Henrywood and Agarwal, Equation (12)

Percentage Accurate: 35.2% → 76.8%

Time: 8.2s

Alternatives: 18

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: 35.2% 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: 76.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{D}{d\_m + d\_m}\\ t_1 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_2 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_1\\ t_3 := \sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;t\_3 \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(1 - \frac{\left(\left(M \cdot \left(t\_0 \cdot \left(t\_0 \cdot M\right)\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (/ D (+ d_m d_m)))
        (t_1
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))
        (t_2
         (* (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0))) t_1))
        (t_3 (* (sqrt (/ 1.0 (* l h))) d_m)))
   (if (<= t_2 0.0)
     (* t_3 t_1)
     (if (<= t_2 5e+234)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m)
       (* t_3 (- 1.0 (/ (* (* (* M (* t_0 (* t_0 M))) 0.5) h) l)))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = D / (d_m + d_m);
	double t_1 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l));
	double t_2 = (pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * t_1;
	double t_3 = sqrt((1.0 / (l * h))) * d_m;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_3 * t_1;
	} else if (t_2 <= 5e+234) {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	} else {
		tmp = t_3 * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = d / (d_m + d_m)
    t_1 = 1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l))
    t_2 = (((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * t_1
    t_3 = sqrt((1.0d0 / (l * h))) * d_m
    if (t_2 <= 0.0d0) then
        tmp = t_3 * t_1
    else if (t_2 <= 5d+234) then
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d_m
    else
        tmp = t_3 * (1.0d0 - ((((m * (t_0 * (t_0 * m))) * 0.5d0) * h) / 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 = D / (d_m + d_m);
	double t_1 = 1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l));
	double t_2 = (Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * t_1;
	double t_3 = Math.sqrt((1.0 / (l * h))) * d_m;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_3 * t_1;
	} else if (t_2 <= 5e+234) {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m;
	} else {
		tmp = t_3 * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = D / (d_m + d_m)
	t_1 = 1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l))
	t_2 = (math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * t_1
	t_3 = math.sqrt((1.0 / (l * h))) * d_m
	tmp = 0
	if t_2 <= 0.0:
		tmp = t_3 * t_1
	elif t_2 <= 5e+234:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m
	else:
		tmp = t_3 * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(D / Float64(d_m + d_m))
	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)))
	t_2 = Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * t_1)
	t_3 = Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(t_3 * t_1);
	elseif (t_2 <= 5e+234)
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m);
	else
		tmp = Float64(t_3 * Float64(1.0 - Float64(Float64(Float64(Float64(M * Float64(t_0 * Float64(t_0 * M))) * 0.5) * h) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = D / (d_m + d_m);
	t_1 = 1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l));
	t_2 = (((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * t_1;
	t_3 = sqrt((1.0 / (l * h))) * d_m;
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = t_3 * t_1;
	elseif (t_2 <= 5e+234)
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	else
		tmp = t_3 * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / 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[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{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]}, Block[{t$95$2 = 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] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(t$95$3 * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 5e+234], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], N[(t$95$3 * N[(1.0 - N[(N[(N[(N[(M * N[(t$95$0 * N[(t$95$0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000003e234

   1. Initial program 35.2%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 24.8

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

   8. Applied rewrites 24.8%

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

   if 5.0000000000000003e234 < (*.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 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. count-2-rev N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 72.6%

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

Alternative 2: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{D}{d\_m + d\_m}\\ t_2 := 1 - \frac{\left(\left(M \cdot \left(t\_1 \cdot \left(t\_1 \cdot M\right)\right)\right) \cdot 0.5\right) \cdot h}{\ell}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}} \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot t\_2\\ \end{array} \end{array} \]

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 (/ D (+ d_m d_m)))
        (t_2 (- 1.0 (/ (* (* (* M (* t_1 (* t_1 M))) 0.5) h) l))))
   (if (<= t_0 0.0)
     (* (/ d_m (sqrt (* l h))) t_2)
     (if (<= t_0 5e+234)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m)
       (* (* (sqrt (/ 1.0 (* l h))) d_m) t_2)))))

