h) * d)) * (1.0d0 - ((t_1 * (t_1 * ((1.0d0 / 2.0d0) * h))) / l))
    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 = (D / d) * (M / 2.0);
	double tmp;
	if (t_0 <= 3e+160) {
		tmp = (1.0 - ((((M / d) * D) * ((((h / l) * (1.0 / 2.0)) * ((D / 2.0) * M)) / d)) / 2.0)) * Math.sqrt(((d / l) * (d / h)));
	} else if (t_0 <= 5e+255) {
		tmp = (Math.sqrt(d) * Math.sqrt((d / h))) / Math.sqrt(l);
	} else {
		tmp = Math.sqrt((((d / l) / h) * d)) * (1.0 - ((t_1 * (t_1 * ((1.0 / 2.0) * h))) / l));
	}
	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 / d) * (M / 2.0)
	tmp = 0
	if t_0 <= 3e+160:
		tmp = (1.0 - ((((M / d) * D) * ((((h / l) * (1.0 / 2.0)) * ((D / 2.0) * M)) / d)) / 2.0)) * math.sqrt(((d / l) * (d / h)))
	elif t_0 <= 5e+255:
		tmp = (math.sqrt(d) * math.sqrt((d / h))) / math.sqrt(l)
	else:
		tmp = math.sqrt((((d / l) / h) * d)) * (1.0 - ((t_1 * (t_1 * ((1.0 / 2.0) * 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 / d) * Float64(M / 2.0))
	tmp = 0.0
	if (t_0 <= 3e+160)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(M / d) * D) * Float64(Float64(Float64(Float64(h / l) * Float64(1.0 / 2.0)) * Float64(Float64(D / 2.0) * M)) / d)) / 2.0)) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (t_0 <= 5e+255)
		tmp = Float64(Float64(sqrt(d) * sqrt(Float64(d / h))) / sqrt(l));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(d / l) / h) * d)) * Float64(1.0 - Float64(Float64(t_1 * Float64(t_1 * Float64(Float64(1.0 / 2.0) * h))) / l)));
	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 / d) * (M / 2.0);
	tmp = 0.0;
	if (t_0 <= 3e+160)
		tmp = (1.0 - ((((M / d) * D) * ((((h / l) * (1.0 / 2.0)) * ((D / 2.0) * M)) / d)) / 2.0)) * sqrt(((d / l) * (d / h)));
	elseif (t_0 <= 5e+255)
		tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l);
	else
		tmp = sqrt((((d / l) / h) * d)) * (1.0 - ((t_1 * (t_1 * ((1.0 / 2.0) * h))) / l));
	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[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 3e+160], N[(N[(1.0 - N[(N[(N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] * N[(N[(N[(N[(h / l), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(D / 2.0), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+255], N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(t$95$1 * N[(t$95$1 * N[(N[(1.0 / 2.0), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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)))) < 2.9999999999999999e160

   1. Initial program 87.9%

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

   4. Applied rewrites 87.9%

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

   6. Applied rewrites 78.1%

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

   8. Applied rewrites 78.7%

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

   10. Applied rewrites 78.7%

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

   if 2.9999999999999999e160 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e255

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 45.7%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 45.7%

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

   if 5.0000000000000002e255 < (*.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 28.2%

