Henrywood and Agarwal, Equation (12)

Percentage Accurate: 36.4% → 76.4%

Time: 12.1s

Alternatives: 14

Speedup: 7.7×

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: 36.4% 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: 76.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d\_m + d\_m}\\ t_1 := {\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_2 := t\_1 \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-252}:\\ \;\;\;\;t\_1 \cdot \left(1 - \left(\frac{1}{2} \cdot \frac{t\_0 \cdot \left(M \cdot D\right)}{d\_m + d\_m}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}} \cdot d\_m\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot \frac{\ell - \frac{0.125}{d\_m} \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d\_m}}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) (+ d_m d_m)))
        (t_1 (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0))))
        (t_2
         (*
          t_1
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))))
   (if (<= t_2 -5e-252)
     (*
      t_1
      (- 1.0 (* (* (/ 1.0 2.0) (/ (* t_0 (* M D)) (+ d_m d_m))) (/ h l))))
     (if (<= t_2 INFINITY)
       (*
        (* (/ (sqrt (/ 1.0 h)) (sqrt l)) d_m)
        (- 1.0 (* (* (/ 1.0 2.0) (* t_0 t_0)) (/ h l))))
       (*
        (* (sqrt (/ 1.0 (* l h))) d_m)
        (/ (- l (* (/ 0.125 d_m) (/ (* (* M D) (* (* M D) h)) d_m))) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (M * D) / (d_m + d_m);
	double t_1 = pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0));
	double t_2 = t_1 * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_2 <= -5e-252) {
		tmp = t_1 * (1.0 - (((1.0 / 2.0) * ((t_0 * (M * D)) / (d_m + d_m))) * (h / l)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = ((sqrt((1.0 / h)) / sqrt(l)) * d_m) * (1.0 - (((1.0 / 2.0) * (t_0 * t_0)) * (h / l)));
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l);
	}
	return tmp;
}

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (M * D) / (d_m + d_m);
	double t_1 = Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0));
	double t_2 = t_1 * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_2 <= -5e-252) {
		tmp = t_1 * (1.0 - (((1.0 / 2.0) * ((t_0 * (M * D)) / (d_m + d_m))) * (h / l)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = ((Math.sqrt((1.0 / h)) / Math.sqrt(l)) * d_m) * (1.0 - (((1.0 / 2.0) * (t_0 * t_0)) * (h / l)));
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d_m) * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l);
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = (M * D) / (d_m + d_m)
	t_1 = math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))
	t_2 = t_1 * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))
	tmp = 0
	if t_2 <= -5e-252:
		tmp = t_1 * (1.0 - (((1.0 / 2.0) * ((t_0 * (M * D)) / (d_m + d_m))) * (h / l)))
	elif t_2 <= math.inf:
		tmp = ((math.sqrt((1.0 / h)) / math.sqrt(l)) * d_m) * (1.0 - (((1.0 / 2.0) * (t_0 * t_0)) * (h / l)))
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d_m) * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l)
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(Float64(M * D) / Float64(d_m + d_m))
	t_1 = Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0)))
	t_2 = Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_2 <= -5e-252)
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * Float64(Float64(t_0 * Float64(M * D)) / Float64(d_m + d_m))) * Float64(h / l))));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / h)) / sqrt(l)) * d_m) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * Float64(t_0 * t_0)) * Float64(h / l))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * Float64(Float64(l - Float64(Float64(0.125 / d_m) * Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * h)) / d_m))) / l));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = (M * D) / (d_m + d_m);
	t_1 = ((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0));
	t_2 = t_1 * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_2 <= -5e-252)
		tmp = t_1 * (1.0 - (((1.0 / 2.0) * ((t_0 * (M * D)) / (d_m + d_m))) * (h / l)));
	elseif (t_2 <= Inf)
		tmp = ((sqrt((1.0 / h)) / sqrt(l)) * d_m) * (1.0 - (((1.0 / 2.0) * (t_0 * t_0)) * (h / l)));
	else
		tmp = (sqrt((1.0 / (l * h))) * d_m) * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l);
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-252], N[(t$95$1 * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(t$95$0 * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(N[(l - N[(N[(0.125 / d$95$m), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000008e-252

   1. Initial program 85.9%

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

   3. Applied rewrites 84.1%

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

   if -5.00000000000000008e-252 < (*.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 79.1%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 82.0%

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

   6. Applied rewrites 82.0%

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

   8. Applied rewrites 88.8%

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

   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 d around 0

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 58.1%

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

   9. Applied rewrites 65.7%

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

   11. Applied rewrites 67.9%

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

Alternative 2: 76.3% accurate, 0.6× speedup

Math

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

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (/ (* M D) (+ d_m d_m))))
   (if (<=
        (*
         (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))
        INFINITY)
     (*
      (* (/ (sqrt (/ 1.0 h)) (sqrt l)) d_m)
      (- 1.0 (* (* (/ 1.0 2.0) (* t_0 t_0)) (/ h l))))
     (*
      (* (sqrt (/ 1.0 (* l h))) d_m)
      (/ (- l (* (/ 0.125 d_m) (/ (* (* M D) (* (* M D) h)) d_m))) l)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.7%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 80.1%

