Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.1% → 77.6%

Time: 8.5s

Alternatives: 15

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.1% 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: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -2.6 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \left(-{\left(\ell \cdot h\right)}^{-0.5}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -2.6e+67)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (-
      1.0
      (* (/ h l) (* (* (* (/ D d) (* 0.5 M)) (* M (/ D (* 2.0 d)))) 0.5)))))
   (if (<= h -5e-310)
     (*
      (* d (- (pow (* l h) -0.5)))
      (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
     (*
      (/ (sqrt d) (sqrt h))
      (*
       (/ (sqrt d) (sqrt l))
       (- 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) 0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -2.6e+67) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (h <= -5e-310) {
		tmp = (d * -pow((l * h), -0.5)) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-2.6d+67)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - ((h / l) * ((((d_1 / d) * (0.5d0 * m)) * (m * (d_1 / (2.0d0 * d)))) * 0.5d0))))
    else if (h <= (-5d-310)) then
        tmp = (d * -((l * h) ** (-0.5d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 - ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -2.6e+67) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (h <= -5e-310) {
		tmp = (d * -Math.pow((l * h), -0.5)) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -2.6e+67:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))))
	elif h <= -5e-310:
		tmp = (d * -math.pow((l * h), -0.5)) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * 0.5))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -2.6e+67)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(M * Float64(D / Float64(2.0 * d)))) * 0.5)))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(d * Float64(-(Float64(l * h) ^ -0.5))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -2.6e+67)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	elseif (h <= -5e-310)
		tmp = (d * -((l * h) ^ -0.5)) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -2.6e+67], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(d * (-N[Power[N[(l * h), $MachinePrecision], -0.5], $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], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.6e67

   1. Initial program 58.9%

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

   3. Applied rewrites 58.5%

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

   5. Applied rewrites 58.5%

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

   7. Applied rewrites 58.5%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 58.5%

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

   if -2.6e67 < h < -4.999999999999985e-310

   1. Initial program 71.6%

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

   3. Applied rewrites 0.0%

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

   4. Taylor expanded in h around -inf

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

   6. Applied rewrites 79.4%

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

   if -4.999999999999985e-310 < h

   1. Initial program 67.5%

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

   3. Applied rewrites 66.9%

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

   5. Applied rewrites 71.2%

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

   7. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 54.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\left(\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot -1}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125\\ \mathbf{elif}\;t\_1 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_0\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -5e+51)
     (*
      (* (/ (* (* (* M D) (* M D)) -1.0) d) (sqrt (/ h (* (* l l) l))))
      -0.125)
     (if (<= t_1 1e+220)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (if (<= t_1 INFINITY) (/ (* t_0 d) h) (/ (* d (- t_0)) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * sqrt((h / ((l * l) * l)))) * -0.125;
	} else if (t_1 <= 1e+220) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * Math.sqrt((h / ((l * l) * l)))) * -0.125;
	} else if (t_1 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -5e+51:
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * math.sqrt((h / ((l * l) * l)))) * -0.125
	elif t_1 <= 1e+220:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = (d * -t_0) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -5e+51)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * -1.0) / d) * sqrt(Float64(h / Float64(Float64(l * l) * l)))) * -0.125);
	elseif (t_1 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_0)) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -5e+51)
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * sqrt((h / ((l * l) * l)))) * -0.125;
	elseif (t_1 <= 1e+220)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = (d * -t_0) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+51], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], If[LessEqual[t$95$1, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5e51

   1. Initial program 85.9%

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

   2. Taylor expanded in h around -inf

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

   4. Applied rewrites 37.0%

      \[\leadsto \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125} \]

   6. Applied rewrites 37.0%

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

   8. Applied rewrites 37.0%

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

   if -5e51 < (*.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)))) < 1e220

   1. Initial program 89.1%

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

   3. Applied rewrites 87.9%

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

   4. Taylor expanded in d around inf

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

   6. Applied rewrites 81.5%

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

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 70.8%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in l around -inf

