Henrywood and Agarwal, Equation (12)

Percentage Accurate: 35.1% → 71.4%

Time: 8.9s

Alternatives: 15

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 71.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\\ t_1 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot t\_0\right) \cdot t\_1 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\left(\frac{\sqrt{d\_m}}{\sqrt{h}} \cdot t\_0\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{\left(d\_m + d\_m\right) \cdot \left(d\_m + d\_m\right)}\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ d_m l) (/ 1.0 2.0)))
        (t_1
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l)))))
   (if (<= (* (* (pow (/ d_m h) (/ 1.0 2.0)) t_0) t_1) 5e+197)
     (* (* (/ (sqrt d_m) (sqrt h)) t_0) t_1)
     (*
      (* d_m (sqrt (/ 1.0 (* h l))))
      (-
       1.0
       (/
        (* (* (* (* D M) (/ (* D M) (* (+ d_m d_m) (+ d_m d_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 = pow((d_m / l), (1.0 / 2.0));
	double t_1 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l));
	double tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * t_0) * t_1) <= 5e+197) {
		tmp = ((sqrt(d_m) / sqrt(h)) * t_0) * t_1;
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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) :: tmp
    t_0 = (d_m / l) ** (1.0d0 / 2.0d0)
    t_1 = 1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l))
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * t_0) * t_1) <= 5d+197) then
        tmp = ((sqrt(d_m) / sqrt(h)) * t_0) * t_1
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * (1.0d0 - (((((d * m) * ((d * m) / ((d_m + d_m) * (d_m + d_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 = Math.pow((d_m / l), (1.0 / 2.0));
	double t_1 = 1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l));
	double tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * t_0) * t_1) <= 5e+197) {
		tmp = ((Math.sqrt(d_m) / Math.sqrt(h)) * t_0) * t_1;
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_m)))) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = math.pow((d_m / l), (1.0 / 2.0))
	t_1 = 1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l))
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * t_0) * t_1) <= 5e+197:
		tmp = ((math.sqrt(d_m) / math.sqrt(h)) * t_0) * t_1
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_m)))) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(d_m / l) ^ Float64(1.0 / 2.0)
	t_1 = Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l)))
	tmp = 0.0
	if (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * t_0) * t_1) <= 5e+197)
		tmp = Float64(Float64(Float64(sqrt(d_m) / sqrt(h)) * t_0) * t_1);
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * M) * Float64(Float64(D * M) / Float64(Float64(d_m + d_m) * Float64(d_m + d_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_m / l) ^ (1.0 / 2.0);
	t_1 = 1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l));
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * t_0) * t_1) <= 5e+197)
		tmp = ((sqrt(d_m) / sqrt(h)) * t_0) * t_1;
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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[Power[N[(d$95$m / l), $MachinePrecision], N[(1.0 / 2.0), $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]}, If[LessEqual[N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], 5e+197], N[(N[(N[(N[Sqrt[d$95$m], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / N[(N[(d$95$m + d$95$m), $MachinePrecision] * N[(d$95$m + d$95$m), $MachinePrecision]), $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)))) < 5.00000000000000009e197

   1. Initial program 35.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 37.1

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

   3. Applied rewrites 37.1%

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

   if 5.00000000000000009e197 < (*.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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 63.7%

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

Alternative 2: 71.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_0 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\left(\sqrt{\frac{d\_m}{\ell}} \cdot \sqrt{\frac{d\_m}{h}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{\left(d\_m + d\_m\right) \cdot \left(d\_m + d\_m\right)}\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l)))))
   (if (<=
        (* (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0))) t_0)
        5e+197)
     (* (* (sqrt (/ d_m l)) (sqrt (/ d_m h))) t_0)
     (*
      (* d_m (sqrt (/ 1.0 (* h l))))
      (-
       1.0
       (/
        (* (* (* (* D M) (/ (* D M) (* (+ d_m d_m) (+ d_m d_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 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l));
	double tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * t_0) <= 5e+197) {
		tmp = (sqrt((d_m / l)) * sqrt((d_m / h))) * t_0;
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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) :: tmp
    t_0 = 1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l))
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * t_0) <= 5d+197) then
        tmp = (sqrt((d_m / l)) * sqrt((d_m / h))) * t_0
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * (1.0d0 - (((((d * m) * ((d * m) / ((d_m + d_m) * (d_m + d_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 = 1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l));
	double tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * t_0) <= 5e+197) {
		tmp = (Math.sqrt((d_m / l)) * Math.sqrt((d_m / h))) * t_0;
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_m)))) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = 1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l))
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * t_0) <= 5e+197:
		tmp = (math.sqrt((d_m / l)) * math.sqrt((d_m / h))) * t_0
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_m)))) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_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 (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * t_0) <= 5e+197)
		tmp = Float64(Float64(sqrt(Float64(d_m / l)) * sqrt(Float64(d_m / h))) * t_0);
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * M) * Float64(Float64(D * M) / Float64(Float64(d_m + d_m) * Float64(d_m + d_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 = 1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l));
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * t_0) <= 5e+197)
		tmp = (sqrt((d_m / l)) * sqrt((d_m / h))) * t_0;
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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[(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]}, 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] * t$95$0), $MachinePrecision], 5e+197], N[(N[(N[Sqrt[N[(d$95$m / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d$95$m / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / N[(N[(d$95$m + d$95$m), $MachinePrecision] * N[(d$95$m + d$95$m), $MachinePrecision]), $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)))) < 5.00000000000000009e197

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 35.1

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

      4. lift-pow.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 32.9

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 32.9

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

   3. Applied rewrites 32.9%

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

   if 5.00000000000000009e197 < (*.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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 63.7%