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

   4. Applied rewrites 35.1%

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

   6. Applied rewrites 37.7%

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

   8. Applied rewrites 42.7%

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

Alternative 3: 59.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq 3 \cdot 10^{+160}:\\ \;\;\;\;\left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{\frac{D}{2}}{d} \cdot M\right) \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right)}{2 \cdot d}\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\left(\left(\frac{M}{2} \cdot h\right) \cdot \frac{1}{2}\right) \cdot \frac{D}{d}\right)}{2 \cdot \ell}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (* (/ d l) (/ d h)))))
   (if (<= t_0 3e+160)
     (*
      (-
       1.0
       (/
        (* (* M D) (* (* (/ (/ D 2.0) d) M) (* (/ h l) (/ 1.0 2.0))))
        (* 2.0 d)))
      t_1)
     (if (<= t_0 5e+255)
       (/ (* (sqrt d) (sqrt (/ d h))) (sqrt l))
       (*
        (-
         1.0
         (/
          (* (* (/ M d) D) (* (* (* (/ M 2.0) h) (/ 1.0 2.0)) (/ D d)))
          (* 2.0 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 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (t_0 <= 3e+160) {
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * t_1;
	} else if (t_0 <= 5e+255) {
		tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l);
	} else {
		tmp = (1.0 - ((((M / d) * D) * ((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d))) / (2.0 * l))) * 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 = sqrt(((d / l) * (d / h)))
    if (t_0 <= 3d+160) then
        tmp = (1.0d0 - (((m * d_1) * ((((d_1 / 2.0d0) / d) * m) * ((h / l) * (1.0d0 / 2.0d0)))) / (2.0d0 * d))) * t_1
    else if (t_0 <= 5d+255) then
        tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l)
    else
        tmp = (1.0d0 - ((((m / d) * d_1) * ((((m / 2.0d0) * h) * (1.0d0 / 2.0d0)) * (d_1 / d))) / (2.0d0 * l))) * 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.sqrt(((d / l) * (d / h)));
	double tmp;
	if (t_0 <= 3e+160) {
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * t_1;
	} else if (t_0 <= 5e+255) {
		tmp = (Math.sqrt(d) * Math.sqrt((d / h))) / Math.sqrt(l);
	} else {
		tmp = (1.0 - ((((M / d) * D) * ((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d))) / (2.0 * l))) * 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.sqrt(((d / l) * (d / h)))
	tmp = 0
	if t_0 <= 3e+160:
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * t_1
	elif t_0 <= 5e+255:
		tmp = (math.sqrt(d) * math.sqrt((d / h))) / math.sqrt(l)
	else:
		tmp = (1.0 - ((((M / d) * D) * ((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d))) / (2.0 * l))) * 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 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (t_0 <= 3e+160)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(M * D) * Float64(Float64(Float64(Float64(D / 2.0) / d) * M) * Float64(Float64(h / l) * Float64(1.0 / 2.0)))) / Float64(2.0 * d))) * t_1);
	elseif (t_0 <= 5e+255)
		tmp = Float64(Float64(sqrt(d) * sqrt(Float64(d / h))) / sqrt(l));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(M / d) * D) * Float64(Float64(Float64(Float64(M / 2.0) * h) * Float64(1.0 / 2.0)) * Float64(D / d))) / Float64(2.0 * l))) * 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 = sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (t_0 <= 3e+160)
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * t_1;
	elseif (t_0 <= 5e+255)
		tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l);
	else
		tmp = (1.0 - ((((M / d) * D) * ((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d))) / (2.0 * l))) * 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[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 3e+160], N[(N[(1.0 - N[(N[(N[(M * D), $MachinePrecision] * N[(N[(N[(N[(D / 2.0), $MachinePrecision] / d), $MachinePrecision] * M), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e+255], N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] * N[(N[(N[(N[(M / 2.0), $MachinePrecision] * h), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $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)))) < 2.9999999999999999e160

   1. Initial program 87.9%

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

   4. Applied rewrites 87.9%

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

   6. Applied rewrites 78.1%

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

   8. Applied rewrites 78.1%

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

   if 2.9999999999999999e160 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e255

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 45.7%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 45.7%

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

   if 5.0000000000000002e255 < (*.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 28.2%

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

   4. Applied rewrites 35.1%

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

   6. Applied rewrites 28.3%

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

   8. Applied rewrites 37.6%

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

Alternative 4: 58.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq 3 \cdot 10^{+160}:\\ \;\;\;\;\left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{\frac{D}{2}}{d} \cdot M\right) \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right)}{2 \cdot d}\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\left(\left(\frac{M}{2} \cdot h\right) \cdot \frac{1}{2}\right) \cdot \frac{D}{d}\right) \cdot \frac{M \cdot D}{\left(2 \cdot d\right) \cdot \ell}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (* (/ d l) (/ d h)))))
   (if (<= t_0 3e+160)
     (*
      (-
       1.0
       (/
        (* (* M D) (* (* (/ (/ D 2.0) d) M) (* (/ h l) (/ 1.0 2.0))))
        (* 2.0 d)))
      t_1)
     (if (<= t_0 5e+255)
       (/ (* (sqrt d) (sqrt (/ d h))) (sqrt l))
       (*
        (-
         1.0
         (*
          (* (* (* (/ M 2.0) h) (/ 1.0 2.0)) (/ D d))
          (/ (* M D) (* (* 2.0 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 = sqrt(((d / l) * (d / h)));
	double tmp;
	if (t_0 <= 3e+160) {
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * t_1;
	} else if (t_0 <= 5e+255) {
		tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l);
	} else {
		tmp = (1.0 - (((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d)) * ((M * D) / ((2.0 * d) * l)))) * 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 = sqrt(((d / l) * (d / h)))
    if (t_0 <= 3d+160) then
        tmp = (1.0d0 - (((m * d_1) * ((((d_1 / 2.0d0) / d) * m) * ((h / l) * (1.0d0 / 2.0d0)))) / (2.0d0 * d))) * t_1
    else if (t_0 <= 5d+255) then
        tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l)
    else
        tmp = (1.0d0 - (((((m / 2.0d0) * h) * (1.0d0 / 2.0d0)) * (d_1 / d)) * ((m * d_1) / ((2.0d0 * d) * l)))) * 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.sqrt(((d / l) * (d / h)));
	double tmp;
	if (t_0 <= 3e+160) {
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * t_1;
	} else if (t_0 <= 5e+255) {
		tmp = (Math.sqrt(d) * Math.sqrt((d / h))) / Math.sqrt(l);
	} else {
		tmp = (1.0 - (((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d)) * ((M * D) / ((2.0 * d) * l)))) * 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.sqrt(((d / l) * (d / h)))
	tmp = 0
	if t_0 <= 3e+160:
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * t_1
	elif t_0 <= 5e+255:
		tmp = (math.sqrt(d) * math.sqrt((d / h))) / math.sqrt(l)
	else:
		tmp = (1.0 - (((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d)) * ((M * D) / ((2.0 * d) * l)))) * 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 = sqrt(Float64(Float64(d / l) * Float64(d / h)))
	tmp = 0.0
	if (t_0 <= 3e+160)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(M * D) * Float64(Float64(Float64(Float64(D / 2.0) / d) * M) * Float64(Float64(h / l) * Float64(1.0 / 2.0)))) / Float64(2.0 * d))) * t_1);
	elseif (t_0 <= 5e+255)
		tmp = Float64(Float64(sqrt(d) * sqrt(Float64(d / h))) / sqrt(l));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M / 2.0) * h) * Float64(1.0 / 2.0)) * Float64(D / d)) * Float64(Float64(M * D) / Float64(Float64(2.0 * d) * l)))) * 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 = sqrt(((d / l) * (d / h)));
	tmp = 0.0;
	if (t_0 <= 3e+160)
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * t_1;
	elseif (t_0 <= 5e+255)
		tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l);
	else
		tmp = (1.0 - (((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d)) * ((M * D) / ((2.0 * d) * l)))) * 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[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 3e+160], N[(N[(1.0 - N[(N[(N[(M * D), $MachinePrecision] * N[(N[(N[(N[(D / 2.0), $MachinePrecision] / d), $MachinePrecision] * M), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 5e+255], N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(N[(M / 2.0), $MachinePrecision] * h), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(N[(2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $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)))) < 2.9999999999999999e160

