Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.9% → 42.9%

Time: 11.2s

Alternatives: 11

Speedup: 3.3×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 24.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

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(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 42.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{\frac{{\left(d\_m \cdot c0\right)}^{2}}{\left(D \cdot w\right) \cdot \left(D \cdot w\right)}}{h}\\ t_1 := \left(d\_m \cdot d\_m\right) \cdot c0\\ \mathbf{if}\;d\_m \leq 5.3 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d\_m \leq 10^{+103}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{2}{\left(h \cdot w\right) \cdot D} \cdot \frac{t\_1}{D}\right)}{w \cdot 2}\\ \mathbf{elif}\;d\_m \leq 1.4 \cdot 10^{+277}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{t\_1}\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (c0 w h D d_m M)
 :precision binary64
 (let* ((t_0 (/ (/ (pow (* d_m c0) 2.0) (* (* D w) (* D w))) h))
        (t_1 (* (* d_m d_m) c0)))
   (if (<= d_m 5.3e-160)
     t_0
     (if (<= d_m 1e+103)
       (/ (* c0 (* (/ 2.0 (* (* h w) D)) (/ t_1 D))) (* w 2.0))
       (if (<= d_m 1.4e+277)
         t_0
         (*
          (/ c0 (* 2.0 w))
          (* (* (* (* M D) (* M D)) (/ (* h w) t_1)) -0.5)))))))

C

d_m = fabs(d);
double code(double c0, double w, double h, double D, double d_m, double M) {
	double t_0 = (pow((d_m * c0), 2.0) / ((D * w) * (D * w))) / h;
	double t_1 = (d_m * d_m) * c0;
	double tmp;
	if (d_m <= 5.3e-160) {
		tmp = t_0;
	} else if (d_m <= 1e+103) {
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_1 / D))) / (w * 2.0);
	} else if (d_m <= 1.4e+277) {
		tmp = t_0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((((M * D) * (M * D)) * ((h * w) / t_1)) * -0.5);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(c0, w, h, d, d_m, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_m
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d_m * c0) ** 2.0d0) / ((d * w) * (d * w))) / h
    t_1 = (d_m * d_m) * c0
    if (d_m <= 5.3d-160) then
        tmp = t_0
    else if (d_m <= 1d+103) then
        tmp = (c0 * ((2.0d0 / ((h * w) * d)) * (t_1 / d))) / (w * 2.0d0)
    else if (d_m <= 1.4d+277) then
        tmp = t_0
    else
        tmp = (c0 / (2.0d0 * w)) * ((((m * d) * (m * d)) * ((h * w) / t_1)) * (-0.5d0))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D, double d_m, double M) {
	double t_0 = (Math.pow((d_m * c0), 2.0) / ((D * w) * (D * w))) / h;
	double t_1 = (d_m * d_m) * c0;
	double tmp;
	if (d_m <= 5.3e-160) {
		tmp = t_0;
	} else if (d_m <= 1e+103) {
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_1 / D))) / (w * 2.0);
	} else if (d_m <= 1.4e+277) {
		tmp = t_0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((((M * D) * (M * D)) * ((h * w) / t_1)) * -0.5);
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(c0, w, h, D, d_m, M):
	t_0 = (math.pow((d_m * c0), 2.0) / ((D * w) * (D * w))) / h
	t_1 = (d_m * d_m) * c0
	tmp = 0
	if d_m <= 5.3e-160:
		tmp = t_0
	elif d_m <= 1e+103:
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_1 / D))) / (w * 2.0)
	elif d_m <= 1.4e+277:
		tmp = t_0
	else:
		tmp = (c0 / (2.0 * w)) * ((((M * D) * (M * D)) * ((h * w) / t_1)) * -0.5)
	return tmp