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 = D / (d_m + d_m);
	double t_2 = 1.0 - ((((M * (t_1 * (t_1 * M))) * 0.5) * h) / l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (d_m / sqrt((l * h))) * t_2;
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * t_2;
	}
	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) :: t_2
    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 = d / (d_m + d_m)
    t_2 = 1.0d0 - ((((m * (t_1 * (t_1 * m))) * 0.5d0) * h) / l)
    if (t_0 <= 0.0d0) then
        tmp = (d_m / sqrt((l * h))) * t_2
    else if (t_0 <= 5d+234) then
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d_m
    else
        tmp = (sqrt((1.0d0 / (l * h))) * d_m) * t_2
    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 = D / (d_m + d_m);
	double t_2 = 1.0 - ((((M * (t_1 * (t_1 * M))) * 0.5) * h) / l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (d_m / Math.sqrt((l * h))) * t_2;
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m;
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d_m) * t_2;
	}
	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 = D / (d_m + d_m)
	t_2 = 1.0 - ((((M * (t_1 * (t_1 * M))) * 0.5) * h) / l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = (d_m / math.sqrt((l * h))) * t_2
	elif t_0 <= 5e+234:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d_m) * t_2
	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(D / Float64(d_m + d_m))
	t_2 = Float64(1.0 - Float64(Float64(Float64(Float64(M * Float64(t_1 * Float64(t_1 * M))) * 0.5) * h) / l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(d_m / sqrt(Float64(l * h))) * t_2);
	elseif (t_0 <= 5e+234)
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * t_2);
	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 = D / (d_m + d_m);
	t_2 = 1.0 - ((((M * (t_1 * (t_1 * M))) * 0.5) * h) / l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (d_m / sqrt((l * h))) * t_2;
	elseif (t_0 <= 5e+234)
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	else
		tmp = (sqrt((1.0 / (l * h))) * d_m) * t_2;
	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[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[(N[(N[(M * N[(t$95$1 * N[(t$95$1 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 5e+234], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. count-2-rev N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-*.f64 72.9

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in d around 0

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

   11. Applied rewrites 72.9%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000003e234

   1. Initial program 35.2%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 24.8

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

   8. Applied rewrites 24.8%

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

   if 5.0000000000000003e234 < (*.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 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. count-2-rev N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 72.6%

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

Alternative 3: 76.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{D}{d\_m + d\_m}\\ t_2 := \frac{d\_m}{\sqrt{\ell \cdot h}} \cdot \left(1 - \frac{\left(\left(M \cdot \left(t\_1 \cdot \left(t\_1 \cdot M\right)\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (/ D (+ d_m d_m)))
        (t_2
         (*
          (/ d_m (sqrt (* l h)))
          (- 1.0 (/ (* (* (* M (* t_1 (* t_1 M))) 0.5) h) l)))))
   (if (<= t_0 0.0)
     t_2
     (if (<= t_0 5e+234) (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m) t_2))))

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 = D / (d_m + d_m);
	double t_2 = (d_m / sqrt((l * h))) * (1.0 - ((((M * (t_1 * (t_1 * M))) * 0.5) * h) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    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 = d / (d_m + d_m)
    t_2 = (d_m / sqrt((l * h))) * (1.0d0 - ((((m * (t_1 * (t_1 * m))) * 0.5d0) * h) / l))
    if (t_0 <= 0.0d0) then
        tmp = t_2
    else if (t_0 <= 5d+234) then
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d_m
    else
        tmp = t_2
    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 = D / (d_m + d_m);
	double t_2 = (d_m / Math.sqrt((l * h))) * (1.0 - ((((M * (t_1 * (t_1 * M))) * 0.5) * h) / l));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m;
	} else {
		tmp = t_2;
	}
	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 = D / (d_m + d_m)
	t_2 = (d_m / math.sqrt((l * h))) * (1.0 - ((((M * (t_1 * (t_1 * M))) * 0.5) * h) / l))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_2
	elif t_0 <= 5e+234:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m
	else:
		tmp = t_2
	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(D / Float64(d_m + d_m))
	t_2 = Float64(Float64(d_m / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(Float64(M * Float64(t_1 * Float64(t_1 * M))) * 0.5) * h) / l)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+234)
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m);
	else
		tmp = t_2;
	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 = D / (d_m + d_m);
	t_2 = (d_m / sqrt((l * h))) * (1.0 - ((((M * (t_1 * (t_1 * M))) * 0.5) * h) / l));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+234)
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	else
		tmp = t_2;
	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[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M * N[(t$95$1 * N[(t$95$1 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 5e+234], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or 5.0000000000000003e234 < (*.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 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. count-2-rev N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-*.f64 72.9