   1. Initial program 87.9%

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

   4. Applied rewrites 87.9%

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

   6. Applied rewrites 78.1%

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

   8. Applied rewrites 78.1%

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

   if 2.9999999999999999e160 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e255

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 45.7%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 45.7%

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

   if 5.0000000000000002e255 < (*.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 28.2%

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

   4. Applied rewrites 35.1%

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

   6. Applied rewrites 28.3%

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

   8. Applied rewrites 36.4%

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

   10. Applied rewrites 37.6%

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

Alternative 5: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot \frac{M}{2}\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) (/ M 2.0)))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* (/ 1.0 2.0) h))) l))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
        5e+255)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) t_1)
     (* (/ (sqrt (* d d)) (sqrt (* l h))) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M / 2.0);
	double t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+255) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_1;
	} else {
		tmp = (sqrt((d * d)) / sqrt((l * h))) * 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_1 / d) * (m / 2.0d0)
    t_1 = 1.0d0 - ((t_0 * (t_0 * ((1.0d0 / 2.0d0) * h))) / l)
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 5d+255) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * t_1
    else
        tmp = (sqrt((d * d)) / sqrt((l * h))) * 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 = (D / d) * (M / 2.0);
	double t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+255) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * t_1;
	} else {
		tmp = (Math.sqrt((d * d)) / Math.sqrt((l * h))) * t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D / d) * (M / 2.0)
	t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l)
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+255:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * t_1
	else:
		tmp = (math.sqrt((d * d)) / math.sqrt((l * h))) * t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * Float64(M / 2.0))
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(Float64(1.0 / 2.0) * h))) / l))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+255)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(d * d)) / sqrt(Float64(l * h))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D / d) * (M / 2.0);
	t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 5e+255)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * t_1;
	else
		tmp = (sqrt((d * d)) / sqrt((l * h))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(N[(1.0 / 2.0), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+255], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e255

   1. Initial program 88.7%

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

   4. Applied rewrites 88.7%

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

   6. Applied rewrites 88.7%

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

   8. Applied rewrites 88.7%

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

   if 5.0000000000000002e255 < (*.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 28.2%

