Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.6% → 74.7%

Time: 10.0s

Alternatives: 12

Speedup: 1.5×

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: 65.6% 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: 74.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d + d}{D \cdot M}\\ \mathbf{if}\;h \leq -5 \cdot 10^{+235}:\\ \;\;\;\;-\frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\sqrt{\frac{-1}{\ell}} \cdot \ell}\\ \mathbf{elif}\;h \leq 1.06 \cdot 10^{-299}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(0.5 \cdot \frac{1}{t\_0 \cdot t\_0}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (+ d d) (* D M))))
   (if (<= h -5e+235)
     (- (/ (sqrt (* (/ d h) (- d))) (* (sqrt (/ -1.0 l)) l)))
     (if (<= h 1.06e-299)
       (*
        (/ (fabs d) (sqrt (* h l)))
        (- 1.0 (* (* 0.5 (/ 1.0 (* t_0 t_0))) (/ h l))))
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (-
         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) {
	double t_0 = (d + d) / (D * M);
	double tmp;
	if (h <= -5e+235) {
		tmp = -(sqrt(((d / h) * -d)) / (sqrt((-1.0 / l)) * l));
	} else if (h <= 1.06e-299) {
		tmp = (fabs(d) / sqrt((h * l))) * (1.0 - ((0.5 * (1.0 / (t_0 * t_0))) * (h / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d + d) / (d_1 * m)
    if (h <= (-5d+235)) then
        tmp = -(sqrt(((d / h) * -d)) / (sqrt(((-1.0d0) / l)) * l))
    else if (h <= 1.06d-299) then
        tmp = (abs(d) / sqrt((h * l))) * (1.0d0 - ((0.5d0 * (1.0d0 / (t_0 * t_0))) * (h / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (d + d) / (D * M);
	double tmp;
	if (h <= -5e+235) {
		tmp = -(Math.sqrt(((d / h) * -d)) / (Math.sqrt((-1.0 / l)) * l));
	} else if (h <= 1.06e-299) {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - ((0.5 * (1.0 / (t_0 * t_0))) * (h / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (d + d) / (D * M)
	tmp = 0
	if h <= -5e+235:
		tmp = -(math.sqrt(((d / h) * -d)) / (math.sqrt((-1.0 / l)) * l))
	elif h <= 1.06e-299:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - ((0.5 * (1.0 / (t_0 * t_0))) * (h / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(d + d) / Float64(D * M))
	tmp = 0.0
	if (h <= -5e+235)
		tmp = Float64(-Float64(sqrt(Float64(Float64(d / h) * Float64(-d))) / Float64(sqrt(Float64(-1.0 / l)) * l)));
	elseif (h <= 1.06e-299)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(0.5 * Float64(1.0 / Float64(t_0 * t_0))) * Float64(h / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (d + d) / (D * M);
	tmp = 0.0;
	if (h <= -5e+235)
		tmp = -(sqrt(((d / h) * -d)) / (sqrt((-1.0 / l)) * l));
	elseif (h <= 1.06e-299)
		tmp = (abs(d) / sqrt((h * l))) * (1.0 - ((0.5 * (1.0 / (t_0 * t_0))) * (h / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(d + d), $MachinePrecision] / N[(D * M), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5e+235], (-N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * (-d)), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), If[LessEqual[h, 1.06e-299], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $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]]]]

Derivation

1. Split input into 3 regimes

2. if h < -5.00000000000000027e235

   1. Initial program 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 34.9

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

   3. Applied rewrites 34.9%

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

   4. Taylor expanded in l around -inf

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow1/2 N/A

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

      4. sqrt-undiv N/A

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

      5. frac-2neg N/A

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

      6. sqrt-unprod N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

   6. Applied rewrites 17.5%

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

   if -5.00000000000000027e235 < h < 1.06e-299

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      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. pow1/2 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. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 55.0

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

   3. Applied rewrites 55.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. sqrt-div N/A

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

      9. pow1/2 N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. unpow2 N/A

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

      17. rem-sqrt-square N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. unpow2 N/A