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

   7. Applied rewrites 17.9%

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

Alternative 3: 53.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\left(\left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot -1}{d}\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125\\ \mathbf{elif}\;t\_1 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_0\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -5e+51)
     (*
      (* (* (* D D) (/ (* (* M M) -1.0) d)) (sqrt (/ h (* (* l l) l))))
      -0.125)
     (if (<= t_1 1e+220)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (if (<= t_1 INFINITY) (/ (* t_0 d) h) (/ (* d (- t_0)) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = (((D * D) * (((M * M) * -1.0) / d)) * sqrt((h / ((l * l) * l)))) * -0.125;
	} else if (t_1 <= 1e+220) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e+51) {
		tmp = (((D * D) * (((M * M) * -1.0) / d)) * Math.sqrt((h / ((l * l) * l)))) * -0.125;
	} else if (t_1 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -5e+51:
		tmp = (((D * D) * (((M * M) * -1.0) / d)) * math.sqrt((h / ((l * l) * l)))) * -0.125
	elif t_1 <= 1e+220:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = (d * -t_0) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -5e+51)
		tmp = Float64(Float64(Float64(Float64(D * D) * Float64(Float64(Float64(M * M) * -1.0) / d)) * sqrt(Float64(h / Float64(Float64(l * l) * l)))) * -0.125);
	elseif (t_1 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_0)) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -5e+51)
		tmp = (((D * D) * (((M * M) * -1.0) / d)) * sqrt((h / ((l * l) * l)))) * -0.125;
	elseif (t_1 <= 1e+220)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = (d * -t_0) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+51], N[(N[(N[(N[(D * D), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * -1.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], If[LessEqual[t$95$1, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5e51

   1. Initial program 85.9%

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

   2. Taylor expanded in h around -inf

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

   4. Applied rewrites 37.0%

      \[\leadsto \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125} \]

   6. Applied rewrites 37.0%

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

   8. Applied rewrites 31.3%

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

   if -5e51 < (*.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)))) < 1e220

   1. Initial program 89.1%

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

   3. Applied rewrites 87.9%

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

   4. Taylor expanded in d around inf

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

   6. Applied rewrites 81.5%

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

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 70.8%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in l around -inf

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

   7. Applied rewrites 17.9%

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

Alternative 4: 52.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := -t\_0\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\left(d \cdot d\right) \cdot t\_1}{d}}{h}\\ \mathbf{elif}\;t\_2 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot t\_1}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1 (- t_0))
        (t_2
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_2 -2e-78)
     (/ (/ (* (* d d) t_1) d) h)
     (if (<= t_2 1e+220)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (if (<= t_2 INFINITY) (/ (* t_0 d) h) (/ (* d t_1) h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = -t_0;
	double t_2 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_2 <= -2e-78) {
		tmp = (((d * d) * t_1) / d) / h;
	} else if (t_2 <= 1e+220) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * t_1) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = -t_0;
	double t_2 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_2 <= -2e-78) {
		tmp = (((d * d) * t_1) / d) / h;
	} else if (t_2 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * t_1) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = -t_0
	t_2 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_2 <= -2e-78:
		tmp = (((d * d) * t_1) / d) / h
	elif t_2 <= 1e+220:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif t_2 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = (d * t_1) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(-t_0)
	t_2 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_2 <= -2e-78)
		tmp = Float64(Float64(Float64(Float64(d * d) * t_1) / d) / h);
	elseif (t_2 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(d * t_1) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = -t_0;
	t_2 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_2 <= -2e-78)
		tmp = (((d * d) * t_1) / d) / h;
	elseif (t_2 <= 1e+220)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (t_2 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = (d * t_1) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-78], N[(N[(N[(N[(d * d), $MachinePrecision] * t$95$1), $MachinePrecision] / d), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$2, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * t$95$1), $MachinePrecision] / h), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-78

   1. Initial program 86.6%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in d around 0

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

   7. Applied rewrites 54.7%

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

   8. Taylor expanded in l around -inf

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

   10. Applied rewrites 27.5%

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

   if -2e-78 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e220

   1. Initial program 88.6%

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

   3. Applied rewrites 87.9%

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

   4. Taylor expanded in d around inf

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

   6. Applied rewrites 85.7%

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

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 70.8%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in l around -inf