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

   4. Applied rewrites 35.1%

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

   6. Applied rewrites 37.7%

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

   8. Applied rewrites 46.3%

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

Alternative 6: 72.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot \frac{M}{2}\\ t_1 := \frac{1}{2} \cdot h\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{t\_0 \cdot \left(\frac{M \cdot D}{d + d} \cdot t\_1\right)}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{t\_0 \cdot \left(t\_0 \cdot t\_1\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) (/ M 2.0))) (t_1 (* (/ 1.0 2.0) h)))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
        5e+255)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (/ (* t_0 (* (/ (* M D) (+ d d)) t_1)) l)))
     (* (/ (sqrt (* d d)) (sqrt (* l h))) (- 1.0 (/ (* t_0 (* t_0 t_1)) l))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M / 2.0);
	double t_1 = (1.0 / 2.0) * h;
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+255) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_0 * (((M * D) / (d + d)) * t_1)) / l));
	} else {
		tmp = (sqrt((d * d)) / sqrt((l * h))) * (1.0 - ((t_0 * (t_0 * t_1)) / l));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_1 / d) * (m / 2.0d0)
    t_1 = (1.0d0 / 2.0d0) * h
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 5d+255) then
        tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((t_0 * (((m * d_1) / (d + d)) * t_1)) / l))
    else
        tmp = (sqrt((d * d)) / sqrt((l * h))) * (1.0d0 - ((t_0 * (t_0 * t_1)) / l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M / 2.0);
	double t_1 = (1.0 / 2.0) * h;
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+255) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * (1.0 - ((t_0 * (((M * D) / (d + d)) * t_1)) / l));
	} else {
		tmp = (Math.sqrt((d * d)) / Math.sqrt((l * h))) * (1.0 - ((t_0 * (t_0 * t_1)) / l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D / d) * (M / 2.0)
	t_1 = (1.0 / 2.0) * h
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+255:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * (1.0 - ((t_0 * (((M * D) / (d + d)) * t_1)) / l))
	else:
		tmp = (math.sqrt((d * d)) / math.sqrt((l * h))) * (1.0 - ((t_0 * (t_0 * t_1)) / l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * Float64(M / 2.0))
	t_1 = Float64(Float64(1.0 / 2.0) * h)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+255)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(t_0 * Float64(Float64(Float64(M * D) / Float64(d + d)) * t_1)) / l)));
	else
		tmp = Float64(Float64(sqrt(Float64(d * d)) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * t_1)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D / d) * (M / 2.0);
	t_1 = (1.0 / 2.0) * h;
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 5e+255)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((t_0 * (((M * D) / (d + d)) * t_1)) / l));
	else
		tmp = (sqrt((d * d)) / sqrt((l * h))) * (1.0 - ((t_0 * (t_0 * t_1)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / 2.0), $MachinePrecision] * h), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+255], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(N[(N[(M * D), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000002e255

   1. Initial program 88.7%

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

   4. Applied rewrites 88.7%

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

   6. Applied rewrites 88.7%

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

   8. Applied rewrites 88.7%

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

   9. Taylor expanded in d around 0

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

   11. Applied rewrites 88.7%

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

   if 5.0000000000000002e255 < (*.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 28.2%

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

   4. Applied rewrites 35.1%

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

   6. Applied rewrites 37.7%

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

   8. Applied rewrites 46.3%

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

Alternative 7: 81.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot \frac{M}{2}\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\\ \mathbf{if}\;d \leq -6.5 \cdot 10^{-279}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) (/ M 2.0)))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* (/ 1.0 2.0) h))) l))))
   (if (<= d -6.5e-279)
     (* (* (sqrt (/ d l)) (/ (sqrt (- d)) (sqrt (- h)))) t_1)
     (* (/ (* (sqrt d) (sqrt d)) (* (sqrt l) (sqrt h))) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M / 2.0);
	double t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	double tmp;
	if (d <= -6.5e-279) {
		tmp = (sqrt((d / l)) * (sqrt(-d) / sqrt(-h))) * t_1;
	} else {
		tmp = ((sqrt(d) * sqrt(d)) / (sqrt(l) * sqrt(h))) * 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_1 / d) * (m / 2.0d0)
    t_1 = 1.0d0 - ((t_0 * (t_0 * ((1.0d0 / 2.0d0) * h))) / l)
    if (d <= (-6.5d-279)) then
        tmp = (sqrt((d / l)) * (sqrt(-d) / sqrt(-h))) * t_1
    else
        tmp = ((sqrt(d) * sqrt(d)) / (sqrt(l) * sqrt(h))) * 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 = (D / d) * (M / 2.0);
	double t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	double tmp;
	if (d <= -6.5e-279) {
		tmp = (Math.sqrt((d / l)) * (Math.sqrt(-d) / Math.sqrt(-h))) * t_1;
	} else {
		tmp = ((Math.sqrt(d) * Math.sqrt(d)) / (Math.sqrt(l) * Math.sqrt(h))) * t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D / d) * (M / 2.0)
	t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l)
	tmp = 0
	if d <= -6.5e-279:
		tmp = (math.sqrt((d / l)) * (math.sqrt(-d) / math.sqrt(-h))) * t_1
	else:
		tmp = ((math.sqrt(d) * math.sqrt(d)) / (math.sqrt(l) * math.sqrt(h))) * t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * Float64(M / 2.0))
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(Float64(1.0 / 2.0) * h))) / l))
	tmp = 0.0
	if (d <= -6.5e-279)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * t_1);
	else
		tmp = Float64(Float64(Float64(sqrt(d) * sqrt(d)) / Float64(sqrt(l) * sqrt(h))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D / d) * (M / 2.0);
	t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	tmp = 0.0;
	if (d <= -6.5e-279)
		tmp = (sqrt((d / l)) * (sqrt(-d) / sqrt(-h))) * t_1;
	else
		tmp = ((sqrt(d) * sqrt(d)) / (sqrt(l) * sqrt(h))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(N[(1.0 / 2.0), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.5e-279], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.4999999999999997e-279

   1. Initial program 75.2%

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

   4. Applied rewrites 75.3%

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

   6. Applied rewrites 73.8%

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

   8. Applied rewrites 73.8%

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

      1. lift-var N/A

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

      2. lift-var N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{\mathsf{Rewrite<=}\left(lift-var-spec, d\right)}{\color{blue}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-var N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-var 86.1