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

      6. div-flip N/A

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

      7. div-flip N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 69.6

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

   7. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.6

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

   9. Applied rewrites 69.6%

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

   if 1.06e-299 < h

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. sqrt-div N/A

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

      12. pow1/2 N/A

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow1/2 N/A

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

      17. sqrt-div N/A

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

      18. pow1/2 N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

      21. frac-times N/A

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

   3. Applied rewrites 40.0%

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

Alternative 2: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d + d}{D \cdot M}\\ t_2 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(0.5 \cdot \frac{1}{t\_1 \cdot t\_1}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ (+ d d) (* D M)))
        (t_2
         (*
          (/ (fabs d) (sqrt (* h l)))
          (- 1.0 (* (* 0.5 (/ 1.0 (* t_1 t_1))) (/ h l))))))
   (if (<= t_0 0.0)
     t_2
     (if (<= t_0 5e+152) (* (sqrt (/ d h)) (* (sqrt (/ d l)) 1.0)) t_2))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (d + d) / (D * M);
	double t_2 = (fabs(d) / sqrt((h * l))) * (1.0 - ((0.5 * (1.0 / (t_1 * t_1))) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+152) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = (d + d) / (d_1 * m)
    t_2 = (abs(d) / sqrt((h * l))) * (1.0d0 - ((0.5d0 * (1.0d0 / (t_1 * t_1))) * (h / l)))
    if (t_0 <= 0.0d0) then
        tmp = t_2
    else if (t_0 <= 5d+152) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * 1.0d0)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (d + d) / (D * M);
	double t_2 = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - ((0.5 * (1.0 / (t_1 * t_1))) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+152) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = (d + d) / (D * M)
	t_2 = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - ((0.5 * (1.0 / (t_1 * t_1))) * (h / l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_2
	elif t_0 <= 5e+152:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * 1.0)
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(d + d) / Float64(D * M))
	t_2 = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(0.5 * Float64(1.0 / Float64(t_1 * t_1))) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+152)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = (d + d) / (D * M);
	t_2 = (abs(d) / sqrt((h * l))) * (1.0 - ((0.5 * (1.0 / (t_1 * t_1))) * (h / l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+152)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * 1.0);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d + d), $MachinePrecision] / N[(D * M), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 5e+152], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5e152 < (*.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 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      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. pow1/2 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. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 55.0

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

   3. Applied rewrites 55.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. sqrt-div N/A

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

      9. pow1/2 N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. unpow2 N/A

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

      17. rem-sqrt-square N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. unpow2 N/A

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

      6. div-flip N/A

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

      7. div-flip N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 69.6

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

   7. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.6

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

   9. Applied rewrites 69.6%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5e152

   1. Initial program 65.6%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 38.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 38.8%

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

Alternative 3: 53.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M \leq 6.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0}\\ \mathbf{elif}\;M \leq 1.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125, h, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{1}{t\_0} \cdot h\right) \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\ell \cdot d} \cdot -0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= M 6.8e-102)
     (/ (fabs d) t_0)
     (if (<= M 1.8e+34)
       (*
        (/ (fabs d) (sqrt (* h l)))
        (fma (* (/ (* (* M M) (* D D)) (* (* d d) l)) -0.125) h 1.0))
       (* (/ (* (* (* (* (/ 1.0 t_0) h) (* M M)) D) D) (* l d)) -0.125)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (M <= 6.8e-102) {
		tmp = fabs(d) / t_0;
	} else if (M <= 1.8e+34) {
		tmp = (fabs(d) / sqrt((h * l))) * fma(((((M * M) * (D * D)) / ((d * d) * l)) * -0.125), h, 1.0);
	} else {
		tmp = ((((((1.0 / t_0) * h) * (M * M)) * D) * D) / (l * d)) * -0.125;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (M <= 6.8e-102)
		tmp = Float64(abs(d) / t_0);
	elseif (M <= 1.8e+34)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * fma(Float64(Float64(Float64(Float64(M * M) * Float64(D * D)) / Float64(Float64(d * d) * l)) * -0.125), h, 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 / t_0) * h) * Float64(M * M)) * D) * D) / Float64(l * d)) * -0.125);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, 6.8e-102], N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[M, 1.8e+34], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(M * M), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * h), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 6.80000000000000026e-102