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

   6. Applied rewrites 80.1%

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

   8. Applied rewrites 86.7%

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

   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 d around 0

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 58.1%

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

   9. Applied rewrites 65.7%

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

   11. Applied rewrites 67.9%

       \[\leadsto \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\right) \cdot \frac{\ell - \frac{0.125}{d} \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d}}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{\ell - \frac{0.125}{d\_m} \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{d\_m}}{\ell}\\ \mathbf{if}\;\ell \leq 7.5 \cdot 10^{-280}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}} \cdot d\_m\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (/ (- l (* (/ 0.125 d_m) (/ (* (* (* M D) (* M D)) h) d_m))) l)))
   (if (<= l 7.5e-280)
     (* (* (sqrt (/ 1.0 (* l h))) d_m) t_0)
     (* (* (/ (sqrt (/ 1.0 l)) (sqrt h)) d_m) t_0))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (l - ((0.125 / d_m) * ((((M * D) * (M * D)) * h) / d_m))) / l;
	double tmp;
	if (l <= 7.5e-280) {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * t_0;
	} else {
		tmp = ((sqrt((1.0 / l)) / sqrt(h)) * d_m) * t_0;
	}
	return tmp;
}

Fortran

d_m =     private
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_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l - ((0.125d0 / d_m) * ((((m * d) * (m * d)) * h) / d_m))) / l
    if (l <= 7.5d-280) then
        tmp = (sqrt((1.0d0 / (l * h))) * d_m) * t_0
    else
        tmp = ((sqrt((1.0d0 / l)) / sqrt(h)) * d_m) * t_0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (l - ((0.125 / d_m) * ((((M * D) * (M * D)) * h) / d_m))) / l;
	double tmp;
	if (l <= 7.5e-280) {
		tmp = (Math.sqrt((1.0 / (l * h))) * d_m) * t_0;
	} else {
		tmp = ((Math.sqrt((1.0 / l)) / Math.sqrt(h)) * d_m) * t_0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = (l - ((0.125 / d_m) * ((((M * D) * (M * D)) * h) / d_m))) / l
	tmp = 0
	if l <= 7.5e-280:
		tmp = (math.sqrt((1.0 / (l * h))) * d_m) * t_0
	else:
		tmp = ((math.sqrt((1.0 / l)) / math.sqrt(h)) * d_m) * t_0
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(Float64(l - Float64(Float64(0.125 / d_m) * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * h) / d_m))) / l)
	tmp = 0.0
	if (l <= 7.5e-280)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * t_0);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / l)) / sqrt(h)) * d_m) * t_0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = (l - ((0.125 / d_m) * ((((M * D) * (M * D)) * h) / d_m))) / l;
	tmp = 0.0;
	if (l <= 7.5e-280)
		tmp = (sqrt((1.0 / (l * h))) * d_m) * t_0;
	else
		tmp = ((sqrt((1.0 / l)) / sqrt(h)) * d_m) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(l - N[(N[(0.125 / d$95$m), $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[l, 7.5e-280], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 7.4999999999999999e-280

   1. Initial program 8.7%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 61.2%

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

   9. Applied rewrites 70.6%

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

   if 7.4999999999999999e-280 < l

   1. Initial program 65.6%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 60.0%

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

   9. Applied rewrites 69.0%

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

   11. Applied rewrites 78.3%

       \[\leadsto \left(\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}} \cdot d\right) \cdot \frac{\ell - \frac{0.125}{d} \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{d}}{\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-161}:\\ \;\;\;\;t\_1 \cdot \frac{\ell - \frac{0.125}{d\_m} \cdot \frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{d\_m}}{\ell}\\ \mathbf{elif}\;t\_0 \leq 10^{+130}:\\ \;\;\;\;{\left(\frac{d\_m}{\ell} \cdot \frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\ell - \frac{0.125}{d\_m} \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d\_m}}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l)))))
        (t_1 (* (sqrt (/ 1.0 (* l h))) d_m)))
   (if (<= t_0 5e-161)
     (* t_1 (/ (- l (* (/ 0.125 d_m) (/ (* (* (* M D) (* M D)) h) d_m))) l))
     (if (<= t_0 1e+130)
       (* (pow (* (/ d_m l) (/ d_m h)) (/ 1.0 2.0)) 1.0)
       (*
        t_1
        (/ (- l (* (/ 0.125 d_m) (/ (* (* M D) (* (* M D) h)) d_m))) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double t_1 = sqrt((1.0 / (l * h))) * d_m;
	double tmp;
	if (t_0 <= 5e-161) {
		tmp = t_1 * ((l - ((0.125 / d_m) * ((((M * D) * (M * D)) * h) / d_m))) / l);
	} else if (t_0 <= 1e+130) {
		tmp = pow(((d_m / l) * (d_m / h)), (1.0 / 2.0)) * 1.0;
	} else {
		tmp = t_1 * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l);
	}
	return tmp;
}