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

   10. Applied rewrites 86.1%

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

   if -6.4999999999999997e-279 < d

   1. Initial program 66.2%

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

   4. Applied rewrites 70.1%

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

   6. Applied rewrites 73.0%

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

   8. Applied rewrites 73.0%

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

      1. lift-var N/A

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

      2. lift-var N/A

         \[\leadsto \left(\sqrt{\frac{\mathsf{Rewrite<=}\left(lift-var-spec, d\right)}{\color{blue}{\ell}}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-var N/A

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

      6. lift-var N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{\mathsf{Rewrite<=}\left(lift-var-spec, d\right)}{\color{blue}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

   10. Applied rewrites 91.4%

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

Alternative 8: 79.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot \frac{M}{2}\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) (/ M 2.0)))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* (/ 1.0 2.0) h))) l)))
        (t_2 (sqrt (/ d l))))
   (if (<= l -4e-310)
     (* (* t_2 (/ (sqrt (- d)) (sqrt (- h)))) t_1)
     (* (* t_2 (/ (sqrt d) (sqrt h))) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M / 2.0);
	double t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	double t_2 = sqrt((d / l));
	double tmp;
	if (l <= -4e-310) {
		tmp = (t_2 * (sqrt(-d) / sqrt(-h))) * t_1;
	} else {
		tmp = (t_2 * (sqrt(d) / sqrt(h))) * 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) :: t_2
    real(8) :: tmp
    t_0 = (d_1 / d) * (m / 2.0d0)
    t_1 = 1.0d0 - ((t_0 * (t_0 * ((1.0d0 / 2.0d0) * h))) / l)
    t_2 = sqrt((d / l))
    if (l <= (-4d-310)) then
        tmp = (t_2 * (sqrt(-d) / sqrt(-h))) * t_1
    else
        tmp = (t_2 * (sqrt(d) / sqrt(h))) * 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 = (D / d) * (M / 2.0);
	double t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (l <= -4e-310) {
		tmp = (t_2 * (Math.sqrt(-d) / Math.sqrt(-h))) * t_1;
	} else {
		tmp = (t_2 * (Math.sqrt(d) / Math.sqrt(h))) * t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D / d) * (M / 2.0)
	t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l)
	t_2 = math.sqrt((d / l))
	tmp = 0
	if l <= -4e-310:
		tmp = (t_2 * (math.sqrt(-d) / math.sqrt(-h))) * t_1
	else:
		tmp = (t_2 * (math.sqrt(d) / math.sqrt(h))) * t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * Float64(M / 2.0))
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(Float64(1.0 / 2.0) * h))) / l))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (l <= -4e-310)
		tmp = Float64(Float64(t_2 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))) * t_1);
	else
		tmp = Float64(Float64(t_2 * Float64(sqrt(d) / sqrt(h))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D / d) * (M / 2.0);
	t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (l <= -4e-310)
		tmp = (t_2 * (sqrt(-d) / sqrt(-h))) * t_1;
	else
		tmp = (t_2 * (sqrt(d) / sqrt(h))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(N[(1.0 / 2.0), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4e-310], N[(N[(t$95$2 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.999999999999988e-310

   1. Initial program 74.0%

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

   4. Applied rewrites 74.1%

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

   6. Applied rewrites 72.6%

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

   8. Applied rewrites 72.6%

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

      1. lift-var N/A

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

      2. lift-var N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{\mathsf{Rewrite<=}\left(lift-var-spec, d\right)}{\color{blue}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-var N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-var 84.7

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

   10. Applied rewrites 84.7%

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

   if -3.999999999999988e-310 < l

   1. Initial program 67.2%

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

   4. Applied rewrites 71.1%

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

   6. Applied rewrites 74.1%

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

   8. Applied rewrites 74.1%

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

      1. lift-var N/A

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

      2. lift-var N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{\mathsf{Rewrite<=}\left(lift-var-spec, d\right)}{\color{blue}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-var N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-var 85.0