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

   7. Applied rewrites 17.9%

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

Alternative 5: 50.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{d \cdot \left(-t\_0\right)}{h}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-245}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (/ (* d (- t_0)) h)))
   (if (<= t_1 2e-245)
     t_2
     (if (<= t_1 1e+220)
       (* (sqrt (/ d h)) (sqrt (/ d l)))
       (if (<= t_1 INFINITY) (/ (* t_0 d) h) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= 2e-245) {
		tmp = t_2;
	} else if (t_1 <= 1e+220) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= 2e-245) {
		tmp = t_2;
	} else if (t_1 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = (d * -t_0) / h
	tmp = 0
	if t_1 <= 2e-245:
		tmp = t_2
	elif t_1 <= 1e+220:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(Float64(d * Float64(-t_0)) / h)
	tmp = 0.0
	if (t_1 <= 2e-245)
		tmp = t_2;
	elseif (t_1 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = (d * -t_0) / h;
	tmp = 0.0;
	if (t_1 <= 2e-245)
		tmp = t_2;
	elseif (t_1 <= 1e+220)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-245], t$95$2, If[LessEqual[t$95$1, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999999e-245 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 55.0%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in l around -inf

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

   7. Applied rewrites 22.2%

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

   if 1.9999999999999999e-245 < (*.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)))) < 1e220

   1. Initial program 98.8%

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in d around inf

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

   6. Applied rewrites 98.1%

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

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 70.8%

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

Alternative 6: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right), -0.125, t\_0 \cdot \left(d \cdot d\right)\right)}{d}}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 1e+220)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (-
        1.0
        (* (/ h l) (* (* (* (/ D d) (* 0.5 M)) (* M (/ D (* 2.0 d)))) 0.5)))))
     (if (<= t_1 INFINITY)
       (/ (* t_0 d) h)
       (/
        (/
         (fma (* (pow (/ h l) 1.5) (* (* M D) (* M D))) -0.125 (* t_0 (* d d)))
         d)
        h)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 1e+220) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (fma((pow((h / l), 1.5) * ((M * D) * (M * D))), -0.125, (t_0 * (d * d))) / d) / h;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(M * Float64(D / Float64(2.0 * d)))) * 0.5)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(fma(Float64((Float64(h / l) ^ 1.5) * Float64(Float64(M * D) * Float64(M * D))), -0.125, Float64(t_0 * Float64(d * d))) / d) / h);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(t$95$0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e220

   1. Initial program 87.6%

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

   3. Applied rewrites 86.7%

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 86.7%

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

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 70.8%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in d around 0

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

   7. Applied rewrites 26.6%

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

   9. Applied rewrites 26.6%

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

Alternative 7: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_0\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 1e+220)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (-
        1.0
        (* (/ h l) (* (* (* (/ D d) (* 0.5 M)) (* M (/ D (* 2.0 d)))) 0.5)))))
     (if (<= t_1 INFINITY) (/ (* t_0 d) h) (/ (* d (- t_0)) h)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 1e+220) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = (d * -t_0) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= 1e+220:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))))
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = (d * -t_0) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(M * Float64(D / Float64(2.0 * d)))) * 0.5)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_0)) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= 1e+220)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = (d * -t_0) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e220

   1. Initial program 87.6%

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

   3. Applied rewrites 86.7%

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.7%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 86.7%

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

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 70.8%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in l around -inf