Fortran

d_m =     private
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_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((1.0d0 / (l * h))) * d_m
    if (t_0 <= 5d-161) then
        tmp = t_1 * ((l - ((0.125d0 / d_m) * ((((m * d) * (m * d)) * h) / d_m))) / l)
    else if (t_0 <= 1d+130) then
        tmp = (((d_m / l) * (d_m / h)) ** (1.0d0 / 2.0d0)) * 1.0d0
    else
        tmp = t_1 * ((l - ((0.125d0 / d_m) * (((m * d) * ((m * d) * h)) / d_m))) / l)
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((1.0 / (l * h))) * d_m;
	double tmp;
	if (t_0 <= 5e-161) {
		tmp = t_1 * ((l - ((0.125 / d_m) * ((((M * D) * (M * D)) * h) / d_m))) / l);
	} else if (t_0 <= 1e+130) {
		tmp = Math.pow(((d_m / l) * (d_m / h)), (1.0 / 2.0)) * 1.0;
	} else {
		tmp = t_1 * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l);
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = (math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))
	t_1 = math.sqrt((1.0 / (l * h))) * d_m
	tmp = 0
	if t_0 <= 5e-161:
		tmp = t_1 * ((l - ((0.125 / d_m) * ((((M * D) * (M * D)) * h) / d_m))) / l)
	elif t_0 <= 1e+130:
		tmp = math.pow(((d_m / l) * (d_m / h)), (1.0 / 2.0)) * 1.0
	else:
		tmp = t_1 * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l)
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m)
	tmp = 0.0
	if (t_0 <= 5e-161)
		tmp = Float64(t_1 * Float64(Float64(l - Float64(Float64(0.125 / d_m) * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * h) / d_m))) / l));
	elseif (t_0 <= 1e+130)
		tmp = Float64((Float64(Float64(d_m / l) * Float64(d_m / h)) ^ Float64(1.0 / 2.0)) * 1.0);
	else
		tmp = Float64(t_1 * Float64(Float64(l - Float64(Float64(0.125 / d_m) * Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * h)) / d_m))) / l));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = (((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((1.0 / (l * h))) * d_m;
	tmp = 0.0;
	if (t_0 <= 5e-161)
		tmp = t_1 * ((l - ((0.125 / d_m) * ((((M * D) * (M * D)) * h) / d_m))) / l);
	elseif (t_0 <= 1e+130)
		tmp = (((d_m / l) * (d_m / h)) ^ (1.0 / 2.0)) * 1.0;
	else
		tmp = t_1 * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l);
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-161], N[(t$95$1 * N[(N[(l - N[(N[(0.125 / d$95$m), $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+130], N[(N[Power[N[(N[(d$95$m / l), $MachinePrecision] * N[(d$95$m / h), $MachinePrecision]), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(t$95$1 * N[(N[(l - N[(N[(0.125 / d$95$m), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.9999999999999999e-161

   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 d around 0

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 52.9%

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

   9. Applied rewrites 72.8%

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

   if 4.9999999999999999e-161 < (*.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)))) < 1.0000000000000001e130

   1. Initial program 99.0%

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

   3. Applied rewrites 57.6%

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 97.7%

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

   if 1.0000000000000001e130 < (*.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 12.9%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 65.1%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 62.5%

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

   9. Applied rewrites 68.8%

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

   11. Applied rewrites 71.7%

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

Alternative 5: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot \frac{\ell - \frac{0.125}{d\_m} \cdot \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)}{d\_m}}{\ell}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+130}:\\ \;\;\;\;{\left(\frac{d\_m}{\ell} \cdot \frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l)))))
        (t_1
         (*
          (* (sqrt (/ 1.0 (* l h))) d_m)
          (/ (- l (* (/ 0.125 d_m) (/ (* (* M D) (* (* M D) h)) d_m))) l))))
   (if (<= t_0 5e-161)
     t_1
     (if (<= t_0 1e+130)
       (* (pow (* (/ d_m l) (/ d_m h)) (/ 1.0 2.0)) 1.0)
       t_1))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double t_1 = (sqrt((1.0 / (l * h))) * d_m) * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l);
	double tmp;
	if (t_0 <= 5e-161) {
		tmp = t_1;
	} else if (t_0 <= 1e+130) {
		tmp = pow(((d_m / l) * (d_m / h)), (1.0 / 2.0)) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