   1. Initial program 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 42.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.9

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

   10. Applied rewrites 42.9%

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

   if 6.80000000000000026e-102 < M < 1.8e34

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      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. pow1/2 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. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 55.0

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

   3. Applied rewrites 55.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. sqrt-div N/A

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

      9. pow1/2 N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. unpow2 N/A

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

      17. rem-sqrt-square N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in h around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. inv-pow N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{0}\right) \]

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   8. Applied rewrites 49.1%

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

   if 1.8e34 < M

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      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. pow1/2 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. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 55.0

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

   6. Applied rewrites 22.4%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 20.5%

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

Alternative 4: 50.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 3.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{\frac{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{\ell}\right)\right) \cdot 0.125}{d}}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= M 3.8e-115)
   (/ (fabs d) (sqrt (* l h)))
   (*
    (/ (fabs d) (sqrt (* h l)))
    (- 1.0 (/ (/ (* (* (* D D) (* (* M M) (/ h l))) 0.125) d) d)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 3.8e-115) {
		tmp = fabs(d) / sqrt((l * h));
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * (1.0 - (((((D * D) * ((M * M) * (h / l))) * 0.125) / d) / d));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 3.8d-115) then
        tmp = abs(d) / sqrt((l * h))
    else
        tmp = (abs(d) / sqrt((h * l))) * (1.0d0 - (((((d_1 * d_1) * ((m * m) * (h / l))) * 0.125d0) / d) / d))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (M <= 3.8e-115) {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - (((((D * D) * ((M * M) * (h / l))) * 0.125) / d) / d));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if M <= 3.8e-115:
		tmp = math.fabs(d) / math.sqrt((l * h))
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - (((((D * D) * ((M * M) * (h / l))) * 0.125) / d) / d))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (M <= 3.8e-115)
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D * D) * Float64(Float64(M * M) * Float64(h / l))) * 0.125) / d) / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (M <= 3.8e-115)
		tmp = abs(d) / sqrt((l * h));
	else
		tmp = (abs(d) / sqrt((h * l))) * (1.0 - (((((D * D) * ((M * M) * (h / l))) * 0.125) / d) / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[M, 3.8e-115], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D * D), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.79999999999999992e-115

   1. Initial program 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 42.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.9

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

   10. Applied rewrites 42.9%

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

   if 3.79999999999999992e-115 < M

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      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. pow1/2 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. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 55.0

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

   3. Applied rewrites 55.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. sqrt-div N/A

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

      9. pow1/2 N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. unpow2 N/A

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

      17. rem-sqrt-square N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in d around 0

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

      1. div-sub N/A

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

      2. pow-div N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left({d}^{0} - \frac{\frac{1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}}}{{d}^{2}}\right) \]

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 48.1%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 - \frac{\left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\ell}\right) \cdot 0.125}{d \cdot d}\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   10. Applied rewrites 54.0%