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in d around 0

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

   11. Applied rewrites 72.9%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000003e234

   1. Initial program 35.2%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 24.8

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

   8. Applied rewrites 24.8%

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

Alternative 4: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{D}{d\_m + d\_m}\\ t_1 := M \cdot t\_0\\ t_2 := {\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ \mathbf{if}\;t\_2 \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+234}:\\ \;\;\;\;t\_2 \cdot \left(1 - \frac{\left(\left(t\_1 \cdot t\_1\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot \left(1 - \frac{\left(\left(M \cdot \left(t\_0 \cdot \left(t\_0 \cdot M\right)\right)\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (/ D (+ d_m d_m)))
        (t_1 (* M t_0))
        (t_2 (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))))
   (if (<=
        (*
         t_2
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))
        5e+234)
     (* t_2 (- 1.0 (/ (* (* (* t_1 t_1) 0.5) h) l)))
     (*
      (* (sqrt (/ 1.0 (* l h))) d_m)
      (- 1.0 (/ (* (* (* M (* t_0 (* t_0 M))) 0.5) h) l))))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = D / (d_m + d_m);
	double t_1 = M * t_0;
	double t_2 = pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0));
	double tmp;
	if ((t_2 * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= 5e+234) {
		tmp = t_2 * (1.0 - ((((t_1 * t_1) * 0.5) * h) / l));
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / 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) :: t_2
    real(8) :: tmp
    t_0 = d / (d_m + d_m)
    t_1 = m * t_0
    t_2 = ((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))
    if ((t_2 * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))) <= 5d+234) then
        tmp = t_2 * (1.0d0 - ((((t_1 * t_1) * 0.5d0) * h) / l))
    else
        tmp = (sqrt((1.0d0 / (l * h))) * d_m) * (1.0d0 - ((((m * (t_0 * (t_0 * m))) * 0.5d0) * h) / 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 = D / (d_m + d_m);
	double t_1 = M * t_0;
	double t_2 = Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0));
	double tmp;
	if ((t_2 * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= 5e+234) {
		tmp = t_2 * (1.0 - ((((t_1 * t_1) * 0.5) * h) / l));
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = D / (d_m + d_m)
	t_1 = M * t_0
	t_2 = math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))
	tmp = 0
	if (t_2 * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= 5e+234:
		tmp = t_2 * (1.0 - ((((t_1 * t_1) * 0.5) * h) / l))
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(D / Float64(d_m + d_m))
	t_1 = Float64(M * t_0)
	t_2 = Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0)))
	tmp = 0.0
	if (Float64(t_2 * 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+234)
		tmp = Float64(t_2 * Float64(1.0 - Float64(Float64(Float64(Float64(t_1 * t_1) * 0.5) * h) / l)));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * Float64(1.0 - Float64(Float64(Float64(Float64(M * Float64(t_0 * Float64(t_0 * M))) * 0.5) * h) / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	t_0 = D / (d_m + d_m);
	t_1 = M * t_0;
	t_2 = ((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0));
	tmp = 0.0;
	if ((t_2 * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)))) <= 5e+234)
		tmp = t_2 * (1.0 - ((((t_1 * t_1) * 0.5) * h) / l));
	else
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((M * (t_0 * (t_0 * M))) * 0.5) * h) / 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[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M * t$95$0), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[N[(t$95$2 * 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+234], N[(t$95$2 * N[(1.0 - N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(M * N[(t$95$0 * N[(t$95$0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000003e234

   1. Initial program 35.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 36.0%

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

   if 5.0000000000000003e234 < (*.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 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. count-2-rev N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 72.6%

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

Alternative 5: 70.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \left(1 - \left(\left(\left(\frac{D \cdot M}{d\_m + d\_m} \cdot \frac{D}{d\_m + d\_m}\right) \cdot M\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \cdot \frac{d\_m}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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
         (*
          (-
           1.0
           (*
            (* (* (* (/ (* D M) (+ d_m d_m)) (/ D (+ d_m d_m))) M) 0.5)
            (/ h l)))
          (/ d_m (sqrt (* l h))))))
   (if (<= t_0 0.0)
     t_1
     (if (<= t_0 5e+234) (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m) 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 = (1.0 - ((((((D * M) / (d_m + d_m)) * (D / (d_m + d_m))) * M) * 0.5) * (h / l))) * (d_m / sqrt((l * h)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	} 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 = (1.0d0 - ((((((d * m) / (d_m + d_m)) * (d / (d_m + d_m))) * m) * 0.5d0) * (h / l))) * (d_m / sqrt((l * h)))
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+234) then
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d_m
    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 = (1.0 - ((((((D * M) / (d_m + d_m)) * (D / (d_m + d_m))) * M) * 0.5) * (h / l))) * (d_m / Math.sqrt((l * h)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m;
	} 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 = (1.0 - ((((((D * M) / (d_m + d_m)) * (D / (d_m + d_m))) * M) * 0.5) * (h / l))) * (d_m / math.sqrt((l * h)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+234:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m
	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(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(D * M) / Float64(d_m + d_m)) * Float64(D / Float64(d_m + d_m))) * M) * 0.5) * Float64(h / l))) * Float64(d_m / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+234)
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m);
	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 = (1.0 - ((((((D * M) / (d_m + d_m)) * (D / (d_m + d_m))) * M) * 0.5) * (h / l))) * (d_m / sqrt((l * h)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+234)
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	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[(1.0 - N[(N[(N[(N[(N[(N[(D * M), $MachinePrecision] / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+234], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0 or 5.0000000000000003e234 < (*.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 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. count-2-rev N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-*.f64 72.9