d_m =     private
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_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))
    t_1 = (sqrt((1.0d0 / (l * h))) * d_m) * ((l - ((0.125d0 / d_m) * (((m * d) * ((m * d) * h)) / d_m))) / l)
    if (t_0 <= 5d-161) then
        tmp = t_1
    else if (t_0 <= 1d+130) then
        tmp = (((d_m / l) * (d_m / h)) ** (1.0d0 / 2.0d0)) * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)));
	double t_1 = (Math.sqrt((1.0 / (l * h))) * d_m) * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l);
	double tmp;
	if (t_0 <= 5e-161) {
		tmp = t_1;
	} else if (t_0 <= 1e+130) {
		tmp = Math.pow(((d_m / l) * (d_m / h)), (1.0 / 2.0)) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = (math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))
	t_1 = (math.sqrt((1.0 / (l * h))) * d_m) * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l)
	tmp = 0
	if t_0 <= 5e-161:
		tmp = t_1
	elif t_0 <= 1e+130:
		tmp = math.pow(((d_m / l) * (d_m / h)), (1.0 / 2.0)) * 1.0
	else:
		tmp = t_1
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * Float64(Float64(l - Float64(Float64(0.125 / d_m) * Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * h)) / d_m))) / l))
	tmp = 0.0
	if (t_0 <= 5e-161)
		tmp = t_1;
	elseif (t_0 <= 1e+130)
		tmp = Float64((Float64(Float64(d_m / l) * Float64(d_m / h)) ^ Float64(1.0 / 2.0)) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = (((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	t_1 = (sqrt((1.0 / (l * h))) * d_m) * ((l - ((0.125 / d_m) * (((M * D) * ((M * D) * h)) / d_m))) / l);
	tmp = 0.0;
	if (t_0 <= 5e-161)
		tmp = t_1;
	elseif (t_0 <= 1e+130)
		tmp = (((d_m / l) * (d_m / h)) ^ (1.0 / 2.0)) * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(N[(l - N[(N[(0.125 / d$95$m), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-161], t$95$1, If[LessEqual[t$95$0, 1e+130], N[(N[Power[N[(N[(d$95$m / l), $MachinePrecision] * N[(d$95$m / h), $MachinePrecision]), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.9999999999999999e-161 or 1.0000000000000001e130 < (*.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 29.5%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 67.8%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 60.1%

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

   9. Applied rewrites 69.8%

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

   11. Applied rewrites 72.3%

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

   if 4.9999999999999999e-161 < (*.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)))) < 1.0000000000000001e130

   1. Initial program 99.0%

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

   3. Applied rewrites 57.6%

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 97.7%

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

Alternative 6: 57.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left(M \cdot D\right) \cdot \left(M \cdot D\right)\\ t_1 := \sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\\ \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d\_m\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+143}:\\ \;\;\;\;t\_1 \cdot \frac{\ell - \frac{0.125}{d\_m} \cdot \left(t\_0 \cdot \frac{h}{d\_m}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{-0.125}{d\_m} \cdot \frac{t\_0 \cdot h}{d\_m}}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* (* M D) (* M D))) (t_1 (* (sqrt (/ 1.0 (* l h))) d_m)))
   (if (<= (* M D) 5e-149)
     (* (/ (sqrt 1.0) (sqrt (* l h))) d_m)
     (if (<= (* M D) 5e+143)
       (* t_1 (/ (- l (* (/ 0.125 d_m) (* t_0 (/ h d_m)))) l))
       (* t_1 (/ (* (/ -0.125 d_m) (/ (* t_0 h) d_m)) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (M * D) * (M * D);
	double t_1 = sqrt((1.0 / (l * h))) * d_m;
	double tmp;
	if ((M * D) <= 5e-149) {
		tmp = (sqrt(1.0) / sqrt((l * h))) * d_m;
	} else if ((M * D) <= 5e+143) {
		tmp = t_1 * ((l - ((0.125 / d_m) * (t_0 * (h / d_m)))) / l);
	} else {
		tmp = t_1 * (((-0.125 / d_m) * ((t_0 * h) / d_m)) / l);
	}
	return tmp;
}