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

Alternative 5: 50.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M \cdot D \leq 10^{+60}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{1}{t\_0} \cdot h\right) \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot D}{\ell \cdot d} \cdot -0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= (* M D) 1e+60)
     (/ (fabs d) t_0)
     (* (/ (* (* (* (* (/ 1.0 t_0) h) (* M M)) D) D) (* l d)) -0.125))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if ((M * D) <= 1e+60) {
		tmp = fabs(d) / t_0;
	} else {
		tmp = ((((((1.0 / t_0) * h) * (M * M)) * D) * D) / (l * d)) * -0.125;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if ((m * d_1) <= 1d+60) then
        tmp = abs(d) / t_0
    else
        tmp = ((((((1.0d0 / t_0) * h) * (m * m)) * d_1) * d_1) / (l * d)) * (-0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if ((M * D) <= 1e+60) {
		tmp = Math.abs(d) / t_0;
	} else {
		tmp = ((((((1.0 / t_0) * h) * (M * M)) * D) * D) / (l * d)) * -0.125;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if (M * D) <= 1e+60:
		tmp = math.fabs(d) / t_0
	else:
		tmp = ((((((1.0 / t_0) * h) * (M * M)) * D) * D) / (l * d)) * -0.125
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (Float64(M * D) <= 1e+60)
		tmp = Float64(abs(d) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 / t_0) * h) * Float64(M * M)) * D) * D) / Float64(l * d)) * -0.125);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if ((M * D) <= 1e+60)
		tmp = abs(d) / t_0;
	else
		tmp = ((((((1.0 / t_0) * h) * (M * M)) * D) * D) / (l * d)) * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 1e+60], N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 / t$95$0), $MachinePrecision] * h), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 9.9999999999999995e59

   1. Initial program 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 42.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.9

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

   10. Applied rewrites 42.9%

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

   if 9.9999999999999995e59 < (*.f64 M D)

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      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. pow1/2 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. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 55.0

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

   6. Applied rewrites 22.4%

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

   7. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 20.5%

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

Alternative 6: 49.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;-\sqrt{\frac{d \cdot d}{\ell \cdot h}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -1e-109)
     (- (sqrt (/ (* d d) (* l h))))
     (if (<= t_0 5e+152)
       (* (sqrt (/ d h)) (* (sqrt (/ d l)) 1.0))
       (/ (fabs d) (sqrt (* l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -sqrt(((d * d) / (l * h)));
	} else if (t_0 <= 5e+152) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * 1.0);
	} else {
		tmp = fabs(d) / sqrt((l * h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d-109)) then
        tmp = -sqrt(((d * d) / (l * h)))
    else if (t_0 <= 5d+152) then
        tmp = sqrt((d / h)) * (sqrt((d / l)) * 1.0d0)
    else
        tmp = abs(d) / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -Math.sqrt(((d * d) / (l * h)));
	} else if (t_0 <= 5e+152) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt((d / l)) * 1.0);
	} else {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e-109:
		tmp = -math.sqrt(((d * d) / (l * h)))
	elif t_0 <= 5e+152:
		tmp = math.sqrt((d / h)) * (math.sqrt((d / l)) * 1.0)
	else:
		tmp = math.fabs(d) / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-109)
		tmp = Float64(-sqrt(Float64(Float64(d * d) / Float64(l * h))));
	elseif (t_0 <= 5e+152)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * 1.0));
	else
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -1e-109)
		tmp = -sqrt(((d * d) / (l * h)));
	elseif (t_0 <= 5e+152)
		tmp = sqrt((d / h)) * (sqrt((d / l)) * 1.0);
	else
		tmp = abs(d) / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-109], (-N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$0, 5e+152], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $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)))) < -9.9999999999999999e-110

   1. Initial program 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. pow2 N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      7. sqrt-div N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. frac-times N/A

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

      11. sqrt-fabs-rev N/A

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

      12. sqrt-prod N/A

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

      13. *-commutative N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-sqrt.f64 N/A

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

      16. *-commutative N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. sqrt-prod N/A

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

      20. rem-square-sqrt N/A

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

   9. Applied rewrites 11.5%

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

   if -9.9999999999999999e-110 < (*.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)))) < 5e152

   1. Initial program 65.6%

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

   2. Taylor expanded in d around inf

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

   4. Applied rewrites 38.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 38.8%

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

   if 5e152 < (*.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 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 42.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.9