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

   8. Applied rewrites 72.9%

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

   10. Applied rewrites 67.4%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000003e234

   1. Initial program 35.2%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 24.8

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

   8. Applied rewrites 24.8%

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

Alternative 6: 62.5% accurate, 0.4× speedup

Math

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

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))))))
   (if (<= t_0 0.0)
     (* (* -0.125 (* (* (* D D) M) (/ M d_m))) (sqrt (/ h (* (* l l) l))))
     (if (<= t_0 INFINITY)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m)
       (*
        (* (sqrt (/ 1.0 (* l h))) d_m)
        (*
         (fma (/ (* (* (* M M) D) D) (* (* d_m d_m) l)) -0.125 (/ 1.0 h))
         h))))))

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 tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * sqrt((h / ((l * l) * l)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (fma(((((M * M) * D) * D) / ((d_m * d_m) * l)), -0.125, (1.0 / h)) * h);
	}
	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))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(-0.125 * Float64(Float64(Float64(D * D) * M) * Float64(M / d_m))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * Float64(fma(Float64(Float64(Float64(Float64(M * M) * D) * D) / Float64(Float64(d_m * d_m) * l)), -0.125, Float64(1.0 / h)) * h));
	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[(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]}, If[LessEqual[t$95$0, 0.0], N[(N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] * M), $MachinePrecision] * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(1.0 / h), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in d around 0

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

      1. unpow-prod-down N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow3 N/A

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

   7. Applied rewrites 27.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 29.0

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

   9. Applied rewrites 29.0%

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

       1. lift-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 30.0

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

   11. Applied rewrites 30.0%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 35.2%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 24.8

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

   8. Applied rewrites 24.8%

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

   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 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in h around inf

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

      1. *-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 54.6%

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

Alternative 7: 60.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d\_m}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot \frac{\ell - \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d\_m \cdot d\_m} \cdot 0.125}{\ell}\\ \end{array} \end{array} \]

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))))))
   (if (<= t_0 0.0)
     (* (* -0.125 (* (* (* D D) M) (/ M d_m))) (sqrt (/ h (* (* l l) l))))
     (if (<= t_0 INFINITY)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m)
       (*
        (* (sqrt (/ 1.0 (* l h))) d_m)
        (/ (- l (* (/ (* (* (* M M) h) (* D D)) (* d_m d_m)) 0.125)) 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 tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * sqrt((h / ((l * l) * l)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * ((l - (((((M * M) * h) * (D * D)) / (d_m * d_m)) * 0.125)) / 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 = (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 tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * Math.sqrt((h / ((l * l) * l)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m;
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d_m) * ((l - (((((M * M) * h) * (D * D)) / (d_m * d_m)) * 0.125)) / 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)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * math.sqrt((h / ((l * l) * l)))
	elif t_0 <= math.inf:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d_m) * ((l - (((((M * M) * h) * (D * D)) / (d_m * d_m)) * 0.125)) / 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))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(-0.125 * Float64(Float64(Float64(D * D) * M) * Float64(M / d_m))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * Float64(Float64(l - Float64(Float64(Float64(Float64(Float64(M * M) * h) * Float64(D * D)) / Float64(d_m * d_m)) * 0.125)) / 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)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * sqrt((h / ((l * l) * l)));
	elseif (t_0 <= Inf)
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	else
		tmp = (sqrt((1.0 / (l * h))) * d_m) * ((l - (((((M * M) * h) * (D * D)) / (d_m * d_m)) * 0.125)) / 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]}, If[LessEqual[t$95$0, 0.0], N[(N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] * M), $MachinePrecision] * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(N[(l - N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] * 0.125), $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)))) < -0.0

   1. Initial program 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in d around 0

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

      1. unpow-prod-down N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow3 N/A

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

   7. Applied rewrites 27.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 29.0

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

   9. Applied rewrites 29.0%

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

       1. lift-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 30.0

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

   11. Applied rewrites 30.0%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 35.2%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 24.8

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

   8. Applied rewrites 24.8%

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

   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 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 49.8%

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

Alternative 8: 60.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d\_m}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot \left(1 - \frac{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d\_m \cdot d\_m} \cdot 0.125\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

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))))))
   (if (<= t_0 0.0)
     (* (* -0.125 (* (* (* D D) M) (/ M d_m))) (sqrt (/ h (* (* l l) l))))
     (if (<= t_0 INFINITY)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m)
       (*
        (* (sqrt (/ 1.0 (* l h))) d_m)
        (- 1.0 (/ (* (* (/ (* (* M M) (* D D)) (* d_m d_m)) 0.125) h) 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 tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * sqrt((h / ((l * l) * l)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((((M * M) * (D * D)) / (d_m * d_m)) * 0.125) * h) / 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 = (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 tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * Math.sqrt((h / ((l * l) * l)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m;
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((((M * M) * (D * D)) / (d_m * d_m)) * 0.125) * h) / 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)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * math.sqrt((h / ((l * l) * l)))
	elif t_0 <= math.inf:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((((M * M) * (D * D)) / (d_m * d_m)) * 0.125) * h) / 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))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(-0.125 * Float64(Float64(Float64(D * D) * M) * Float64(M / d_m))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * M) * Float64(D * D)) / Float64(d_m * d_m)) * 0.125) * h) / 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)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * sqrt((h / ((l * l) * l)));
	elseif (t_0 <= Inf)
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	else
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((((M * M) * (D * D)) / (d_m * d_m)) * 0.125) * h) / 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]}, If[LessEqual[t$95$0, 0.0], N[(N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] * M), $MachinePrecision] * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in d around 0