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

   7. Applied rewrites 17.9%

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

Alternative 8: 70.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{D}{2 \cdot d}\\ t_2 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 10^{+220}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\left(t\_1 \cdot M\right) \cdot M\right) \cdot t\_1\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{t\_2 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(-t\_2\right)}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ D (* 2.0 d)))
        (t_2 (sqrt (/ h l))))
   (if (<= t_0 1e+220)
     (*
      (sqrt (/ d h))
      (* (sqrt (/ d l)) (- 1.0 (* (/ h l) (* (* (* (* t_1 M) M) t_1) 0.5)))))
     (if (<= t_0 INFINITY) (/ (* t_2 d) h) (/ (* d (- t_2)) h)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = D / (2.0 * d);
	double t_2 = sqrt((h / l));
	double tmp;
	if (t_0 <= 1e+220) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((t_1 * M) * M) * t_1) * 0.5))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (t_2 * d) / h;
	} else {
		tmp = (d * -t_2) / h;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = D / (2.0 * d);
	double t_2 = Math.sqrt((h / l));
	double tmp;
	if (t_0 <= 1e+220) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - ((h / l) * ((((t_1 * M) * M) * t_1) * 0.5))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (t_2 * d) / h;
	} else {
		tmp = (d * -t_2) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = D / (2.0 * d)
	t_2 = math.sqrt((h / l))
	tmp = 0
	if t_0 <= 1e+220:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - ((h / l) * ((((t_1 * M) * M) * t_1) * 0.5))))
	elif t_0 <= math.inf:
		tmp = (t_2 * d) / h
	else:
		tmp = (d * -t_2) / h
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(D / Float64(2.0 * d))
	t_2 = sqrt(Float64(h / l))
	tmp = 0.0
	if (t_0 <= 1e+220)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(t_1 * M) * M) * t_1) * 0.5)))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(t_2 * d) / h);
	else
		tmp = Float64(Float64(d * Float64(-t_2)) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = D / (2.0 * d);
	t_2 = sqrt((h / l));
	tmp = 0.0;
	if (t_0 <= 1e+220)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((t_1 * M) * M) * t_1) * 0.5))));
	elseif (t_0 <= Inf)
		tmp = (t_2 * d) / h;
	else
		tmp = (d * -t_2) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 1e+220], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(t$95$1 * M), $MachinePrecision] * M), $MachinePrecision] * t$95$1), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(t$95$2 * d), $MachinePrecision] / h), $MachinePrecision], N[(N[(d * (-t$95$2)), $MachinePrecision] / h), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1e220

   1. Initial program 87.6%

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

   3. Applied rewrites 86.7%

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

   5. Applied rewrites 86.7%

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

   7. Applied rewrites 86.7%

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

   9. Applied rewrites 84.9%

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

   if 1e220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 56.3%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 70.8%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 19.6%

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

   5. Taylor expanded in l around -inf

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

   7. Applied rewrites 17.9%

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

Alternative 9: 47.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{d \cdot \left(-t\_0\right)}{h}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-78}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (/ (* d (- t_0)) h)))
   (if (<= t_1 -2e-78) t_2 (if (<= t_1 INFINITY) (/ (* t_0 d) h) t_2))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= -2e-78) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (d * -t_0) / h;
	double tmp;
	if (t_1 <= -2e-78) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = (d * -t_0) / h
	tmp = 0
	if t_1 <= -2e-78:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (t_0 * d) / h
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(Float64(d * Float64(-t_0)) / h)
	tmp = 0.0
	if (t_1 <= -2e-78)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = (d * -t_0) / h;
	tmp = 0.0;
	if (t_1 <= -2e-78)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (t_0 * d) / h;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-78], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-78 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 55.8%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 42.6%

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

   5. Taylor expanded in l around -inf

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

   7. Applied rewrites 20.8%

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

   if -2e-78 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 78.7%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 57.8%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 74.8%

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

Alternative 10: 41.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (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))))
      2e-245)
   (* (sqrt (/ (/ 1.0 l) h)) d)
   (/ (* (sqrt (/ h l)) d) h)))

C

double code(double d, double h, double l, double M, double D) {
	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)))) <= 2e-245) {
		tmp = sqrt(((1.0 / l) / h)) * d;
	} else {
		tmp = (sqrt((h / l)) * d) / 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 (((((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)))) <= 2d-245) then
        tmp = sqrt(((1.0d0 / l) / h)) * d
    else
        tmp = (sqrt((h / l)) * d) / 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 (((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)))) <= 2e-245) {
		tmp = Math.sqrt(((1.0 / l) / h)) * d;
	} else {
		tmp = (Math.sqrt((h / l)) * d) / h;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	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)))) <= 2e-245:
		tmp = math.sqrt(((1.0 / l) / h)) * d
	else:
		tmp = (math.sqrt((h / l)) * d) / h
	return tmp