Fortran

d_m =     private
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_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (m * d) * (m * d)
    t_1 = sqrt((1.0d0 / (l * h))) * d_m
    if ((m * d) <= 5d-149) then
        tmp = (sqrt(1.0d0) / sqrt((l * h))) * d_m
    else if ((m * d) <= 5d+143) then
        tmp = t_1 * ((l - ((0.125d0 / d_m) * (t_0 * (h / d_m)))) / l)
    else
        tmp = t_1 * ((((-0.125d0) / d_m) * ((t_0 * h) / d_m)) / l)
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (M * D) * (M * D);
	double t_1 = Math.sqrt((1.0 / (l * h))) * d_m;
	double tmp;
	if ((M * D) <= 5e-149) {
		tmp = (Math.sqrt(1.0) / Math.sqrt((l * h))) * d_m;
	} else if ((M * D) <= 5e+143) {
		tmp = t_1 * ((l - ((0.125 / d_m) * (t_0 * (h / d_m)))) / l);
	} else {
		tmp = t_1 * (((-0.125 / d_m) * ((t_0 * h) / d_m)) / l);
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = (M * D) * (M * D)
	t_1 = math.sqrt((1.0 / (l * h))) * d_m
	tmp = 0
	if (M * D) <= 5e-149:
		tmp = (math.sqrt(1.0) / math.sqrt((l * h))) * d_m
	elif (M * D) <= 5e+143:
		tmp = t_1 * ((l - ((0.125 / d_m) * (t_0 * (h / d_m)))) / l)
	else:
		tmp = t_1 * (((-0.125 / d_m) * ((t_0 * h) / d_m)) / l)
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(Float64(M * D) * Float64(M * D))
	t_1 = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m)
	tmp = 0.0
	if (Float64(M * D) <= 5e-149)
		tmp = Float64(Float64(sqrt(1.0) / sqrt(Float64(l * h))) * d_m);
	elseif (Float64(M * D) <= 5e+143)
		tmp = Float64(t_1 * Float64(Float64(l - Float64(Float64(0.125 / d_m) * Float64(t_0 * Float64(h / d_m)))) / l));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(-0.125 / d_m) * Float64(Float64(t_0 * h) / d_m)) / l));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = (M * D) * (M * D);
	t_1 = sqrt((1.0 / (l * h))) * d_m;
	tmp = 0.0;
	if ((M * D) <= 5e-149)
		tmp = (sqrt(1.0) / sqrt((l * h))) * d_m;
	elseif ((M * D) <= 5e+143)
		tmp = t_1 * ((l - ((0.125 / d_m) * (t_0 * (h / d_m)))) / l);
	else
		tmp = t_1 * (((-0.125 / d_m) * ((t_0 * h) / d_m)) / l);
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 5e-149], N[(N[(N[Sqrt[1.0], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], If[LessEqual[N[(M * D), $MachinePrecision], 5e+143], N[(t$95$1 * N[(N[(l - N[(N[(0.125 / d$95$m), $MachinePrecision] * N[(t$95$0 * N[(h / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(-0.125 / d$95$m), $MachinePrecision] * N[(N[(t$95$0 * h), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 4.99999999999999968e-149

   1. Initial program 37.3%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 50.3%

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

   6. Applied rewrites 50.5%

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

   if 4.99999999999999968e-149 < (*.f64 M D) < 5.00000000000000012e143

   1. Initial program 34.5%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 71.0%

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

   11. Applied rewrites 73.4%

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

   if 5.00000000000000012e143 < (*.f64 M D)

   1. Initial program 35.6%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 52.0%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 51.1%

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

   12. Applied rewrites 63.1%

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

Alternative 7: 55.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\\ t_1 := \left(M \cdot D\right) \cdot \left(M \cdot D\right)\\ \mathbf{if}\;M \cdot D \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d\_m\\ \mathbf{elif}\;M \cdot D \leq 10^{+35}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\frac{-0.125 \cdot t\_1}{\left(d\_m \cdot d\_m\right) \cdot \ell}, h, 1\right)\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\frac{t\_1 \cdot -0.125}{d\_m} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{-0.125}{d\_m} \cdot \frac{t\_1 \cdot h}{d\_m}}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ 1.0 (* l h))) d_m)) (t_1 (* (* M D) (* M D))))
   (if (<= (* M D) 2.5e-141)
     (* (/ (sqrt 1.0) (sqrt (* l h))) d_m)
     (if (<= (* M D) 1e+35)
       (* t_0 (fma (/ (* -0.125 t_1) (* (* d_m d_m) l)) h 1.0))
       (if (<= (* M D) 5e+260)
         (* (/ (* t_1 -0.125) d_m) (sqrt (/ h (* (* l l) l))))
         (* t_0 (/ (* (/ -0.125 d_m) (/ (* t_1 h) d_m)) l)))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h))) * d_m;
	double t_1 = (M * D) * (M * D);
	double tmp;
	if ((M * D) <= 2.5e-141) {
		tmp = (sqrt(1.0) / sqrt((l * h))) * d_m;
	} else if ((M * D) <= 1e+35) {
		tmp = t_0 * fma(((-0.125 * t_1) / ((d_m * d_m) * l)), h, 1.0);
	} else if ((M * D) <= 5e+260) {
		tmp = ((t_1 * -0.125) / d_m) * sqrt((h / ((l * l) * l)));
	} else {
		tmp = t_0 * (((-0.125 / d_m) * ((t_1 * h) / d_m)) / l);
	}
	return tmp;
}

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m)
	t_1 = Float64(Float64(M * D) * Float64(M * D))
	tmp = 0.0
	if (Float64(M * D) <= 2.5e-141)
		tmp = Float64(Float64(sqrt(1.0) / sqrt(Float64(l * h))) * d_m);
	elseif (Float64(M * D) <= 1e+35)
		tmp = Float64(t_0 * fma(Float64(Float64(-0.125 * t_1) / Float64(Float64(d_m * d_m) * l)), h, 1.0));
	elseif (Float64(M * D) <= 5e+260)
		tmp = Float64(Float64(Float64(t_1 * -0.125) / d_m) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(-0.125 / d_m) * Float64(Float64(t_1 * h) / d_m)) / l));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 2.5e-141], N[(N[(N[Sqrt[1.0], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], If[LessEqual[N[(M * D), $MachinePrecision], 1e+35], N[(t$95$0 * N[(N[(N[(-0.125 * t$95$1), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(M * D), $MachinePrecision], 5e+260], N[(N[(N[(t$95$1 * -0.125), $MachinePrecision] / d$95$m), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(-0.125 / d$95$m), $MachinePrecision] * N[(N[(t$95$1 * h), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 M D) < 2.5e-141