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

      1. unpow-prod-down N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow3 N/A

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

   7. Applied rewrites 27.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 29.0

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

   9. Applied rewrites 29.0%

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

       1. lift-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 30.0

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

   11. Applied rewrites 30.0%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 35.2%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 24.8

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

   8. Applied rewrites 24.8%

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

   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 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. count-2-rev N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   6. Applied rewrites 72.6%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 50.7

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

   9. Applied rewrites 50.7%

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

Alternative 9: 58.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d\_m}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot \left(1 - \left(\frac{\left(\left(M \cdot M\right) \cdot D\right) \cdot D}{d\_m \cdot d\_m} \cdot 0.125\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

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))))))
   (if (<= t_0 0.0)
     (* (* -0.125 (* (* (* D D) M) (/ M d_m))) (sqrt (/ h (* (* l l) l))))
     (if (<= t_0 INFINITY)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m)
       (*
        (* (sqrt (/ 1.0 (* l h))) d_m)
        (- 1.0 (* (* (/ (* (* (* M M) D) D) (* d_m d_m)) 0.125) (/ h 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 tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * sqrt((h / ((l * l) * l)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	} else {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((((M * M) * D) * D) / (d_m * d_m)) * 0.125) * (h / 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 = (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 tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * Math.sqrt((h / ((l * l) * l)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m;
	} else {
		tmp = (Math.sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((((M * M) * D) * D) / (d_m * d_m)) * 0.125) * (h / 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)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * math.sqrt((h / ((l * l) * l)))
	elif t_0 <= math.inf:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m
	else:
		tmp = (math.sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((((M * M) * D) * D) / (d_m * d_m)) * 0.125) * (h / 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))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(-0.125 * Float64(Float64(Float64(D * D) * M) * Float64(M / d_m))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m);
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(M * M) * D) * D) / Float64(d_m * d_m)) * 0.125) * Float64(h / 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)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * sqrt((h / ((l * l) * l)));
	elseif (t_0 <= Inf)
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	else
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (1.0 - ((((((M * M) * D) * D) / (d_m * d_m)) * 0.125) * (h / 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]}, If[LessEqual[t$95$0, 0.0], N[(N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] * M), $MachinePrecision] * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in d around 0

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

      1. unpow-prod-down N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow3 N/A

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

   7. Applied rewrites 27.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 29.0

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

   9. Applied rewrites 29.0%

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

       1. lift-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 30.0

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

   11. Applied rewrites 30.0%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 35.2%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 24.8

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

   8. Applied rewrites 24.8%

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

   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 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 51.1

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

   7. Applied rewrites 51.1%

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

Alternative 10: 55.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(-0.125 \cdot \left(\left(\left(D \cdot D\right) \cdot M\right) \cdot \frac{M}{d\_m}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\_m\\ \end{array} \end{array} \]

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))))))
   (if (<= t_0 0.0)
     (* (* -0.125 (* (* (* D D) M) (/ M d_m))) (sqrt (/ h (* (* l l) l))))
     (if (<= t_0 5e+234)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m)
       (* (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) {
	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 tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * sqrt((h / ((l * l) * l)));
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	} else {
		tmp = sqrt(((1.0 / l) / h)) * 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) :: t_0
    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)))
    if (t_0 <= 0.0d0) then
        tmp = ((-0.125d0) * (((d * d) * m) * (m / d_m))) * sqrt((h / ((l * l) * l)))
    else if (t_0 <= 5d+234) then
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d_m
    else
        tmp = sqrt(((1.0d0 / l) / h)) * 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 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 tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * Math.sqrt((h / ((l * l) * l)));
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m;
	} else {
		tmp = Math.sqrt(((1.0 / l) / h)) * d_m;
	}
	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)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * math.sqrt((h / ((l * l) * l)))
	elif t_0 <= 5e+234:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m
	else:
		tmp = math.sqrt(((1.0 / l) / h)) * d_m
	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))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(-0.125 * Float64(Float64(Float64(D * D) * M) * Float64(M / d_m))) * sqrt(Float64(h / Float64(Float64(l * l) * l))));
	elseif (t_0 <= 5e+234)
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m);
	else
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d_m);
	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)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (-0.125 * (((D * D) * M) * (M / d_m))) * sqrt((h / ((l * l) * l)));
	elseif (t_0 <= 5e+234)
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	else
		tmp = sqrt(((1.0 / l) / h)) * d_m;
	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]}, If[LessEqual[t$95$0, 0.0], N[(N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] * M), $MachinePrecision] * N[(M / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+234], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.2%