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

   10. Applied rewrites 42.9%

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

Alternative 7: 48.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;-\sqrt{\frac{d \cdot d}{\ell \cdot h}}\\ \mathbf{elif}\;t\_0 \leq 10^{+126}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\frac{h}{\ell}} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -1e-109)
     (- (sqrt (/ (* d d) (* l h))))
     (if (<= t_0 1e+126)
       (/ (fabs d) (* (sqrt (/ h l)) l))
       (/ (fabs d) (sqrt (* l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -sqrt(((d * d) / (l * h)));
	} else if (t_0 <= 1e+126) {
		tmp = fabs(d) / (sqrt((h / l)) * l);
	} else {
		tmp = fabs(d) / sqrt((l * h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d-109)) then
        tmp = -sqrt(((d * d) / (l * h)))
    else if (t_0 <= 1d+126) then
        tmp = abs(d) / (sqrt((h / l)) * l)
    else
        tmp = abs(d) / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -Math.sqrt(((d * d) / (l * h)));
	} else if (t_0 <= 1e+126) {
		tmp = Math.abs(d) / (Math.sqrt((h / l)) * l);
	} else {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e-109:
		tmp = -math.sqrt(((d * d) / (l * h)))
	elif t_0 <= 1e+126:
		tmp = math.fabs(d) / (math.sqrt((h / l)) * l)
	else:
		tmp = math.fabs(d) / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-109)
		tmp = Float64(-sqrt(Float64(Float64(d * d) / Float64(l * h))));
	elseif (t_0 <= 1e+126)
		tmp = Float64(abs(d) / Float64(sqrt(Float64(h / l)) * l));
	else
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -1e-109)
		tmp = -sqrt(((d * d) / (l * h)));
	elseif (t_0 <= 1e+126)
		tmp = abs(d) / (sqrt((h / l)) * l);
	else
		tmp = abs(d) / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-109], (-N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$0, 1e+126], N[(N[Abs[d], $MachinePrecision] / N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $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)))) < -9.9999999999999999e-110

   1. Initial program 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. pow2 N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      7. sqrt-div N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. frac-times N/A

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

      11. sqrt-fabs-rev N/A

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

      12. sqrt-prod N/A

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

      13. *-commutative N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-sqrt.f64 N/A

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

      16. *-commutative N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. sqrt-prod N/A

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

      20. rem-square-sqrt N/A

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

   9. Applied rewrites 11.5%

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

   if -9.9999999999999999e-110 < (*.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)))) < 9.99999999999999925e125

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      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. pow1/2 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. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 55.0

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

   3. Applied rewrites 55.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. sqrt-div N/A

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

      9. pow1/2 N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow1/2 N/A

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

      16. unpow2 N/A

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

      17. rem-sqrt-square N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in l around inf

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

      1. rem-sqrt-square-rev N/A

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

      2. pow2 N/A

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

      3. sqrt-div N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-/.f64 23.0

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

   8. Applied rewrites 23.0%

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

   if 9.99999999999999925e125 < (*.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 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 42.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.9

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

   10. Applied rewrites 42.9%

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

Alternative 8: 46.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;-\sqrt{\frac{d \cdot d}{\ell \cdot h}}\\ \mathbf{elif}\;t\_0 \leq 10^{+116}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -1e-109)
     (- (sqrt (/ (* d d) (* l h))))
     (if (<= t_0 1e+116)
       (/ (* (sqrt (/ h l)) d) h)
       (/ (fabs d) (sqrt (* l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -sqrt(((d * d) / (l * h)));
	} else if (t_0 <= 1e+116) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = fabs(d) / sqrt((l * h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d-109)) then
        tmp = -sqrt(((d * d) / (l * h)))
    else if (t_0 <= 1d+116) then
        tmp = (sqrt((h / l)) * d) / h
    else
        tmp = abs(d) / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -Math.sqrt(((d * d) / (l * h)));
	} else if (t_0 <= 1e+116) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e-109:
		tmp = -math.sqrt(((d * d) / (l * h)))
	elif t_0 <= 1e+116:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = math.fabs(d) / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-109)
		tmp = Float64(-sqrt(Float64(Float64(d * d) / Float64(l * h))));
	elseif (t_0 <= 1e+116)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -1e-109)
		tmp = -sqrt(((d * d) / (l * h)));
	elseif (t_0 <= 1e+116)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = abs(d) / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-109], (-N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$0, 1e+116], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $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)))) < -9.9999999999999999e-110