   1. Initial program 37.3%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 50.3%

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

   6. Applied rewrites 50.5%

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

   if 2.5e-141 < (*.f64 M D) < 9.9999999999999997e34

   1. Initial program 33.3%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 66.5%

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

   5. Taylor expanded in h around inf

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

   7. Applied rewrites 66.5%

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

   if 9.9999999999999997e34 < (*.f64 M D) < 4.9999999999999996e260

   1. Initial program 35.5%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 45.0%

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

   if 4.9999999999999996e260 < (*.f64 M D)

   1. Initial program 36.2%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 56.8%

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

   12. Applied rewrites 73.0%

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

Alternative 8: 55.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\\ t_1 := \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h\\ \mathbf{if}\;M \cdot D \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d\_m\\ \mathbf{elif}\;M \cdot D \leq 10^{+100}:\\ \;\;\;\;t\_0 \cdot \frac{\mathsf{fma}\left(\frac{t\_1}{d\_m \cdot d\_m}, -0.125, \ell\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{-0.125}{d\_m} \cdot \frac{t\_1}{d\_m}}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ 1.0 (* l h))) d_m)) (t_1 (* (* (* M D) (* M D)) h)))
   (if (<= (* M D) 2.5e-141)
     (* (/ (sqrt 1.0) (sqrt (* l h))) d_m)
     (if (<= (* M D) 1e+100)
       (* t_0 (/ (fma (/ t_1 (* d_m d_m)) -0.125 l) l))
       (* t_0 (/ (* (/ -0.125 d_m) (/ t_1 d_m)) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h))) * d_m;
	double t_1 = ((M * D) * (M * D)) * h;
	double tmp;
	if ((M * D) <= 2.5e-141) {
		tmp = (sqrt(1.0) / sqrt((l * h))) * d_m;
	} else if ((M * D) <= 1e+100) {
		tmp = t_0 * (fma((t_1 / (d_m * d_m)), -0.125, l) / l);
	} else {
		tmp = t_0 * (((-0.125 / d_m) * (t_1 / d_m)) / l);
	}
	return tmp;
}

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m)
	t_1 = Float64(Float64(Float64(M * D) * Float64(M * D)) * h)
	tmp = 0.0
	if (Float64(M * D) <= 2.5e-141)
		tmp = Float64(Float64(sqrt(1.0) / sqrt(Float64(l * h))) * d_m);
	elseif (Float64(M * D) <= 1e+100)
		tmp = Float64(t_0 * Float64(fma(Float64(t_1 / Float64(d_m * d_m)), -0.125, l) / l));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(-0.125 / d_m) * Float64(t_1 / d_m)) / l));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 2.5e-141], N[(N[(N[Sqrt[1.0], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], If[LessEqual[N[(M * D), $MachinePrecision], 1e+100], N[(t$95$0 * N[(N[(N[(t$95$1 / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] * -0.125 + l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(-0.125 / d$95$m), $MachinePrecision] * N[(t$95$1 / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 2.5e-141

   1. Initial program 37.3%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 50.3%

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

   6. Applied rewrites 50.5%

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

   if 2.5e-141 < (*.f64 M D) < 1.00000000000000002e100

   1. Initial program 34.4%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 67.2%

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

   9. Applied rewrites 67.2%

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

   if 1.00000000000000002e100 < (*.f64 M D)

   1. Initial program 35.4%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 49.1%

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

   12. Applied rewrites 60.3%

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

Alternative 9: 53.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\\ t_1 := \left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h\\ \mathbf{if}\;M \cdot D \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d\_m\\ \mathbf{elif}\;M \cdot D \leq 10^{+100}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-0.125, \frac{t\_1}{\left(d\_m \cdot d\_m\right) \cdot \ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{-0.125}{d\_m} \cdot \frac{t\_1}{d\_m}}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ 1.0 (* l h))) d_m)) (t_1 (* (* (* M D) (* M D)) h)))
   (if (<= (* M D) 2.5e-141)
     (* (/ (sqrt 1.0) (sqrt (* l h))) d_m)
     (if (<= (* M D) 1e+100)
       (* t_0 (fma -0.125 (/ t_1 (* (* d_m d_m) l)) 1.0))
       (* t_0 (/ (* (/ -0.125 d_m) (/ t_1 d_m)) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = sqrt((1.0 / (l * h))) * d_m;
	double t_1 = ((M * D) * (M * D)) * h;
	double tmp;
	if ((M * D) <= 2.5e-141) {
		tmp = (sqrt(1.0) / sqrt((l * h))) * d_m;
	} else if ((M * D) <= 1e+100) {
		tmp = t_0 * fma(-0.125, (t_1 / ((d_m * d_m) * l)), 1.0);
	} else {
		tmp = t_0 * (((-0.125 / d_m) * (t_1 / d_m)) / l);
	}
	return tmp;
}