Julia

function code(d, h, l, M, D)
	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)))) <= 2e-245)
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d);
	else
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	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)))) <= 2e-245)
		tmp = sqrt(((1.0 / l) / h)) * d;
	else
		tmp = (sqrt((h / l)) * d) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := 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], 2e-245], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $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)))) < 1.9999999999999999e-245

   1. Initial program 80.0%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 17.3%

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

   6. Applied rewrites 17.3%

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

   8. Applied rewrites 17.3%

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

   if 1.9999999999999999e-245 < (*.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 58.6%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 56.6%

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

Alternative 11: 75.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-160}:\\ \;\;\;\;\frac{\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}\right) \cdot -0.125}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -7.5e-119)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (-
      1.0
      (* (/ h l) (* (* (* (/ D d) (* 0.5 M)) (* M (/ D (* 2.0 d)))) 0.5)))))
   (if (<= d 1.9e-160)
     (/ (* (* (pow (/ h l) 1.5) (/ (pow (* D M) 2.0) d)) -0.125) h)
     (*
      (/ (sqrt d) (sqrt h))
      (*
       (/ (sqrt d) (sqrt l))
       (- 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) 0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -7.5e-119) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (d <= 1.9e-160) {
		tmp = ((pow((h / l), 1.5) * (pow((D * M), 2.0) / d)) * -0.125) / h;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	}
	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 (d <= (-7.5d-119)) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0d0 - ((h / l) * ((((d_1 / d) * (0.5d0 * m)) * (m * (d_1 / (2.0d0 * d)))) * 0.5d0))))
    else if (d <= 1.9d-160) then
        tmp = ((((h / l) ** 1.5d0) * (((d_1 * m) ** 2.0d0) / d)) * (-0.125d0)) / h
    else
        tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0d0 - ((h / l) * ((((m / 2.0d0) * (d_1 / d)) ** 2.0d0) * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -7.5e-119) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (d <= 1.9e-160) {
		tmp = ((Math.pow((h / l), 1.5) * (Math.pow((D * M), 2.0) / d)) * -0.125) / h;
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((Math.sqrt(d) / Math.sqrt(l)) * (1.0 - ((h / l) * (Math.pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -7.5e-119:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))))
	elif d <= 1.9e-160:
		tmp = ((math.pow((h / l), 1.5) * (math.pow((D * M), 2.0) / d)) * -0.125) / h
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((math.sqrt(d) / math.sqrt(l)) * (1.0 - ((h / l) * (math.pow(((M / 2.0) * (D / d)), 2.0) * 0.5))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -7.5e-119)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(M * Float64(D / Float64(2.0 * d)))) * 0.5)))));
	elseif (d <= 1.9e-160)
		tmp = Float64(Float64(Float64((Float64(h / l) ^ 1.5) * Float64((Float64(D * M) ^ 2.0) / d)) * -0.125) / h);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -7.5e-119)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	elseif (d <= 1.9e-160)
		tmp = ((((h / l) ^ 1.5) * (((D * M) ^ 2.0) / d)) * -0.125) / h;
	else
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - ((h / l) * ((((M / 2.0) * (D / d)) ^ 2.0) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -7.5e-119], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e-160], N[(N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.50000000000000044e-119