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

   10. Applied rewrites 85.0%

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

Alternative 9: 74.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot \frac{M}{2}\\ t_1 := 1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_2 \cdot \sqrt{\frac{d}{h}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) (/ M 2.0)))
        (t_1 (- 1.0 (/ (* t_0 (* t_0 (* (/ 1.0 2.0) h))) l)))
        (t_2 (sqrt (/ d l))))
   (if (<= h -5e-310)
     (* (* t_2 (sqrt (/ d h))) t_1)
     (* (* t_2 (/ (sqrt d) (sqrt h))) t_1))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M / 2.0);
	double t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	double t_2 = sqrt((d / l));
	double tmp;
	if (h <= -5e-310) {
		tmp = (t_2 * sqrt((d / h))) * t_1;
	} else {
		tmp = (t_2 * (sqrt(d) / sqrt(h))) * 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) :: t_2
    real(8) :: tmp
    t_0 = (d_1 / d) * (m / 2.0d0)
    t_1 = 1.0d0 - ((t_0 * (t_0 * ((1.0d0 / 2.0d0) * h))) / l)
    t_2 = sqrt((d / l))
    if (h <= (-5d-310)) then
        tmp = (t_2 * sqrt((d / h))) * t_1
    else
        tmp = (t_2 * (sqrt(d) / sqrt(h))) * 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 = (D / d) * (M / 2.0);
	double t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (h <= -5e-310) {
		tmp = (t_2 * Math.sqrt((d / h))) * t_1;
	} else {
		tmp = (t_2 * (Math.sqrt(d) / Math.sqrt(h))) * t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (D / d) * (M / 2.0)
	t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l)
	t_2 = math.sqrt((d / l))
	tmp = 0
	if h <= -5e-310:
		tmp = (t_2 * math.sqrt((d / h))) * t_1
	else:
		tmp = (t_2 * (math.sqrt(d) / math.sqrt(h))) * t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * Float64(M / 2.0))
	t_1 = Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(Float64(1.0 / 2.0) * h))) / l))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -5e-310)
		tmp = Float64(Float64(t_2 * sqrt(Float64(d / h))) * t_1);
	else
		tmp = Float64(Float64(t_2 * Float64(sqrt(d) / sqrt(h))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (D / d) * (M / 2.0);
	t_1 = 1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l);
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (h <= -5e-310)
		tmp = (t_2 * sqrt((d / h))) * t_1;
	else
		tmp = (t_2 * (sqrt(d) / sqrt(h))) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 * N[(t$95$0 * N[(N[(1.0 / 2.0), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -5e-310], N[(N[(t$95$2 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 74.0%

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

   4. Applied rewrites 74.1%

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

   6. Applied rewrites 72.6%

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

   8. Applied rewrites 72.6%

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

   if -4.999999999999985e-310 < h

   1. Initial program 67.2%

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

   4. Applied rewrites 71.1%

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

   6. Applied rewrites 74.1%

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

   8. Applied rewrites 74.1%

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

      1. lift-var N/A

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

      2. lift-var N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{\mathsf{Rewrite<=}\left(lift-var-spec, d\right)}{\color{blue}{h}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \]

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-var N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-var 85.0

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

   10. Applied rewrites 85.0%

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

Alternative 10: 56.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq 1.9 \cdot 10^{-272}:\\ \;\;\;\;\left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\frac{\frac{D}{2}}{d} \cdot M\right) \cdot \left(\frac{h}{\ell} \cdot \frac{1}{2}\right)\right)}{2 \cdot d}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{\left(M \cdot D\right) \cdot \left(\left(\left(\frac{M}{2} \cdot h\right) \cdot \frac{1}{2}\right) \cdot \frac{D}{d}\right)}{\left(2 \cdot d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h 1.9e-272)
   (*
    (-
     1.0
     (/
      (* (* M D) (* (* (/ (/ D 2.0) d) M) (* (/ h l) (/ 1.0 2.0))))
      (* 2.0 d)))
    (sqrt (* (/ d l) (/ d h))))
   (/
    (*
     (-
      1.0
      (/
       (* (* M D) (* (* (* (/ M 2.0) h) (/ 1.0 2.0)) (/ D d)))
       (* (* 2.0 d) l)))
     (sqrt (* (/ d l) d)))
    (sqrt h))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 1.9e-272) {
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = ((1.0 - (((M * D) * ((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d))) / ((2.0 * d) * l))) * sqrt(((d / l) * d))) / sqrt(h);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= 1.9d-272) then
        tmp = (1.0d0 - (((m * d_1) * ((((d_1 / 2.0d0) / d) * m) * ((h / l) * (1.0d0 / 2.0d0)))) / (2.0d0 * d))) * sqrt(((d / l) * (d / h)))
    else
        tmp = ((1.0d0 - (((m * d_1) * ((((m / 2.0d0) * h) * (1.0d0 / 2.0d0)) * (d_1 / d))) / ((2.0d0 * d) * l))) * sqrt(((d / l) * d))) / sqrt(h)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= 1.9e-272) {
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = ((1.0 - (((M * D) * ((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d))) / ((2.0 * d) * l))) * Math.sqrt(((d / l) * d))) / Math.sqrt(h);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= 1.9e-272:
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * math.sqrt(((d / l) * (d / h)))
	else:
		tmp = ((1.0 - (((M * D) * ((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d))) / ((2.0 * d) * l))) * math.sqrt(((d / l) * d))) / math.sqrt(h)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= 1.9e-272)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(M * D) * Float64(Float64(Float64(Float64(D / 2.0) / d) * M) * Float64(Float64(h / l) * Float64(1.0 / 2.0)))) / Float64(2.0 * d))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(Float64(M * D) * Float64(Float64(Float64(Float64(M / 2.0) * h) * Float64(1.0 / 2.0)) * Float64(D / d))) / Float64(Float64(2.0 * d) * l))) * sqrt(Float64(Float64(d / l) * d))) / sqrt(h));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= 1.9e-272)
		tmp = (1.0 - (((M * D) * ((((D / 2.0) / d) * M) * ((h / l) * (1.0 / 2.0)))) / (2.0 * d))) * sqrt(((d / l) * (d / h)));
	else
		tmp = ((1.0 - (((M * D) * ((((M / 2.0) * h) * (1.0 / 2.0)) * (D / d))) / ((2.0 * d) * l))) * sqrt(((d / l) * d))) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, 1.9e-272], N[(N[(1.0 - N[(N[(N[(M * D), $MachinePrecision] * N[(N[(N[(N[(D / 2.0), $MachinePrecision] / d), $MachinePrecision] * M), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(N[(M * D), $MachinePrecision] * N[(N[(N[(N[(M / 2.0), $MachinePrecision] * h), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 1.89999999999999985e-272