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m)
	t_1 = Float64(Float64(Float64(M * D) * Float64(M * D)) * h)
	tmp = 0.0
	if (Float64(M * D) <= 2.5e-141)
		tmp = Float64(Float64(sqrt(1.0) / sqrt(Float64(l * h))) * d_m);
	elseif (Float64(M * D) <= 1e+100)
		tmp = Float64(t_0 * fma(-0.125, Float64(t_1 / Float64(Float64(d_m * d_m) * l)), 1.0));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(-0.125 / d_m) * Float64(t_1 / d_m)) / l));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 2.5e-141], N[(N[(N[Sqrt[1.0], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], If[LessEqual[N[(M * D), $MachinePrecision], 1e+100], N[(t$95$0 * N[(-0.125 * N[(t$95$1 / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(-0.125 / d$95$m), $MachinePrecision] * N[(t$95$1 / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 2.5e-141

   1. Initial program 37.3%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 50.3%

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

   6. Applied rewrites 50.5%

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

   if 2.5e-141 < (*.f64 M D) < 1.00000000000000002e100

   1. Initial program 34.4%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 67.2%

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

   if 1.00000000000000002e100 < (*.f64 M D)

   1. Initial program 35.4%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 49.1%

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

   12. Applied rewrites 60.3%

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

Alternative 10: 51.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left(M \cdot D\right) \cdot \left(M \cdot D\right)\\ \mathbf{if}\;M \cdot D \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot d\_m\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\frac{t\_0 \cdot -0.125}{d\_m} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot \frac{\frac{-0.125}{d\_m} \cdot \frac{t\_0 \cdot h}{d\_m}}{\ell}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* (* M D) (* M D))))
   (if (<= (* M D) 4e+30)
     (* (sqrt (/ (/ 1.0 h) l)) d_m)
     (if (<= (* M D) 5e+260)
       (* (/ (* t_0 -0.125) d_m) (sqrt (/ h (* (* l l) l))))
       (*
        (* (sqrt (/ 1.0 (* l h))) d_m)
        (/ (* (/ -0.125 d_m) (/ (* t_0 h) d_m)) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (M * D) * (M * D);
	double tmp;
	if ((M * D) <= 4e+30) {
		tmp = sqrt(((1.0 / h) / l)) * d_m;
	} else if ((M * D) <= 5e+260) {
		tmp = ((t_0 * -0.125) / d_m) * sqrt((h / ((l * l) * l)));
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (((-0.125 / d_m) * ((t_0 * h) / d_m)) / l);
	}
	return tmp;
}

Fortran

d_m =     private
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_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m * d) * (m * d)
    if ((m * d) <= 4d+30) then
        tmp = sqrt(((1.0d0 / h) / l)) * d_m
    else if ((m * d) <= 5d+260) then
        tmp = ((t_0 * (-0.125d0)) / d_m) * sqrt((h / ((l * l) * l)))
    else
        tmp = (sqrt((1.0d0 / (l * h))) * d_m) * ((((-0.125d0) / d_m) * ((t_0 * h) / d_m)) / l)
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (M * D) * (M * D);
	double tmp;
	if ((M * D) <= 4e+30) {
		tmp = Math.sqrt(((1.0 / h) / l)) * d_m;
	} else if ((M * D) <= 5e+260) {
		tmp = ((t_0 * -0.125) / d_m) * Math.sqrt((h / ((l * l) * l)));
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d_m) * (((-0.125 / d_m) * ((t_0 * h) / d_m)) / l);
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = (M * D) * (M * D)
	tmp = 0
	if (M * D) <= 4e+30:
		tmp = math.sqrt(((1.0 / h) / l)) * d_m
	elif (M * D) <= 5e+260:
		tmp = ((t_0 * -0.125) / d_m) * math.sqrt((h / ((l * l) * l)))
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d_m) * (((-0.125 / d_m) * ((t_0 * h) / d_m)) / l)
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(Float64(M * D) * Float64(M * D))
	tmp = 0.0
	if (Float64(M * D) <= 4e+30)
		tmp = Float64(sqrt(Float64(Float64(1.0 / h) / l)) * d_m);
	elseif (Float64(M * D) <= 5e+260)
		tmp = Float64(Float64(Float64(t_0 * -0.125) / d_m) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * Float64(Float64(Float64(-0.125 / d_m) * Float64(Float64(t_0 * h) / d_m)) / l));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = (M * D) * (M * D);
	tmp = 0.0;
	if ((M * D) <= 4e+30)
		tmp = sqrt(((1.0 / h) / l)) * d_m;
	elseif ((M * D) <= 5e+260)
		tmp = ((t_0 * -0.125) / d_m) * sqrt((h / ((l * l) * l)));
	else
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (((-0.125 / d_m) * ((t_0 * h) / d_m)) / l);
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 4e+30], N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision], If[LessEqual[N[(M * D), $MachinePrecision], 5e+260], N[(N[(N[(t$95$0 * -0.125), $MachinePrecision] / d$95$m), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(N[(N[(-0.125 / d$95$m), $MachinePrecision] * N[(N[(t$95$0 * h), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 4.0000000000000001e30