   1. Initial program 76.2%

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

   3. Applied rewrites 75.8%

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 75.8%

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

   if -7.50000000000000044e-119 < d < 1.8999999999999999e-160

   1. Initial program 45.6%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in d around 0

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

   7. Applied rewrites 57.8%

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

   if 1.8999999999999999e-160 < d

   1. Initial program 74.4%

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

   3. Applied rewrites 74.1%

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 75.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right) \cdot \left(M \cdot \frac{D}{2 \cdot d}\right)\right) \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({\left(\frac{h}{\ell}\right)}^{1.5} \cdot \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right), -0.125, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{d}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.5e-27)
   (*
    (sqrt (/ d h))
    (*
     (sqrt (/ d l))
     (-
      1.0
      (* (/ h l) (* (* (* (/ D d) (* 0.5 M)) (* M (/ D (* 2.0 d)))) 0.5)))))
   (if (<= d -1e-309)
     (/
      (/
       (fma
        (* (pow (/ h l) 1.5) (* (* M D) (* M D)))
        -0.125
        (* (sqrt (/ h l)) (* d d)))
       d)
      h)
     (*
      (/ (sqrt d) (sqrt h))
      (*
       (/ (sqrt d) (sqrt l))
       (- 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) 0.5))))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.5e-27) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - ((h / l) * ((((D / d) * (0.5 * M)) * (M * (D / (2.0 * d)))) * 0.5))));
	} else if (d <= -1e-309) {
		tmp = (fma((pow((h / l), 1.5) * ((M * D) * (M * D))), -0.125, (sqrt((h / l)) * (d * d))) / d) / h;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((sqrt(d) / sqrt(l)) * (1.0 - ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * 0.5))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.5e-27)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(Float64(Float64(D / d) * Float64(0.5 * M)) * Float64(M * Float64(D / Float64(2.0 * d)))) * 0.5)))));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(fma(Float64((Float64(h / l) ^ 1.5) * Float64(Float64(M * D) * Float64(M * D))), -0.125, Float64(sqrt(Float64(h / l)) * Float64(d * d))) / d) / h);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(sqrt(d) / sqrt(l)) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * 0.5)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.5e-27], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(0.5 * M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(N[(N[(N[Power[N[(h / l), $MachinePrecision], 1.5], $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / h), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.5000000000000001e-27

   1. Initial program 76.8%

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

   3. Applied rewrites 76.6%

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

   5. Applied rewrites 76.6%

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

   7. Applied rewrites 76.6%

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

   8. Taylor expanded in M around 0

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

   10. Applied rewrites 76.6%

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

   if -2.5000000000000001e-27 < d < -1.000000000000002e-309

   1. Initial program 54.5%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 48.6%

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

   5. Taylor expanded in d around 0

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

   7. Applied rewrites 54.1%

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

   9. Applied rewrites 54.1%

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

   if -1.000000000000002e-309 < d

   1. Initial program 67.5%

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

   3. Applied rewrites 66.9%

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

   5. Applied rewrites 71.2%

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

   7. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot 0.5\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+126}:\\ \;\;\;\;\left(-{\left(\ell \cdot h\right)}^{-0.5}\right) \cdot d\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot -1}{d} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\right) \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.05e+126)
   (* (- (pow (* l h) -0.5)) d)
   (if (<= d -1.45e-31)
     (* (sqrt (/ d h)) (sqrt (/ d l)))
     (if (<= d -1e-309)
       (*
        (* (/ (* (* (* M D) (* M D)) -1.0) d) (sqrt (/ h (* (* l l) l))))
        -0.125)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.05e+126) {
		tmp = -pow((l * h), -0.5) * d;
	} else if (d <= -1.45e-31) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (d <= -1e-309) {
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * sqrt((h / ((l * l) * l)))) * -0.125;
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-1.05d+126)) then
        tmp = -((l * h) ** (-0.5d0)) * d
    else if (d <= (-1.45d-31)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (d <= (-1d-309)) then
        tmp = (((((m * d_1) * (m * d_1)) * (-1.0d0)) / d) * sqrt((h / ((l * l) * l)))) * (-0.125d0)
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.05e+126) {
		tmp = -Math.pow((l * h), -0.5) * d;
	} else if (d <= -1.45e-31) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (d <= -1e-309) {
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * Math.sqrt((h / ((l * l) * l)))) * -0.125;
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -1.05e+126:
		tmp = -math.pow((l * h), -0.5) * d
	elif d <= -1.45e-31:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif d <= -1e-309:
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * math.sqrt((h / ((l * l) * l)))) * -0.125
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.05e+126)
		tmp = Float64(Float64(-(Float64(l * h) ^ -0.5)) * d);
	elseif (d <= -1.45e-31)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (d <= -1e-309)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * -1.0) / d) * sqrt(Float64(h / Float64(Float64(l * l) * l)))) * -0.125);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -1.05e+126)
		tmp = -((l * h) ^ -0.5) * d;
	elseif (d <= -1.45e-31)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (d <= -1e-309)
		tmp = (((((M * D) * (M * D)) * -1.0) / d) * sqrt((h / ((l * l) * l)))) * -0.125;
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.05e+126], N[((-N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[d, -1.45e-31], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1e-309], N[(N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.05e126