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

Alternative 3: 71.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := 1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_0 \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\frac{d\_m \cdot \sqrt{\frac{h}{\ell}}}{h} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{\left(d\_m + d\_m\right) \cdot \left(d\_m + d\_m\right)}\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l)))))
   (if (<=
        (* (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0))) t_0)
        2e+144)
     (* (/ (* d_m (sqrt (/ h l))) h) t_0)
     (*
      (* d_m (sqrt (/ 1.0 (* h l))))
      (-
       1.0
       (/
        (* (* (* (* D M) (/ (* D M) (* (+ d_m d_m) (+ d_m d_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 = 1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l));
	double tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * t_0) <= 2e+144) {
		tmp = ((d_m * sqrt((h / l))) / h) * t_0;
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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) :: tmp
    t_0 = 1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l))
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * t_0) <= 2d+144) then
        tmp = ((d_m * sqrt((h / l))) / h) * t_0
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * (1.0d0 - (((((d * m) * ((d * m) / ((d_m + d_m) * (d_m + d_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 = 1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l));
	double tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * t_0) <= 2e+144) {
		tmp = ((d_m * Math.sqrt((h / l))) / h) * t_0;
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_m)))) * 0.5) * h) / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	t_0 = 1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l))
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * t_0) <= 2e+144:
		tmp = ((d_m * math.sqrt((h / l))) / h) * t_0
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_m)))) * 0.5) * h) / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_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 (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * t_0) <= 2e+144)
		tmp = Float64(Float64(Float64(d_m * sqrt(Float64(h / l))) / h) * t_0);
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * M) * Float64(Float64(D * M) / Float64(Float64(d_m + d_m) * Float64(d_m + d_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 = 1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l));
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * t_0) <= 2e+144)
		tmp = ((d_m * sqrt((h / l))) / h) * t_0;
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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[(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]}, 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] * t$95$0), $MachinePrecision], 2e+144], N[(N[(N[(d$95$m * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / N[(N[(d$95$m + d$95$m), $MachinePrecision] * N[(d$95$m + d$95$m), $MachinePrecision]), $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)))) < 2.00000000000000005e144

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

   7. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 36.0

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

   9. Applied rewrites 36.0%

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

   if 2.00000000000000005e144 < (*.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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 63.7%

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

Alternative 4: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{D}{d\_m + d\_m}\\ \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\frac{d\_m}{h}} \cdot \left(\frac{\sqrt{d\_m}}{\sqrt{\ell}} \cdot \left(1 - \left(\left(\left(\left(t\_0 \cdot M\right) \cdot t\_0\right) \cdot M\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{\left(d\_m + d\_m\right) \cdot \left(d\_m + d\_m\right)}\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (/ D (+ d_m d_m))))
   (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+144)
     (*
      (sqrt (/ d_m h))
      (*
       (/ (sqrt d_m) (sqrt l))
       (- 1.0 (* (* (* (* (* t_0 M) t_0) M) 0.5) (/ h l)))))
     (*
      (* d_m (sqrt (/ 1.0 (* h l))))
      (-
       1.0
       (/
        (* (* (* (* D M) (/ (* D M) (* (+ d_m d_m) (+ d_m d_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 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+144) {
		tmp = sqrt((d_m / h)) * ((sqrt(d_m) / sqrt(l)) * (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))));
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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) :: tmp
    t_0 = d / (d_m + d_m)
    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+144) then
        tmp = sqrt((d_m / h)) * ((sqrt(d_m) / sqrt(l)) * (1.0d0 - (((((t_0 * m) * t_0) * m) * 0.5d0) * (h / l))))
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * (1.0d0 - (((((d * m) * ((d * m) / ((d_m + d_m) * (d_m + d_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 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+144) {
		tmp = Math.sqrt((d_m / h)) * ((Math.sqrt(d_m) / Math.sqrt(l)) * (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))));
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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)
	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+144:
		tmp = math.sqrt((d_m / h)) * ((math.sqrt(d_m) / math.sqrt(l)) * (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))))
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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))
	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+144)
		tmp = Float64(sqrt(Float64(d_m / h)) * Float64(Float64(sqrt(d_m) / sqrt(l)) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(t_0 * M) * t_0) * M) * 0.5) * Float64(h / l)))));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * M) * Float64(Float64(D * M) / Float64(Float64(d_m + d_m) * Float64(d_m + d_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);
	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+144)
		tmp = sqrt((d_m / h)) * ((sqrt(d_m) / sqrt(l)) * (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))));
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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]}, 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+144], N[(N[Sqrt[N[(d$95$m / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[d$95$m], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(t$95$0 * M), $MachinePrecision] * t$95$0), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / N[(N[(d$95$m + d$95$m), $MachinePrecision] * N[(d$95$m + d$95$m), $MachinePrecision]), $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)))) < 2.00000000000000005e144

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   5. Applied rewrites 32.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 33.9

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

   7. Applied rewrites 33.9%

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

   if 2.00000000000000005e144 < (*.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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 63.7%