   1. Initial program 73.9%

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

   4. Applied rewrites 74.0%

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

   6. Applied rewrites 60.7%

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

   8. Applied rewrites 62.2%

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

   if 1.89999999999999985e-272 < h

   1. Initial program 66.9%

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

   4. Applied rewrites 71.1%

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

   6. Applied rewrites 63.5%

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

   8. Applied rewrites 65.2%

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

Alternative 11: 68.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double d, double h, double l, double M, double D) {
	return (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((D / d) * (M / 2.0)) * (((M * D) / (d + d)) * ((1.0 / 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 = (sqrt((d / l)) * sqrt((d / h))) * (1.0d0 - ((((d_1 / d) * (m / 2.0d0)) * (((m * d_1) / (d + d)) * ((1.0d0 / 2.0d0) * h))) / l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

4. Applied rewrites 72.6%

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

6. Applied rewrites 73.4%

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

8. Applied rewrites 73.4%

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

9. Taylor expanded in d around 0

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

11. Applied rewrites 73.3%

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

Alternative 12: 58.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot \frac{M}{2}\\ \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \cdot \left(1 - \frac{t\_0 \cdot \left(t\_0 \cdot \left(\frac{1}{2} \cdot h\right)\right)}{\ell}\right) \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ D d) (/ M 2.0))))
   (*
    (sqrt (* (/ (/ d l) h) d))
    (- 1.0 (/ (* t_0 (* t_0 (* (/ 1.0 2.0) h))) l)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M / 2.0);
	return sqrt((((d / l) / h) * d)) * (1.0 - ((t_0 * (t_0 * ((1.0 / 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
    real(8) :: t_0
    t_0 = (d_1 / d) * (m / 2.0d0)
    code = sqrt((((d / l) / h) * d)) * (1.0d0 - ((t_0 * (t_0 * ((1.0d0 / 2.0d0) * h))) / l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (D / d) * (M / 2.0);
	return Math.sqrt((((d / l) / h) * d)) * (1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l));
}

Python

def code(d, h, l, M, D):
	t_0 = (D / d) * (M / 2.0)
	return math.sqrt((((d / l) / h) * d)) * (1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l))

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * Float64(M / 2.0))
	return Float64(sqrt(Float64(Float64(Float64(d / l) / h) * d)) * Float64(1.0 - Float64(Float64(t_0 * Float64(t_0 * Float64(Float64(1.0 / 2.0) * h))) / l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	t_0 = (D / d) * (M / 2.0);
	tmp = sqrt((((d / l) / h) * d)) * (1.0 - ((t_0 * (t_0 * ((1.0 / 2.0) * h))) / l));
end

Wolfram

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

Derivation

1. Initial program 70.5%

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

4. Applied rewrites 72.6%

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

6. Applied rewrites 73.4%

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

8. Applied rewrites 63.4%

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

Alternative 13: 55.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.00034:\\ \;\;\;\;\left(1 + \frac{\left(M \cdot D\right) \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{2}}{\left(2 \cdot d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 0.00034)
   (*
    (+
     1.0
     (/
      (* (* M D) (/ (* (* (/ M d) D) (* (/ -1.0 2.0) h)) 2.0))
      (* (* 2.0 d) l)))
    (sqrt (* (/ d l) (/ d h))))
   (/ (* (sqrt d) (sqrt (/ d h))) (sqrt l))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 0.00034) {
		tmp = (1.0 + (((M * D) * ((((M / d) * D) * ((-1.0 / 2.0) * h)) / 2.0)) / ((2.0 * d) * l))) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 0.00034d0) then
        tmp = (1.0d0 + (((m * d_1) * ((((m / d) * d_1) * (((-1.0d0) / 2.0d0) * h)) / 2.0d0)) / ((2.0d0 * d) * l))) * sqrt(((d / l) * (d / h)))
    else
        tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 0.00034) {
		tmp = (1.0 + (((M * D) * ((((M / d) * D) * ((-1.0 / 2.0) * h)) / 2.0)) / ((2.0 * d) * l))) * Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = (Math.sqrt(d) * Math.sqrt((d / h))) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 0.00034:
		tmp = (1.0 + (((M * D) * ((((M / d) * D) * ((-1.0 / 2.0) * h)) / 2.0)) / ((2.0 * d) * l))) * math.sqrt(((d / l) * (d / h)))
	else:
		tmp = (math.sqrt(d) * math.sqrt((d / h))) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 0.00034)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(M * D) * Float64(Float64(Float64(Float64(M / d) * D) * Float64(Float64(-1.0 / 2.0) * h)) / 2.0)) / Float64(Float64(2.0 * d) * l))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(Float64(sqrt(d) * sqrt(Float64(d / h))) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 0.00034)
		tmp = (1.0 + (((M * D) * ((((M / d) * D) * ((-1.0 / 2.0) * h)) / 2.0)) / ((2.0 * d) * l))) * sqrt(((d / l) * (d / h)));
	else
		tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 0.00034], N[(N[(1.0 + N[(N[(N[(M * D), $MachinePrecision] * N[(N[(N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision] * N[(N[(-1.0 / 2.0), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.4e-4