   1. Initial program 74.1%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 6.5%

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

   5. Taylor expanded in h around -inf

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

   7. Applied rewrites 66.2%

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

   if -1.05e126 < d < -1.45e-31

   1. Initial program 79.9%

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

   3. Applied rewrites 79.2%

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

   4. Taylor expanded in d around inf

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

   6. Applied rewrites 48.4%

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

   if -1.45e-31 < d < -1.000000000000002e-309

   1. Initial program 54.1%

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

   2. Taylor expanded in h around -inf

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

   4. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\left(\frac{{\left(D \cdot M\right)}^{2} \cdot -1}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot -0.125} \]

   6. Applied rewrites 45.1%

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

   8. Applied rewrites 45.1%

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

   if -1.000000000000002e-309 < d

   1. Initial program 67.5%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 44.4%

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

   6. Applied rewrites 44.5%

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

   8. Applied rewrites 52.4%

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

Alternative 14: 43.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -5.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-122}:\\ \;\;\;\;\frac{d \cdot \left(-t\_0\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))))
   (if (<= d -5.6e-108)
     (/ (* t_0 d) h)
     (if (<= d 7e-122)
       (/ (* d (- t_0)) h)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double tmp;
	if (d <= -5.6e-108) {
		tmp = (t_0 * d) / h;
	} else if (d <= 7e-122) {
		tmp = (d * -t_0) / h;
	} else {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h / l))
    if (d <= (-5.6d-108)) then
        tmp = (t_0 * d) / h
    else if (d <= 7d-122) then
        tmp = (d * -t_0) / h
    else
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((h / l));
	double tmp;
	if (d <= -5.6e-108) {
		tmp = (t_0 * d) / h;
	} else if (d <= 7e-122) {
		tmp = (d * -t_0) / h;
	} else {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h / l))
	tmp = 0
	if d <= -5.6e-108:
		tmp = (t_0 * d) / h
	elif d <= 7e-122:
		tmp = (d * -t_0) / h
	else:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	tmp = 0.0
	if (d <= -5.6e-108)
		tmp = Float64(Float64(t_0 * d) / h);
	elseif (d <= 7e-122)
		tmp = Float64(Float64(d * Float64(-t_0)) / h);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h / l));
	tmp = 0.0;
	if (d <= -5.6e-108)
		tmp = (t_0 * d) / h;
	elseif (d <= 7e-122)
		tmp = (d * -t_0) / h;
	else
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -5.6e-108], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[d, 7e-122], N[(N[(d * (-t$95$0)), $MachinePrecision] / h), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.6e-108

   1. Initial program 76.5%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 46.9%

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

   if -5.6e-108 < d < 7.0000000000000003e-122

   1. Initial program 46.9%

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

   2. Taylor expanded in h around 0

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in l around -inf

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

   7. Applied rewrites 17.7%

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

   if 7.0000000000000003e-122 < d

   1. Initial program 76.0%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 52.7%

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

   6. Applied rewrites 52.9%

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

   8. Applied rewrites 62.7%

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

Alternative 15: 27.4% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.1%

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

2. Taylor expanded in d around inf

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

4. Applied rewrites 27.4%

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

6. Applied rewrites 27.4%

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

Reproduce

herbie shell --seed 2025106 
(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)))))