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

   2. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.0

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in d around 0

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

      1. unpow-prod-down N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow3 N/A

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

   7. Applied rewrites 27.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 29.0

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

   9. Applied rewrites 29.0%

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

       1. lift-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 30.0

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

   11. Applied rewrites 30.0%

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

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000003e234

   1. Initial program 35.2%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 42.3

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

   6. Applied rewrites 42.3%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-sqrt.f64 24.8

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

   8. Applied rewrites 24.8%

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

   if 5.0000000000000003e234 < (*.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 35.2%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.4

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

   6. Applied rewrites 42.4%

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

Alternative 11: 48.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M \leq 2.05 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot D\right)\right) \cdot -0.125}{d\_m}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<= M 2.05e-69)
   (* (sqrt (/ (/ 1.0 l) h)) d_m)
   (/ (* (* (sqrt (/ h (* (* l l) l))) (* (* (* M M) D) D)) -0.125) d_m)))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (M <= 2.05e-69) {
		tmp = sqrt(((1.0 / l) / h)) * d_m;
	} else {
		tmp = ((sqrt((h / ((l * l) * l))) * (((M * M) * D) * D)) * -0.125) / 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 (m <= 2.05d-69) then
        tmp = sqrt(((1.0d0 / l) / h)) * d_m
    else
        tmp = ((sqrt((h / ((l * l) * l))) * (((m * m) * d) * d)) * (-0.125d0)) / 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 (M <= 2.05e-69) {
		tmp = Math.sqrt(((1.0 / l) / h)) * d_m;
	} else {
		tmp = ((Math.sqrt((h / ((l * l) * l))) * (((M * M) * D) * D)) * -0.125) / d_m;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if M <= 2.05e-69:
		tmp = math.sqrt(((1.0 / l) / h)) * d_m
	else:
		tmp = ((math.sqrt((h / ((l * l) * l))) * (((M * M) * D) * D)) * -0.125) / d_m
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (M <= 2.05e-69)
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d_m);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(h / Float64(Float64(l * l) * l))) * Float64(Float64(Float64(M * M) * D) * D)) * -0.125) / 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 (M <= 2.05e-69)
		tmp = sqrt(((1.0 / l) / h)) * d_m;
	else
		tmp = ((sqrt((h / ((l * l) * l))) * (((M * M) * D) * D)) * -0.125) / d_m;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[M, 2.05e-69], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / d$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.04999999999999995e-69

   1. Initial program 35.2%

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

   2. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.4

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

   6. Applied rewrites 42.4%

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

   if 2.04999999999999995e-69 < M

   1. Initial program 35.2%

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

   2. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + {d}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}}{\color{blue}{d}} \]

   4. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right), \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}, \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(d \cdot d\right)\right)}{d}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)}{d} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) \cdot \frac{-1}{8}}{d} \]

   7. Applied rewrites 29.9%

      \[\leadsto \frac{\left(\sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot D\right)\right) \cdot -0.125}{d} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 47.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;M \leq 2.6 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d\_m}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<= M 2.6e-58)
   (* (sqrt (/ (/ 1.0 l) h)) d_m)
   (* (* -0.125 (* (* D D) (/ (* M M) 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 <= 2.6e-58) {
		tmp = sqrt(((1.0 / l) / h)) * d_m;
	} else {
		tmp = (-0.125 * ((D * D) * ((M * M) / 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 <= 2.6d-58) then
        tmp = sqrt(((1.0d0 / l) / h)) * d_m
    else
        tmp = ((-0.125d0) * ((d * d) * ((m * m) / 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 <= 2.6e-58) {
		tmp = Math.sqrt(((1.0 / l) / h)) * d_m;
	} else {
		tmp = (-0.125 * ((D * D) * ((M * M) / 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 <= 2.6e-58:
		tmp = math.sqrt(((1.0 / l) / h)) * d_m
	else:
		tmp = (-0.125 * ((D * D) * ((M * M) / 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 (M <= 2.6e-58)
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d_m);
	else
		tmp = Float64(Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(M * M) / 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 <= 2.6e-58)
		tmp = sqrt(((1.0 / l) / h)) * d_m;
	else
		tmp = (-0.125 * ((D * D) * ((M * M) / 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[M, 2.6e-58], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision], N[(N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / N[(N[(l * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.60000000000000007e-58

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 42.1

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. associate-/r* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      5. lower-/.f64 42.4

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   6. Applied rewrites 42.4%

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   if 2.60000000000000007e-58 < M

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d}\right) \cdot \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \]

      4. associate-/l* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{\color{blue}{{\ell}^{3}}}} \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\color{blue}{\ell}}^{3}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{{M}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{\color{blue}{3}}}} \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}} \]

      13. unpow3 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      14. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{2} \cdot \ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{2} \cdot \ell}} \]