   1. Initial program 36.6%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 50.5%

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

   6. Applied rewrites 50.9%

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

   if 4.0000000000000001e30 < (*.f64 M D) < 4.9999999999999996e260

   1. Initial program 35.7%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 44.9%

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

   if 4.9999999999999996e260 < (*.f64 M D)

   1. Initial program 36.2%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 56.8%

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

   12. Applied rewrites 73.0%

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

Alternative 11: 50.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot -0.125}{d\_m} \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<= (* M D) 4e+30)
   (* (sqrt (/ (/ 1.0 h) l)) d_m)
   (* (/ (* (* (* M D) (* M D)) -0.125) d_m) (sqrt (/ h (* (* l l) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if ((M * D) <= 4e+30) {
		tmp = sqrt(((1.0 / h) / l)) * d_m;
	} else {
		tmp = ((((M * D) * (M * D)) * -0.125) / d_m) * sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Fortran

d_m =     private
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_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((m * d) <= 4d+30) then
        tmp = sqrt(((1.0d0 / h) / l)) * d_m
    else
        tmp = ((((m * d) * (m * d)) * (-0.125d0)) / d_m) * sqrt((h / ((l * l) * l)))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if ((M * D) <= 4e+30) {
		tmp = Math.sqrt(((1.0 / h) / l)) * d_m;
	} else {
		tmp = ((((M * D) * (M * D)) * -0.125) / d_m) * Math.sqrt((h / ((l * l) * l)));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if (M * D) <= 4e+30:
		tmp = math.sqrt(((1.0 / h) / l)) * d_m
	else:
		tmp = ((((M * D) * (M * D)) * -0.125) / d_m) * math.sqrt((h / ((l * l) * l)))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (Float64(M * D) <= 4e+30)
		tmp = Float64(sqrt(Float64(Float64(1.0 / h) / l)) * d_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * -0.125) / d_m) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if ((M * D) <= 4e+30)
		tmp = sqrt(((1.0 / h) / l)) * d_m;
	else
		tmp = ((((M * D) * (M * D)) * -0.125) / d_m) * sqrt((h / ((l * l) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[N[(M * D), $MachinePrecision], 4e+30], N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision], N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / d$95$m), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 4.0000000000000001e30

   1. Initial program 36.6%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 50.5%

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

   6. Applied rewrites 50.9%

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

   if 4.0000000000000001e30 < (*.f64 M D)

   1. Initial program 35.9%

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

   2. Taylor expanded in d around 0

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

   4. Applied rewrites 50.3%

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

Alternative 12: 47.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot d\_m\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))
      -5e-252)
   (* (sqrt (/ 1.0 (* l h))) (- d_m))
   (* (sqrt (/ (/ 1.0 h) l)) d_m)))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e-252) {
		tmp = sqrt((1.0 / (l * h))) * -d_m;
	} else {
		tmp = sqrt(((1.0 / h) / l)) * d_m;
	}
	return tmp;
}

Fortran

d_m =     private
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_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))) <= (-5d-252)) then
        tmp = sqrt((1.0d0 / (l * h))) * -d_m
    else
        tmp = sqrt(((1.0d0 / h) / l)) * d_m
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e-252) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d_m;
	} else {
		tmp = Math.sqrt(((1.0 / h) / l)) * d_m;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -5e-252:
		tmp = math.sqrt((1.0 / (l * h))) * -d_m
	else:
		tmp = math.sqrt(((1.0 / h) / l)) * d_m
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l)))) <= -5e-252)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d_m));
	else
		tmp = Float64(sqrt(Float64(Float64(1.0 / h) / l)) * d_m);
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)))) <= -5e-252)
		tmp = sqrt((1.0 / (l * h))) * -d_m;
	else
		tmp = sqrt(((1.0 / h) / l)) * d_m;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-252], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d$95$m)), $MachinePrecision], N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000008e-252

   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 l around -inf

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

   4. Applied rewrites 22.0%

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

   if -5.00000000000000008e-252 < (*.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 26.1%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 52.3%

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

Alternative 13: 43.5% accurate, 7.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot d\_m \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D) :precision binary64 (* (sqrt (/ (/ 1.0 h) l)) d_m))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

2. Taylor expanded in d around inf

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

4. Applied rewrites 43.2%

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

6. Applied rewrites 43.5%

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

Alternative 14: 43.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D) :precision binary64 (* (sqrt (/ 1.0 (* l h))) d_m))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

2. Taylor expanded in d around inf

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

4. Applied rewrites 43.2%

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

Reproduce

herbie shell --seed 2025129 
(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)))))