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

Alternative 5: 70.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{D}{d\_m + d\_m}\\ \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{d\_m}{h}} \cdot \left(\sqrt{\frac{d\_m}{\ell}} \cdot \left(1 - \left(\left(\left(\left(t\_0 \cdot M\right) \cdot t\_0\right) \cdot M\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{\left(d\_m + d\_m\right) \cdot \left(d\_m + d\_m\right)}\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (/ D (+ d_m d_m))))
   (if (<=
        (*
         (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
         (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d_m)) 2.0)) (/ h l))))
        5e+197)
     (*
      (sqrt (/ d_m h))
      (* (sqrt (/ d_m l)) (- 1.0 (* (* (* (* (* t_0 M) t_0) M) 0.5) (/ h l)))))
     (*
      (* d_m (sqrt (/ 1.0 (* h l))))
      (-
       1.0
       (/
        (* (* (* (* D M) (/ (* D M) (* (+ d_m d_m) (+ d_m d_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 tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= 5e+197) {
		tmp = sqrt((d_m / h)) * (sqrt((d_m / l)) * (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))));
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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) :: tmp
    t_0 = d / (d_m + d_m)
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))) <= 5d+197) then
        tmp = sqrt((d_m / h)) * (sqrt((d_m / l)) * (1.0d0 - (((((t_0 * m) * t_0) * m) * 0.5d0) * (h / l))))
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * (1.0d0 - (((((d * m) * ((d * m) / ((d_m + d_m) * (d_m + d_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 tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= 5e+197) {
		tmp = Math.sqrt((d_m / h)) * (Math.sqrt((d_m / l)) * (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))));
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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)
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d_m)), 2.0)) * (h / l)))) <= 5e+197:
		tmp = math.sqrt((d_m / h)) * (math.sqrt((d_m / l)) * (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))))
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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))
	tmp = 0.0
	if (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l)))) <= 5e+197)
		tmp = Float64(sqrt(Float64(d_m / h)) * Float64(sqrt(Float64(d_m / l)) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(t_0 * M) * t_0) * M) * 0.5) * Float64(h / l)))));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * M) * Float64(Float64(D * M) / Float64(Float64(d_m + d_m) * Float64(d_m + d_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);
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d_m)) ^ 2.0)) * (h / l)))) <= 5e+197)
		tmp = sqrt((d_m / h)) * (sqrt((d_m / l)) * (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))));
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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]}, If[LessEqual[N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+197], N[(N[Sqrt[N[(d$95$m / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d$95$m / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(t$95$0 * M), $MachinePrecision] * t$95$0), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / N[(N[(d$95$m + d$95$m), $MachinePrecision] * N[(d$95$m + d$95$m), $MachinePrecision]), $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)))) < 5.00000000000000009e197

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   5. Applied rewrites 32.1%

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

   if 5.00000000000000009e197 < (*.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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 63.7%

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

Alternative 6: 69.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{D}{d\_m + d\_m}\\ t_1 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(1 - \left(\left(\left(\left(t\_0 \cdot M\right) \cdot t\_0\right) \cdot M\right) \cdot 0.5\right) \cdot \frac{h}{\ell}\right) \cdot \left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{d\_m}{h}} \cdot \sqrt{\frac{d\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{\left(d\_m + d\_m\right) \cdot \left(d\_m + d\_m\right)}\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

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
         (*
          (* (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_1 0.0)
     (*
      (- 1.0 (* (* (* (* (* t_0 M) t_0) M) 0.5) (/ h l)))
      (* (sqrt (/ 1.0 (* l h))) d_m))
     (if (<= t_1 5e+197)
       (* (sqrt (/ d_m h)) (sqrt (/ d_m l)))
       (*
        (* d_m (sqrt (/ 1.0 (* h l))))
        (-
         1.0
         (/
          (* (* (* (* D M) (/ (* D M) (* (+ d_m d_m) (+ d_m d_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 = (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_1 <= 0.0) {
		tmp = (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))) * (sqrt((1.0 / (l * h))) * d_m);
	} else if (t_1 <= 5e+197) {
		tmp = sqrt((d_m / h)) * sqrt((d_m / l));
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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) :: tmp
    t_0 = d / (d_m + d_m)
    t_1 = (((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_1 <= 0.0d0) then
        tmp = (1.0d0 - (((((t_0 * m) * t_0) * m) * 0.5d0) * (h / l))) * (sqrt((1.0d0 / (l * h))) * d_m)
    else if (t_1 <= 5d+197) then
        tmp = sqrt((d_m / h)) * sqrt((d_m / l))
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * (1.0d0 - (((((d * m) * ((d * m) / ((d_m + d_m) * (d_m + d_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 = (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_1 <= 0.0) {
		tmp = (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))) * (Math.sqrt((1.0 / (l * h))) * d_m);
	} else if (t_1 <= 5e+197) {
		tmp = Math.sqrt((d_m / h)) * Math.sqrt((d_m / l));
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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 = (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_1 <= 0.0:
		tmp = (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))) * (math.sqrt((1.0 / (l * h))) * d_m)
	elif t_1 <= 5e+197:
		tmp = math.sqrt((d_m / h)) * math.sqrt((d_m / l))
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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(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_1 <= 0.0)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(t_0 * M) * t_0) * M) * 0.5) * Float64(h / l))) * Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m));
	elseif (t_1 <= 5e+197)
		tmp = Float64(sqrt(Float64(d_m / h)) * sqrt(Float64(d_m / l)));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * M) * Float64(Float64(D * M) / Float64(Float64(d_m + d_m) * Float64(d_m + d_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 = (((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_1 <= 0.0)
		tmp = (1.0 - (((((t_0 * M) * t_0) * M) * 0.5) * (h / l))) * (sqrt((1.0 / (l * h))) * d_m);
	elseif (t_1 <= 5e+197)
		tmp = sqrt((d_m / h)) * sqrt((d_m / l));
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_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[(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$1, 0.0], N[(N[(1.0 - N[(N[(N[(N[(N[(t$95$0 * M), $MachinePrecision] * t$95$0), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+197], N[(N[Sqrt[N[(d$95$m / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d$95$m / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / N[(N[(d$95$m + d$95$m), $MachinePrecision] * N[(d$95$m + d$95$m), $MachinePrecision]), $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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 69.3

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

   8. Applied rewrites 69.3%

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

   10. Applied rewrites 68.2%

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

   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)))) < 5.00000000000000009e197

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 10.2

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

   6. Applied rewrites 10.2%

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

      1. times-frac 10.2

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

      2. lift-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 10.2

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

      6. metadata-eval 10.2

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

      7. pow1/2 10.2

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

      8. metadata-eval 10.2

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 10.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 10.2