      16. unpow2 N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

      17. lower-*.f64 27.6

          \[\leadsto \left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}} \]

   4. Applied rewrites 27.6%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{M \cdot M}{d}\right)\right) \cdot \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 46.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\left(d\_m \cdot d\_m\right) \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)}{d\_m}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\_m\\ \end{array} \end{array} \]

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))))))
   (if (<= t_0 0.0)
     (/ (* (* d_m d_m) (- (sqrt (/ 1.0 (* l h))))) d_m)
     (if (<= t_0 5e+234)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m)
       (* (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) {
	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 tmp;
	if (t_0 <= 0.0) {
		tmp = ((d_m * d_m) * -sqrt((1.0 / (l * h)))) / d_m;
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	} else {
		tmp = sqrt(((1.0 / l) / h)) * 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) :: t_0
    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)))
    if (t_0 <= 0.0d0) then
        tmp = ((d_m * d_m) * -sqrt((1.0d0 / (l * h)))) / d_m
    else if (t_0 <= 5d+234) then
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d_m
    else
        tmp = sqrt(((1.0d0 / l) / h)) * 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 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 tmp;
	if (t_0 <= 0.0) {
		tmp = ((d_m * d_m) * -Math.sqrt((1.0 / (l * h)))) / d_m;
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m;
	} else {
		tmp = Math.sqrt(((1.0 / l) / h)) * d_m;
	}
	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)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = ((d_m * d_m) * -math.sqrt((1.0 / (l * h)))) / d_m
	elif t_0 <= 5e+234:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m
	else:
		tmp = math.sqrt(((1.0 / l) / h)) * d_m
	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))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(d_m * d_m) * Float64(-sqrt(Float64(1.0 / Float64(l * h))))) / d_m);
	elseif (t_0 <= 5e+234)
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m);
	else
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d_m);
	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)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = ((d_m * d_m) * -sqrt((1.0 / (l * h)))) / d_m;
	elseif (t_0 <= 5e+234)
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	else
		tmp = sqrt(((1.0 / l) / h)) * d_m;
	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]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(d$95$m * d$95$m), $MachinePrecision] * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / d$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e+234], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + {d}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right) + {d}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}}{\color{blue}{d}} \]

   4. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125 \cdot \left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right), \sqrt{\frac{h}{\left(\ell \cdot \ell\right) \cdot \ell}}, \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(d \cdot d\right)\right)}{d}} \]

   5. Taylor expanded in l around -inf

      \[\leadsto \frac{\left({d}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}{d} \]
   6. Step-by-step derivation

      1. sqrt-pow2 N/A

         \[\leadsto \frac{\left({d}^{2} \cdot {-1}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}{d} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left({d}^{2} \cdot {-1}^{1}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}{d} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left({d}^{2} \cdot -1\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}}{d} \]

      4. associate-*l* N/A

         \[\leadsto \frac{{d}^{2} \cdot \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{d} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{{d}^{2} \cdot \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{d} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{d} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}{d} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right)}{d} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)}{d} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)}{d} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)}{d} \]

      13. lift-*.f64 11.1

          \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)}{d} \]

   7. Applied rewrites 11.1%

      \[\leadsto \frac{\left(d \cdot d\right) \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)}{d} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000003e234

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 42.1

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 42.3

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 42.3%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lift-sqrt.f64 24.8

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 24.8%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   if 5.0000000000000003e234 < (*.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 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 42.1

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. associate-/r* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      5. lower-/.f64 42.4

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   6. Applied rewrites 42.4%

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 45.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;d\_m \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\_m\\ \end{array} \end{array} \]

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))))))
   (if (<= t_0 0.0)
     (* d_m (- (sqrt (/ 1.0 (* l h)))))
     (if (<= t_0 5e+234)
       (* (/ 1.0 (* (sqrt l) (sqrt h))) d_m)
       (* (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) {
	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 tmp;
	if (t_0 <= 0.0) {
		tmp = d_m * -sqrt((1.0 / (l * h)));
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	} else {
		tmp = sqrt(((1.0 / l) / h)) * 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) :: t_0
    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)))
    if (t_0 <= 0.0d0) then
        tmp = d_m * -sqrt((1.0d0 / (l * h)))
    else if (t_0 <= 5d+234) then
        tmp = (1.0d0 / (sqrt(l) * sqrt(h))) * d_m
    else
        tmp = sqrt(((1.0d0 / l) / h)) * 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 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 tmp;
	if (t_0 <= 0.0) {
		tmp = d_m * -Math.sqrt((1.0 / (l * h)));
	} else if (t_0 <= 5e+234) {
		tmp = (1.0 / (Math.sqrt(l) * Math.sqrt(h))) * d_m;
	} else {
		tmp = Math.sqrt(((1.0 / l) / h)) * d_m;
	}
	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)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = d_m * -math.sqrt((1.0 / (l * h)))
	elif t_0 <= 5e+234:
		tmp = (1.0 / (math.sqrt(l) * math.sqrt(h))) * d_m
	else:
		tmp = math.sqrt(((1.0 / l) / h)) * d_m
	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))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(d_m * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	elseif (t_0 <= 5e+234)
		tmp = Float64(Float64(1.0 / Float64(sqrt(l) * sqrt(h))) * d_m);
	else
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * d_m);
	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)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = d_m * -sqrt((1.0 / (l * h)));
	elseif (t_0 <= 5e+234)
		tmp = (1.0 / (sqrt(l) * sqrt(h))) * d_m;
	else
		tmp = sqrt(((1.0 / l) / h)) * d_m;
	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]}, If[LessEqual[t$95$0, 0.0], N[(d$95$m * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$0, 5e+234], N[(N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d$95$m), $MachinePrecision], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -0.0