   1. Initial program 72.4%

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

   4. Applied rewrites 75.1%

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

   6. Applied rewrites 60.5%

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

   8. Applied rewrites 58.5%

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

   10. Applied rewrites 61.1%

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

   if 3.4e-4 < l

   1. Initial program 64.5%

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

   4. Applied rewrites 64.7%

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

   6. Applied rewrites 67.8%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 64.7%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.00034:\\ \;\;\;\;\left(1 + \frac{\left(M \cdot D\right) \cdot \frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{-1}{2} \cdot h\right)}{2}}{\left(2 \cdot d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.62 \cdot 10^{-131}:\\ \;\;\;\;\left(1 + \frac{\left(M \cdot D\right) \cdot \frac{\left(\frac{-1}{2} \cdot h\right) \cdot \left(M \cdot D\right)}{2 \cdot d}}{\left(2 \cdot d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.62e-131)
   (*
    (+
     1.0
     (/
      (* (* M D) (/ (* (* (/ -1.0 2.0) h) (* M D)) (* 2.0 d)))
      (* (* 2.0 d) l)))
    (sqrt (* (/ d l) (/ d h))))
   (/ (* (sqrt d) (sqrt (/ d h))) (sqrt l))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.62e-131) {
		tmp = (1.0 + (((M * D) * ((((-1.0 / 2.0) * h) * (M * D)) / (2.0 * d))) / ((2.0 * d) * l))) * sqrt(((d / l) * (d / h)));
	} else {
		tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 1.62d-131) then
        tmp = (1.0d0 + (((m * d_1) * (((((-1.0d0) / 2.0d0) * h) * (m * d_1)) / (2.0d0 * d))) / ((2.0d0 * d) * l))) * sqrt(((d / l) * (d / h)))
    else
        tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.62e-131) {
		tmp = (1.0 + (((M * D) * ((((-1.0 / 2.0) * h) * (M * D)) / (2.0 * d))) / ((2.0 * d) * l))) * Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = (Math.sqrt(d) * Math.sqrt((d / h))) / Math.sqrt(l);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.62e-131:
		tmp = (1.0 + (((M * D) * ((((-1.0 / 2.0) * h) * (M * D)) / (2.0 * d))) / ((2.0 * d) * l))) * math.sqrt(((d / l) * (d / h)))
	else:
		tmp = (math.sqrt(d) * math.sqrt((d / h))) / math.sqrt(l)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.62e-131)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(M * D) * Float64(Float64(Float64(Float64(-1.0 / 2.0) * h) * Float64(M * D)) / Float64(2.0 * d))) / Float64(Float64(2.0 * d) * l))) * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	else
		tmp = Float64(Float64(sqrt(d) * sqrt(Float64(d / h))) / sqrt(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.62e-131)
		tmp = (1.0 + (((M * D) * ((((-1.0 / 2.0) * h) * (M * D)) / (2.0 * d))) / ((2.0 * d) * l))) * sqrt(((d / l) * (d / h)));
	else
		tmp = (sqrt(d) * sqrt((d / h))) / sqrt(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.62e-131], N[(N[(1.0 + N[(N[(N[(M * D), $MachinePrecision] * N[(N[(N[(N[(-1.0 / 2.0), $MachinePrecision] * h), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.61999999999999988e-131

   1. Initial program 74.3%

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

   4. Applied rewrites 76.8%

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

   6. Applied rewrites 63.1%

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

   8. Applied rewrites 60.6%

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

   10. Applied rewrites 62.5%

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

   if 1.61999999999999988e-131 < l

   1. Initial program 63.8%

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

   4. Applied rewrites 65.1%

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

   6. Applied rewrites 68.0%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 60.6%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.62 \cdot 10^{-131}:\\ \;\;\;\;\left(1 + \frac{\left(M \cdot D\right) \cdot \frac{\left(\frac{-1}{2} \cdot h\right) \cdot \left(M \cdot D\right)}{2 \cdot d}}{\left(2 \cdot d\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d} \cdot \sqrt{\frac{d}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 22.3% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

4. Applied rewrites 72.6%

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

6. Applied rewrites 36.6%

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

7. Taylor expanded in d around inf

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

9. Applied rewrites 28.1%

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

Reproduce

herbie shell --seed 2025021 
(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)))))