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

   8. Applied rewrites 10.2%

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

   9. Taylor expanded in l around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-/.f64 20.2

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

   11. Applied rewrites 20.2%

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

   if 5.00000000000000009e197 < (*.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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 63.7%

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

Alternative 7: 67.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \frac{\left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{\left(d\_m + d\_m\right) \cdot \left(d\_m + d\_m\right)}\right) \cdot 0.5\right) \cdot h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{d\_m} \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\frac{d\_m}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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
         (*
          (* d_m (sqrt (/ 1.0 (* h l))))
          (-
           1.0
           (/
            (* (* (* (* D M) (/ (* D M) (* (+ d_m d_m) (+ d_m d_m)))) 0.5) h)
            l)))))
   (if (<= t_0 -1e-105)
     t_1
     (if (<= t_0 5e+197)
       (* (sqrt d_m) (* (sqrt (/ 1.0 h)) (sqrt (/ d_m l))))
       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 = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_m)))) * 0.5) * h) / l));
	double tmp;
	if (t_0 <= -1e-105) {
		tmp = t_1;
	} else if (t_0 <= 5e+197) {
		tmp = sqrt(d_m) * (sqrt((1.0 / h)) * sqrt((d_m / l)));
	} 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 = (d_m * sqrt((1.0d0 / (h * l)))) * (1.0d0 - (((((d * m) * ((d * m) / ((d_m + d_m) * (d_m + d_m)))) * 0.5d0) * h) / l))
    if (t_0 <= (-1d-105)) then
        tmp = t_1
    else if (t_0 <= 5d+197) then
        tmp = sqrt(d_m) * (sqrt((1.0d0 / h)) * sqrt((d_m / l)))
    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 = (d_m * Math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_m)))) * 0.5) * h) / l));
	double tmp;
	if (t_0 <= -1e-105) {
		tmp = t_1;
	} else if (t_0 <= 5e+197) {
		tmp = Math.sqrt(d_m) * (Math.sqrt((1.0 / h)) * Math.sqrt((d_m / l)));
	} 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 = (d_m * math.sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_m)))) * 0.5) * h) / l))
	tmp = 0
	if t_0 <= -1e-105:
		tmp = t_1
	elif t_0 <= 5e+197:
		tmp = math.sqrt(d_m) * (math.sqrt((1.0 / h)) * math.sqrt((d_m / l)))
	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(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * M) * Float64(Float64(D * M) / Float64(Float64(d_m + d_m) * Float64(d_m + d_m)))) * 0.5) * h) / l)))
	tmp = 0.0
	if (t_0 <= -1e-105)
		tmp = t_1;
	elseif (t_0 <= 5e+197)
		tmp = Float64(sqrt(d_m) * Float64(sqrt(Float64(1.0 / h)) * sqrt(Float64(d_m / l))));
	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 = (d_m * sqrt((1.0 / (h * l)))) * (1.0 - (((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_m)))) * 0.5) * h) / l));
	tmp = 0.0;
	if (t_0 <= -1e-105)
		tmp = t_1;
	elseif (t_0 <= 5e+197)
		tmp = sqrt(d_m) * (sqrt((1.0 / h)) * sqrt((d_m / l)));
	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[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / N[(N[(d$95$m + d$95$m), $MachinePrecision] * N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-105], t$95$1, If[LessEqual[t$95$0, 5e+197], N[(N[Sqrt[d$95$m], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d$95$m / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.99999999999999965e-106 or 5.00000000000000009e197 < (*.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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 63.7%

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

   if -9.99999999999999965e-106 < (*.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)))) < 5.00000000000000009e197

   1. Initial program 35.1%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 19.1

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

   4. Applied rewrites 19.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 21.7

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

   6. Applied rewrites 21.7%

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

   7. Taylor expanded in h around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 22.5

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

   9. Applied rewrites 22.5%

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

Alternative 8: 52.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot 1\\ \mathbf{if}\;d\_m \leq 3 \cdot 10^{-214}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d\_m \leq 9 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{\left(d\_m + d\_m\right) \cdot \left(d\_m + d\_m\right)}\right) \cdot -0.5, \frac{h}{\ell}, 1\right) \cdot \sqrt{d\_m \cdot \frac{d\_m}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (let* ((t_0 (* (* d_m (sqrt (/ 1.0 (* h l)))) 1.0)))
   (if (<= d_m 3e-214)
     t_0
     (if (<= d_m 9e+104)
       (*
        (fma
         (* (* (* D M) (/ (* D M) (* (+ d_m d_m) (+ d_m d_m)))) -0.5)
         (/ h l)
         1.0)
        (sqrt (* d_m (/ d_m (* l h)))))
       t_0))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double t_0 = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	double tmp;
	if (d_m <= 3e-214) {
		tmp = t_0;
	} else if (d_m <= 9e+104) {
		tmp = fma((((D * M) * ((D * M) / ((d_m + d_m) * (d_m + d_m)))) * -0.5), (h / l), 1.0) * sqrt((d_m * (d_m / (l * h))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	t_0 = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * 1.0)
	tmp = 0.0
	if (d_m <= 3e-214)
		tmp = t_0;
	elseif (d_m <= 9e+104)
		tmp = Float64(fma(Float64(Float64(Float64(D * M) * Float64(Float64(D * M) / Float64(Float64(d_m + d_m) * Float64(d_m + d_m)))) * -0.5), Float64(h / l), 1.0) * sqrt(Float64(d_m * Float64(d_m / Float64(l * h)))));
	else
		tmp = t_0;
	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[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[d$95$m, 3e-214], t$95$0, If[LessEqual[d$95$m, 9e+104], N[(N[(N[(N[(N[(D * M), $MachinePrecision] * N[(N[(D * M), $MachinePrecision] / N[(N[(d$95$m + d$95$m), $MachinePrecision] * N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d$95$m * N[(d$95$m / N[(l * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < 2.99999999999999994e-214 or 8.9999999999999997e104 < d