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 55.2

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. metadata-eval 55.2

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.2%

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      3. associate-*l* N/A

         \[\leadsto d \cdot \color{blue}{\left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto d \cdot \color{blue}{\left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. *-commutative N/A

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

      10. lift-*.f64 9.8

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

   6. Applied rewrites 9.8%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000003e234

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 42.1

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      4. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\ell \cdot h}} \cdot d \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{h \cdot \ell}} \cdot d \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      10. lift-*.f64 42.3

          \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

   6. Applied rewrites 42.3%

      \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell \cdot h}} \cdot d \]

      3. sqrt-prod N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

      6. lift-sqrt.f64 24.8

         \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   8. Applied rewrites 24.8%

      \[\leadsto \frac{1}{\sqrt{\ell} \cdot \sqrt{h}} \cdot d \]

   if 5.0000000000000003e234 < (*.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 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 42.1

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. associate-/r* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      5. lower-/.f64 42.4

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   6. Applied rewrites 42.4%

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 45.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-226}:\\ \;\;\;\;d\_m \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\_m\\ \end{array} \end{array} \]

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))))
      -2e-226)
   (* d_m (- (sqrt (/ 1.0 (* l h)))))
   (* (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) {
	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)))) <= -2e-226) {
		tmp = d_m * -sqrt((1.0 / (l * h)));
	} else {
		tmp = sqrt(((1.0 / l) / h)) * 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)))) <= (-2d-226)) then
        tmp = d_m * -sqrt((1.0d0 / (l * h)))
    else
        tmp = sqrt(((1.0d0 / l) / h)) * 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)))) <= -2e-226) {
		tmp = d_m * -Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = Math.sqrt(((1.0 / l) / h)) * 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)))) <= -2e-226:
		tmp = d_m * -math.sqrt((1.0 / (l * h)))
	else:
		tmp = math.sqrt(((1.0 / l) / h)) * 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)))) <= -2e-226)
		tmp = Float64(d_m * Float64(-sqrt(Float64(1.0 / Float64(l * h)))));
	else
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * 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)))) <= -2e-226)
		tmp = d_m * -sqrt((1.0 / (l * h)));
	else
		tmp = sqrt(((1.0 / l) / h)) * 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], -2e-226], N[(d$95$m * (-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $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)))) < -1.99999999999999984e-226

   1. Initial program 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\color{blue}{\left(\frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto {\left(\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 55.2

          \[\leadsto {\left(\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. metadata-eval 55.2

          \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 55.2%

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{0.5}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \left(d \cdot -1\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      3. associate-*l* N/A

         \[\leadsto d \cdot \color{blue}{\left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto d \cdot \color{blue}{\left(-1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto d \cdot \left(\mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}}\right)\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. *-commutative N/A

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

      10. lift-*.f64 9.8

          \[\leadsto d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \]

   6. Applied rewrites 9.8%

      \[\leadsto \color{blue}{d \cdot \left(-\sqrt{\frac{1}{\ell \cdot h}}\right)} \]

   if -1.99999999999999984e-226 < (*.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 35.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      6. lower-*.f64 42.1

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

      3. associate-/r* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

      5. lower-/.f64 42.4

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   6. Applied rewrites 42.4%

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 42.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d\_m \end{array} \]

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(Float64(1.0 / 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[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   6. lower-*.f64 42.1

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

4. Applied rewrites 42.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   2. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   3. associate-/r* N/A

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

   5. lower-/.f64 42.4

      \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]

6. Applied rewrites 42.4%

   \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot d \]
7. Add Preprocessing

Alternative 17: 42.4% 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 35.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   6. lower-*.f64 42.1

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

4. Applied rewrites 42.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   3. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. associate-/r* N/A

      \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot d \]

   5. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot d \]

   6. lower-/.f64 42.4

      \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot d \]

6. Applied rewrites 42.4%

   \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot d \]
7. Add Preprocessing

Alternative 18: 42.1% 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 35.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

   6. lower-*.f64 42.1

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot d \]

4. Applied rewrites 42.1%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025124 
(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)))))