   1. Initial program 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. pow2 N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      7. sqrt-div N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. frac-times N/A

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

      11. sqrt-fabs-rev N/A

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

      12. sqrt-prod N/A

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

      13. *-commutative N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-sqrt.f64 N/A

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

      16. *-commutative N/A

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

      17. sqrt-prod N/A

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

      18. *-commutative N/A

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

      19. sqrt-prod N/A

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

      20. rem-square-sqrt N/A

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

   9. Applied rewrites 11.5%

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

   if -9.9999999999999999e-110 < (*.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)))) < 1.00000000000000002e116

   1. Initial program 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lift-/.f64 37.3

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

   7. Applied rewrites 37.3%

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

   if 1.00000000000000002e116 < (*.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 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 42.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.9

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

   10. Applied rewrites 42.9%

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

Alternative 9: 45.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+116}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -1e-109)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (if (<= t_0 1e+116)
       (/ (* (sqrt (/ h l)) d) h)
       (/ (fabs d) (sqrt (* l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else if (t_0 <= 1e+116) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = fabs(d) / sqrt((l * h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d-109)) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else if (t_0 <= 1d+116) then
        tmp = (sqrt((h / l)) * d) / h
    else
        tmp = abs(d) / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else if (t_0 <= 1e+116) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e-109:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	elif t_0 <= 1e+116:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = math.fabs(d) / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-109)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	elseif (t_0 <= 1e+116)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -1e-109)
		tmp = -d * sqrt((1.0 / (h * l)));
	elseif (t_0 <= 1e+116)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = abs(d) / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-109], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+116], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $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)))) < -9.9999999999999999e-110

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      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. pow1/2 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. sqrt-unprod N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 55.0

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

   3. Applied rewrites 55.0%

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

   4. Taylor expanded in d around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 26.4

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

   6. Applied rewrites 26.4%

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

   if -9.9999999999999999e-110 < (*.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)))) < 1.00000000000000002e116

   1. Initial program 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lift-/.f64 37.3

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

   7. Applied rewrites 37.3%

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

   if 1.00000000000000002e116 < (*.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 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 42.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.9

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

   10. Applied rewrites 42.9%

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

Alternative 10: 44.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;-\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+116}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -1e-109)
     (- (/ (fabs d) (sqrt (* h l))))
     (if (<= t_0 1e+116)
       (/ (* (sqrt (/ h l)) d) h)
       (/ (fabs d) (sqrt (* l h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -(fabs(d) / sqrt((h * l)));
	} else if (t_0 <= 1e+116) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = fabs(d) / sqrt((l * h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d-109)) then
        tmp = -(abs(d) / sqrt((h * l)))
    else if (t_0 <= 1d+116) then
        tmp = (sqrt((h / l)) * d) / h
    else
        tmp = abs(d) / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -(Math.abs(d) / Math.sqrt((h * l)));
	} else if (t_0 <= 1e+116) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e-109:
		tmp = -(math.fabs(d) / math.sqrt((h * l)))
	elif t_0 <= 1e+116:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = math.fabs(d) / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-109)
		tmp = Float64(-Float64(abs(d) / sqrt(Float64(h * l))));
	elseif (t_0 <= 1e+116)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -1e-109)
		tmp = -(abs(d) / sqrt((h * l)));
	elseif (t_0 <= 1e+116)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = abs(d) / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-109], (-N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$0, 1e+116], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $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)))) < -9.9999999999999999e-110

   1. Initial program 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

   if -9.9999999999999999e-110 < (*.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)))) < 1.00000000000000002e116

   1. Initial program 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lift-/.f64 37.3

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

   7. Applied rewrites 37.3%

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

   if 1.00000000000000002e116 < (*.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 65.6%

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

   2. Taylor expanded in 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

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

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

      12. rem-sqrt-square N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 9.6

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 42.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.9

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   10. Applied rewrites 42.9%