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 43.4%

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

   if 2.99999999999999994e-214 < d < 8.9999999999999997e104

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 35.1

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

   3. Applied rewrites 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 37.0

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 43.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 43.5

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

   5. Applied rewrites 43.5%

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

Alternative 9: 49.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\left(\sqrt{d\_m} \cdot \sqrt{h}\right) \cdot \left(-1 \cdot \left(d\_m \cdot \sqrt{\frac{1}{d\_m \cdot \ell}}\right)\right)}{h}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{d\_m} \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\frac{d\_m}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot 1\\ \end{array} \end{array} \]

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))))))
   (if (<= t_0 -2e-202)
     (/
      (* (* (sqrt d_m) (sqrt h)) (* -1.0 (* d_m (sqrt (/ 1.0 (* d_m l))))))
      h)
     (if (<= t_0 5e+197)
       (* (sqrt d_m) (* (sqrt (/ 1.0 h)) (sqrt (/ d_m l))))
       (* (* d_m (sqrt (/ 1.0 (* h l)))) 1.0)))))

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 <= -2e-202) {
		tmp = ((sqrt(d_m) * sqrt(h)) * (-1.0 * (d_m * sqrt((1.0 / (d_m * l)))))) / h;
	} else if (t_0 <= 5e+197) {
		tmp = sqrt(d_m) * (sqrt((1.0 / h)) * sqrt((d_m / l)));
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	}
	return tmp;
}

Fortran

d_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((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 <= (-2d-202)) then
        tmp = ((sqrt(d_m) * sqrt(h)) * ((-1.0d0) * (d_m * sqrt((1.0d0 / (d_m * l)))))) / h
    else if (t_0 <= 5d+197) then
        tmp = sqrt(d_m) * (sqrt((1.0d0 / h)) * sqrt((d_m / l)))
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * 1.0d0
    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 <= -2e-202) {
		tmp = ((Math.sqrt(d_m) * Math.sqrt(h)) * (-1.0 * (d_m * Math.sqrt((1.0 / (d_m * l)))))) / h;
	} else if (t_0 <= 5e+197) {
		tmp = Math.sqrt(d_m) * (Math.sqrt((1.0 / h)) * Math.sqrt((d_m / l)));
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * 1.0;
	}
	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 <= -2e-202:
		tmp = ((math.sqrt(d_m) * math.sqrt(h)) * (-1.0 * (d_m * math.sqrt((1.0 / (d_m * l)))))) / h
	elif t_0 <= 5e+197:
		tmp = math.sqrt(d_m) * (math.sqrt((1.0 / h)) * math.sqrt((d_m / l)))
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * 1.0
	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 <= -2e-202)
		tmp = Float64(Float64(Float64(sqrt(d_m) * sqrt(h)) * Float64(-1.0 * Float64(d_m * sqrt(Float64(1.0 / Float64(d_m * l)))))) / h);
	elseif (t_0 <= 5e+197)
		tmp = Float64(sqrt(d_m) * Float64(sqrt(Float64(1.0 / h)) * sqrt(Float64(d_m / l))));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * 1.0);
	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 <= -2e-202)
		tmp = ((sqrt(d_m) * sqrt(h)) * (-1.0 * (d_m * sqrt((1.0 / (d_m * l)))))) / h;
	elseif (t_0 <= 5e+197)
		tmp = sqrt(d_m) * (sqrt((1.0 / h)) * sqrt((d_m / l)));
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(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, -2e-202], N[(N[(N[(N[Sqrt[d$95$m], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(-1.0 * N[(d$95$m * N[Sqrt[N[(1.0 / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 5e+197], N[(N[Sqrt[d$95$m], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d$95$m / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.0000000000000001e-202

   1. Initial program 35.1%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 19.1

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

   4. Applied rewrites 19.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 21.7

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

   6. Applied rewrites 21.7%

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

   7. Taylor expanded in d around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 8.1

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

   9. Applied rewrites 8.1%

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

   if -2.0000000000000001e-202 < (*.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)))) < 5.00000000000000009e197

   1. Initial program 35.1%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 19.1

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

   4. Applied rewrites 19.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 21.7

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

   6. Applied rewrites 21.7%

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

   7. Taylor expanded in h around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 22.5

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

   9. Applied rewrites 22.5%

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

   if 5.00000000000000009e197 < (*.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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 43.4%

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

Alternative 10: 48.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{d\_m \cdot h} \cdot \left(-1 \cdot \left(d\_m \cdot \sqrt{\frac{1}{d\_m \cdot \ell}}\right)\right)}{h}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{d\_m} \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\frac{d\_m}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot 1\\ \end{array} \end{array} \]

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))))))
   (if (<= t_0 -2e-202)
     (/ (* (sqrt (* d_m h)) (* -1.0 (* d_m (sqrt (/ 1.0 (* d_m l)))))) h)
     (if (<= t_0 5e+197)
       (* (sqrt d_m) (* (sqrt (/ 1.0 h)) (sqrt (/ d_m l))))
       (* (* d_m (sqrt (/ 1.0 (* h l)))) 1.0)))))

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 <= -2e-202) {
		tmp = (sqrt((d_m * h)) * (-1.0 * (d_m * sqrt((1.0 / (d_m * l)))))) / h;
	} else if (t_0 <= 5e+197) {
		tmp = sqrt(d_m) * (sqrt((1.0 / h)) * sqrt((d_m / l)));
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	}
	return tmp;
}

Fortran

d_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((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 <= (-2d-202)) then
        tmp = (sqrt((d_m * h)) * ((-1.0d0) * (d_m * sqrt((1.0d0 / (d_m * l)))))) / h
    else if (t_0 <= 5d+197) then
        tmp = sqrt(d_m) * (sqrt((1.0d0 / h)) * sqrt((d_m / l)))
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * 1.0d0
    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 <= -2e-202) {
		tmp = (Math.sqrt((d_m * h)) * (-1.0 * (d_m * Math.sqrt((1.0 / (d_m * l)))))) / h;
	} else if (t_0 <= 5e+197) {
		tmp = Math.sqrt(d_m) * (Math.sqrt((1.0 / h)) * Math.sqrt((d_m / l)));
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * 1.0;
	}
	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 <= -2e-202:
		tmp = (math.sqrt((d_m * h)) * (-1.0 * (d_m * math.sqrt((1.0 / (d_m * l)))))) / h
	elif t_0 <= 5e+197:
		tmp = math.sqrt(d_m) * (math.sqrt((1.0 / h)) * math.sqrt((d_m / l)))
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * 1.0
	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 <= -2e-202)
		tmp = Float64(Float64(sqrt(Float64(d_m * h)) * Float64(-1.0 * Float64(d_m * sqrt(Float64(1.0 / Float64(d_m * l)))))) / h);
	elseif (t_0 <= 5e+197)
		tmp = Float64(sqrt(d_m) * Float64(sqrt(Float64(1.0 / h)) * sqrt(Float64(d_m / l))));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * 1.0);
	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 <= -2e-202)
		tmp = (sqrt((d_m * h)) * (-1.0 * (d_m * sqrt((1.0 / (d_m * l)))))) / h;
	elseif (t_0 <= 5e+197)
		tmp = sqrt(d_m) * (sqrt((1.0 / h)) * sqrt((d_m / l)));
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(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, -2e-202], N[(N[(N[Sqrt[N[(d$95$m * h), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * N[(d$95$m * N[Sqrt[N[(1.0 / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 5e+197], N[(N[Sqrt[d$95$m], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d$95$m / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.0000000000000001e-202

   1. Initial program 35.1%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 19.1

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

   4. Applied rewrites 19.1%

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

   5. Taylor expanded in d around -inf

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{1}{d \cdot \ell}}\right)\right)}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 7.3

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

   7. Applied rewrites 7.3%

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

   if -2.0000000000000001e-202 < (*.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)))) < 5.00000000000000009e197

   1. Initial program 35.1%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 19.1

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

   4. Applied rewrites 19.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 21.7

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

   6. Applied rewrites 21.7%

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

   7. Taylor expanded in h around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 22.5

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

   9. Applied rewrites 22.5%

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

   if 5.00000000000000009e197 < (*.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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 43.4%

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

Alternative 11: 48.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-202}:\\ \;\;\;\;-1 \cdot \frac{d\_m \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{d\_m} \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\frac{d\_m}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot 1\\ \end{array} \end{array} \]

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))))))
   (if (<= t_0 -2e-202)
     (* -1.0 (/ (* d_m (sqrt (/ h l))) h))
     (if (<= t_0 5e+197)
       (* (sqrt d_m) (* (sqrt (/ 1.0 h)) (sqrt (/ d_m l))))
       (* (* d_m (sqrt (/ 1.0 (* h l)))) 1.0)))))

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 <= -2e-202) {
		tmp = -1.0 * ((d_m * sqrt((h / l))) / h);
	} else if (t_0 <= 5e+197) {
		tmp = sqrt(d_m) * (sqrt((1.0 / h)) * sqrt((d_m / l)));
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	}
	return tmp;
}

Fortran

d_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((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 <= (-2d-202)) then
        tmp = (-1.0d0) * ((d_m * sqrt((h / l))) / h)
    else if (t_0 <= 5d+197) then
        tmp = sqrt(d_m) * (sqrt((1.0d0 / h)) * sqrt((d_m / l)))
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * 1.0d0
    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 <= -2e-202) {
		tmp = -1.0 * ((d_m * Math.sqrt((h / l))) / h);
	} else if (t_0 <= 5e+197) {
		tmp = Math.sqrt(d_m) * (Math.sqrt((1.0 / h)) * Math.sqrt((d_m / l)));
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * 1.0;
	}
	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 <= -2e-202:
		tmp = -1.0 * ((d_m * math.sqrt((h / l))) / h)
	elif t_0 <= 5e+197:
		tmp = math.sqrt(d_m) * (math.sqrt((1.0 / h)) * math.sqrt((d_m / l)))
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * 1.0
	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 <= -2e-202)
		tmp = Float64(-1.0 * Float64(Float64(d_m * sqrt(Float64(h / l))) / h));
	elseif (t_0 <= 5e+197)
		tmp = Float64(sqrt(d_m) * Float64(sqrt(Float64(1.0 / h)) * sqrt(Float64(d_m / l))));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * 1.0);
	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 <= -2e-202)
		tmp = -1.0 * ((d_m * sqrt((h / l))) / h);
	elseif (t_0 <= 5e+197)
		tmp = sqrt(d_m) * (sqrt((1.0 / h)) * sqrt((d_m / l)));
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(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, -2e-202], N[(-1.0 * N[(N[(d$95$m * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+197], N[(N[Sqrt[d$95$m], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d$95$m / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.0000000000000001e-202

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 10.2

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

   6. Applied rewrites 10.2%

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

   7. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 26.1

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

   9. Applied rewrites 26.1%

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

   if -2.0000000000000001e-202 < (*.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)))) < 5.00000000000000009e197