       \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 42.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-109}:\\ \;\;\;\;-\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      -1e-109)
   (- (/ (fabs d) (sqrt (* h l))))
   (/ (fabs d) (sqrt (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-109) {
		tmp = -(fabs(d) / sqrt((h * l)));
	} else {
		tmp = fabs(d) / sqrt((l * h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-1d-109)) then
        tmp = -(abs(d) / sqrt((h * l)))
    else
        tmp = abs(d) / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-109) {
		tmp = -(Math.abs(d) / Math.sqrt((h * l)));
	} else {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -1e-109:
		tmp = -(math.fabs(d) / math.sqrt((h * l)))
	else:
		tmp = math.fabs(d) / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -1e-109)
		tmp = Float64(-Float64(abs(d) / sqrt(Float64(h * l))));
	else
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -1e-109)
		tmp = -(abs(d) / sqrt((h * l)));
	else
		tmp = abs(d) / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-109], (-N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $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)))) < -9.9999999999999999e-110

   1. Initial program 65.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in 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

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot \left(h \cdot d\right)}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\frac{1}{2}}}{\sqrt{h \cdot \ell}} \]

      5. metadata-eval N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

      9. metadata-eval N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\frac{1}{2}}}{\sqrt{h \cdot \ell}} \]

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

          \[\leadsto -\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      12. rem-sqrt-square N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      13. lower-fabs.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      15. lower-*.f64 9.6

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   7. Applied rewrites 9.6%

      \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   if -9.9999999999999999e-110 < (*.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 65.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in 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

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

   4. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot \left(h \cdot d\right)}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      3. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      4. pow1/2 N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\frac{1}{2}}}{\sqrt{h \cdot \ell}} \]

      5. metadata-eval N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

      7. lower-/.f64 N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

      9. metadata-eval N/A

         \[\leadsto -\frac{{\left({d}^{2}\right)}^{\frac{1}{2}}}{\sqrt{h \cdot \ell}} \]

      10. pow1/2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      11. unpow2 N/A

          \[\leadsto -\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      12. rem-sqrt-square N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      13. lower-fabs.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      15. lower-*.f64 9.6

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   7. Applied rewrites 9.6%

      \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   8. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   9. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.9

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.9

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   10. Applied rewrites 42.9%

       \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 42.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ (fabs d) (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return fabs(d) / sqrt((l * h));
}

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 = abs(d) / sqrt((l * h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return Math.abs(d) / Math.sqrt((l * h));
}

Python

def code(d, h, l, M, D):
	return math.fabs(d) / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(abs(d) / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = abs(d) / sqrt((l * h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in 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

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

4. Applied rewrites 21.4%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell} \cdot \left(h \cdot d\right)}}{h}} \]

5. Taylor expanded in h around -inf

   \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\right) \]

   2. lower-neg.f64 N/A

      \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

   3. sqrt-div N/A

      \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

   4. pow1/2 N/A

      \[\leadsto -\frac{{\left({d}^{2}\right)}^{\frac{1}{2}}}{\sqrt{h \cdot \ell}} \]

   5. metadata-eval N/A

      \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

   6. lift-/.f64 N/A

      \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

   7. lower-/.f64 N/A

      \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

   8. lift-/.f64 N/A

      \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

   9. metadata-eval N/A

      \[\leadsto -\frac{{\left({d}^{2}\right)}^{\frac{1}{2}}}{\sqrt{h \cdot \ell}} \]

   10. pow1/2 N/A

       \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

   11. unpow2 N/A

       \[\leadsto -\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

   12. rem-sqrt-square N/A

       \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   13. lower-fabs.f64 N/A

       \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   14. lower-sqrt.f64 N/A

       \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   15. lower-*.f64 9.6

       \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

7. Applied rewrites 9.6%

   \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

8. Taylor expanded in h around inf

   \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
9. Step-by-step derivation

   1. sqrt-div N/A

      \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

   2. pow2 N/A

      \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

   3. rem-sqrt-square-rev N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   7. lift-fabs.f64 42.9

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   10. lower-*.f64 42.9

       \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

10. Applied rewrites 42.9%

    \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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)))))