   1. Initial program 35.1%

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

   2. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 19.1

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

   4. Applied rewrites 19.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 21.7

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

   6. Applied rewrites 21.7%

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

   7. Taylor expanded in h around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 22.5

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

   9. Applied rewrites 22.5%

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

   if 5.00000000000000009e197 < (*.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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 43.4%

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

Alternative 12: 48.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-202}:\\ \;\;\;\;-1 \cdot \frac{d\_m \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{d\_m}{h}} \cdot \sqrt{\frac{d\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot 1\\ \end{array} \end{array} \]

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))))))
   (if (<= t_0 -2e-202)
     (* -1.0 (/ (* d_m (sqrt (/ h l))) h))
     (if (<= t_0 5e+197)
       (* (sqrt (/ d_m h)) (sqrt (/ d_m l)))
       (* (* d_m (sqrt (/ 1.0 (* h l)))) 1.0)))))

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 <= -2e-202) {
		tmp = -1.0 * ((d_m * sqrt((h / l))) / h);
	} else if (t_0 <= 5e+197) {
		tmp = sqrt((d_m / h)) * sqrt((d_m / l));
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	}
	return tmp;
}

Fortran

d_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((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 <= (-2d-202)) then
        tmp = (-1.0d0) * ((d_m * sqrt((h / l))) / h)
    else if (t_0 <= 5d+197) then
        tmp = sqrt((d_m / h)) * sqrt((d_m / l))
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * 1.0d0
    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 <= -2e-202) {
		tmp = -1.0 * ((d_m * Math.sqrt((h / l))) / h);
	} else if (t_0 <= 5e+197) {
		tmp = Math.sqrt((d_m / h)) * Math.sqrt((d_m / l));
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * 1.0;
	}
	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 <= -2e-202:
		tmp = -1.0 * ((d_m * math.sqrt((h / l))) / h)
	elif t_0 <= 5e+197:
		tmp = math.sqrt((d_m / h)) * math.sqrt((d_m / l))
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * 1.0
	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 <= -2e-202)
		tmp = Float64(-1.0 * Float64(Float64(d_m * sqrt(Float64(h / l))) / h));
	elseif (t_0 <= 5e+197)
		tmp = Float64(sqrt(Float64(d_m / h)) * sqrt(Float64(d_m / l)));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * 1.0);
	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 <= -2e-202)
		tmp = -1.0 * ((d_m * sqrt((h / l))) / h);
	elseif (t_0 <= 5e+197)
		tmp = sqrt((d_m / h)) * sqrt((d_m / l));
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(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, -2e-202], N[(-1.0 * N[(N[(d$95$m * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+197], N[(N[Sqrt[N[(d$95$m / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d$95$m / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.0000000000000001e-202

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 10.2

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

   6. Applied rewrites 10.2%

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

   7. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 26.1

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

   9. Applied rewrites 26.1%

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

   if -2.0000000000000001e-202 < (*.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)))) < 5.00000000000000009e197

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 10.2

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

   6. Applied rewrites 10.2%

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

      1. times-frac 10.2

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

      2. lift-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 10.2

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

      6. metadata-eval 10.2

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

      7. pow1/2 10.2

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

      8. metadata-eval 10.2

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 10.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 10.2

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

   8. Applied rewrites 10.2%

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

   9. Taylor expanded in l around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-/.f64 20.2

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

   11. Applied rewrites 20.2%

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

   if 5.00000000000000009e197 < (*.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 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   4. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 43.4%

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

Alternative 13: 48.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;-\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{d\_m}{h}} \cdot \sqrt{\frac{d\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(d\_m \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot 1\\ \end{array} \end{array} \]

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))))))
   (if (<= t_0 0.0)
     (- (* (sqrt (/ 1.0 (* l h))) d_m))
     (if (<= t_0 5e+197)
       (* (sqrt (/ d_m h)) (sqrt (/ d_m l)))
       (* (* d_m (sqrt (/ 1.0 (* h l)))) 1.0)))))

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 = -(sqrt((1.0 / (l * h))) * d_m);
	} else if (t_0 <= 5e+197) {
		tmp = sqrt((d_m / h)) * sqrt((d_m / l));
	} else {
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	}
	return tmp;
}

Fortran

d_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((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 = -(sqrt((1.0d0 / (l * h))) * d_m)
    else if (t_0 <= 5d+197) then
        tmp = sqrt((d_m / h)) * sqrt((d_m / l))
    else
        tmp = (d_m * sqrt((1.0d0 / (h * l)))) * 1.0d0
    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 = -(Math.sqrt((1.0 / (l * h))) * d_m);
	} else if (t_0 <= 5e+197) {
		tmp = Math.sqrt((d_m / h)) * Math.sqrt((d_m / l));
	} else {
		tmp = (d_m * Math.sqrt((1.0 / (h * l)))) * 1.0;
	}
	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 = -(math.sqrt((1.0 / (l * h))) * d_m)
	elif t_0 <= 5e+197:
		tmp = math.sqrt((d_m / h)) * math.sqrt((d_m / l))
	else:
		tmp = (d_m * math.sqrt((1.0 / (h * l)))) * 1.0
	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(sqrt(Float64(1.0 / Float64(l * h))) * d_m));
	elseif (t_0 <= 5e+197)
		tmp = Float64(sqrt(Float64(d_m / h)) * sqrt(Float64(d_m / l)));
	else
		tmp = Float64(Float64(d_m * sqrt(Float64(1.0 / Float64(h * l)))) * 1.0);
	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 = -(sqrt((1.0 / (l * h))) * d_m);
	elseif (t_0 <= 5e+197)
		tmp = sqrt((d_m / h)) * sqrt((d_m / l));
	else
		tmp = (d_m * sqrt((1.0 / (h * l)))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(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[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]), If[LessEqual[t$95$0, 5e+197], N[(N[Sqrt[N[(d$95$m / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d$95$m / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d$95$m * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 42.9

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

   3. Applied rewrites 42.9%

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

   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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 10.2

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

   6. Applied rewrites 10.2%

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

      1. times-frac 10.2

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

      2. lift-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 10.2

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

      6. metadata-eval 10.2

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

      7. pow1/2 10.2

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

      8. metadata-eval 10.2

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 10.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 10.2

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

   8. Applied rewrites 10.2%

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

   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)))) < 5.00000000000000009e197

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 42.9

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 10.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. times-frac 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. sqrt-unprod N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. pow1/2 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lift-/.f64 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. metadata-eval 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. pow1/2 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. metadata-eval 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. lower-neg.f64 10.2

          \[\leadsto -d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto -d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      13. *-commutative N/A

          \[\leadsto -\sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      14. lower-*.f64 10.2

          \[\leadsto -\sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   8. Applied rewrites 10.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       3. lower-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\color{blue}{d}}{\ell}} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

       5. lower-/.f64 20.2

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 20.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   if 5.00000000000000009e197 < (*.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 35.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 42.9

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lower-*.f64 69.5

         \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. Taylor expanded in d around inf

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 43.4%

      \[\leadsto \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 25.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{-238}:\\ \;\;\;\;-\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d\_m}{h}} \cdot \sqrt{\frac{d\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (d_m h l M D)
 :precision binary64
 (if (<= l 1.05e-238)
   (- (* (sqrt (/ 1.0 (* l h))) d_m))
   (* (sqrt (/ d_m h)) (sqrt (/ d_m l)))))

C

d_m = fabs(d);
double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.05e-238) {
		tmp = -(sqrt((1.0 / (l * h))) * d_m);
	} else {
		tmp = sqrt((d_m / h)) * sqrt((d_m / l));
	}
	return tmp;
}

Fortran

d_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d_m, h, l, m, d)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8) :: tmp
    if (l <= 1.05d-238) then
        tmp = -(sqrt((1.0d0 / (l * h))) * d_m)
    else
        tmp = sqrt((d_m / h)) * sqrt((d_m / l))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double d_m, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.05e-238) {
		tmp = -(Math.sqrt((1.0 / (l * h))) * d_m);
	} else {
		tmp = Math.sqrt((d_m / h)) * Math.sqrt((d_m / l));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(d_m, h, l, M, D):
	tmp = 0
	if l <= 1.05e-238:
		tmp = -(math.sqrt((1.0 / (l * h))) * d_m)
	else:
		tmp = math.sqrt((d_m / h)) * math.sqrt((d_m / l))
	return tmp

Julia

d_m = abs(d)
function code(d_m, h, l, M, D)
	tmp = 0.0
	if (l <= 1.05e-238)
		tmp = Float64(-Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m));
	else
		tmp = Float64(sqrt(Float64(d_m / h)) * sqrt(Float64(d_m / l)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(d_m, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.05e-238)
		tmp = -(sqrt((1.0 / (l * h))) * d_m);
	else
		tmp = sqrt((d_m / h)) * sqrt((d_m / l));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[d$95$m_, h_, l_, M_, D_] := If[LessEqual[l, 1.05e-238], (-N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision]), N[(N[Sqrt[N[(d$95$m / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d$95$m / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.0500000000000001e-238

   1. Initial program 35.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 42.9

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 10.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. times-frac 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. sqrt-unprod N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. pow1/2 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lift-/.f64 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. metadata-eval 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. pow1/2 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. metadata-eval 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. lower-neg.f64 10.2

          \[\leadsto -d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto -d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      13. *-commutative N/A

          \[\leadsto -\sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      14. lower-*.f64 10.2

          \[\leadsto -\sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   8. Applied rewrites 10.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   if 1.0500000000000001e-238 < l

   1. Initial program 35.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. metadata-eval N/A

         \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-*.f64 42.9

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 10.2%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. times-frac 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. sqrt-unprod N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. pow1/2 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lift-/.f64 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      6. metadata-eval 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      7. pow1/2 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      8. metadata-eval 10.2

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      11. lower-neg.f64 10.2

          \[\leadsto -d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      12. lift-*.f64 N/A

          \[\leadsto -d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      13. *-commutative N/A

          \[\leadsto -\sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

      14. lower-*.f64 10.2

          \[\leadsto -\sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   8. Applied rewrites 10.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}} \]

       3. lower-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{\color{blue}{d}}{\ell}} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

       5. lower-/.f64 20.2

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

   11. Applied rewrites 20.2%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 10.2% accurate, 7.1× 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(-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 35.1%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. lift-pow.f64 N/A

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. pow-prod-down N/A

      \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. metadata-eval N/A

      \[\leadsto {\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   7. unpow1/2 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   9. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   10. lift-/.f64 N/A

       \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   11. frac-times N/A

       \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \sqrt{\frac{\color{blue}{d \cdot d}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   14. lower-*.f64 42.9

       \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Applied rewrites 42.9%

   \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

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. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   5. lower-*.f64 10.2

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

6. Applied rewrites 10.2%

   \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
7. Step-by-step derivation

   1. times-frac 10.2

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   3. sqrt-unprod N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   4. pow1/2 N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   5. lift-/.f64 10.2

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. metadata-eval 10.2

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   7. pow1/2 10.2

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   8. metadata-eval 10.2

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   11. lower-neg.f64 10.2

       \[\leadsto -d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   12. lift-*.f64 N/A

       \[\leadsto -d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   13. *-commutative N/A

       \[\leadsto -\sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

   14. lower-*.f64 10.2

       \[\leadsto -\sqrt{\frac{1}{h \cdot \ell}} \cdot d \]

8. Applied rewrites 10.2%

   \[\leadsto \color{blue}{-\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(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)))))