Julia

d_m = abs(d)
function code(c0, w, h, D, d_m, M)
	t_0 = Float64(Float64((Float64(d_m * c0) ^ 2.0) / Float64(Float64(D * w) * Float64(D * w))) / h)
	t_1 = Float64(Float64(d_m * d_m) * c0)
	tmp = 0.0
	if (d_m <= 5.3e-160)
		tmp = t_0;
	elseif (d_m <= 1e+103)
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 / Float64(Float64(h * w) * D)) * Float64(t_1 / D))) / Float64(w * 2.0));
	elseif (d_m <= 1.4e+277)
		tmp = t_0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(Float64(h * w) / t_1)) * -0.5));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(c0, w, h, D, d_m, M)
	t_0 = (((d_m * c0) ^ 2.0) / ((D * w) * (D * w))) / h;
	t_1 = (d_m * d_m) * c0;
	tmp = 0.0;
	if (d_m <= 5.3e-160)
		tmp = t_0;
	elseif (d_m <= 1e+103)
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_1 / D))) / (w * 2.0);
	elseif (d_m <= 1.4e+277)
		tmp = t_0;
	else
		tmp = (c0 / (2.0 * w)) * ((((M * D) * (M * D)) * ((h * w) / t_1)) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D_, d$95$m_, M_] := Block[{t$95$0 = N[(N[(N[Power[N[(d$95$m * c0), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(D * w), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d$95$m * d$95$m), $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[d$95$m, 5.3e-160], t$95$0, If[LessEqual[d$95$m, 1e+103], N[(N[(c0 * N[(N[(2.0 / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d$95$m, 1.4e+277], t$95$0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[(h * w), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 5.3000000000000001e-160 or 1e103 < d < 1.39999999999999993e277

   1. Initial program 20.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h} \cdot {w}^{2}} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      12. lower-*.f64 34.2

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

   5. Applied rewrites 34.2%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right) \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot \color{blue}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot h} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2}}}{\color{blue}{h}} \]

   7. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{h}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{h} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{h} \]

      3. unpow2 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot w\right) \cdot \left(D \cdot w\right)}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot w\right) \cdot \left(D \cdot w\right)}}{h} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot w\right) \cdot \left(D \cdot w\right)}}{h} \]

      6. lift-*.f64 42.2

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot w\right) \cdot \left(D \cdot w\right)}}{h} \]

   9. Applied rewrites 42.2%

      \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot w\right) \cdot \left(D \cdot w\right)}}{h} \]

   if 5.3000000000000001e-160 < d < 1e103

   1. Initial program 37.4%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot w\right) \cdot \color{blue}{{D}^{2}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot {\color{blue}{D}}^{2}} \]

      10. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot \left(D \cdot \color{blue}{D}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      15. lower-*.f64 54.6

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

   5. Applied rewrites 54.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{2 \cdot w}} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{2 \cdot w}} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{2 \cdot w}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{2 \cdot w}} \]

   7. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{2}{\left(h \cdot w\right) \cdot D} \cdot \frac{\left(d \cdot d\right) \cdot c0}{D}\right)}{w \cdot 2}} \]

   if 1.39999999999999993e277 < d

   1. Initial program 26.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}}{w}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}}{\color{blue}{w}} \]

   5. Applied rewrites 21.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{h}, 2, \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{\left(d \cdot d\right) \cdot c0} \cdot -0.5\right)}{w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{{d}^{2} \cdot c0} \cdot \frac{-1}{2}\right) \]

      12. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      14. lift-*.f64 47.8

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot -0.5\right) \]

   8. Applied rewrites 47.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \color{blue}{-0.5}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      6. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({M}^{2} \cdot {D}^{2}\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      11. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{{d}^{2} \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({M}^{2} \cdot {D}^{2}\right) \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      15. unpow-prod-down N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      20. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{{d}^{2} \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      21. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      23. lift-*.f64 47.9

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]

   10. Applied rewrites 47.9%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       3. unpow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       6. lift-*.f64 47.9

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]

   12. Applied rewrites 47.9%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.3 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot w\right) \cdot \left(D \cdot w\right)}}{h}\\ \mathbf{elif}\;d \leq 10^{+103}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{2}{\left(h \cdot w\right) \cdot D} \cdot \frac{\left(d \cdot d\right) \cdot c0}{D}\right)}{w \cdot 2}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot w\right) \cdot \left(D \cdot w\right)}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left(d\_m \cdot d\_m\right) \cdot c0\\ t_1 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{2}{\left(h \cdot w\right) \cdot D} \cdot \frac{t\_0}{D}\right)}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{t\_0} \cdot -0.5\right)}{w \cdot 2}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (c0 w h D d_m M)
 :precision binary64
 (let* ((t_0 (* (* d_m d_m) c0))
        (t_1 (/ (* c0 (* d_m d_m)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (/ (* c0 (* (/ 2.0 (* (* h w) D)) (/ t_0 D))) (* w 2.0))
     (/ (* c0 (* (/ (* (pow (* M D) 2.0) (* h w)) t_0) -0.5)) (* w 2.0)))))

C

d_m = fabs(d);
double code(double c0, double w, double h, double D, double d_m, double M) {
	double t_0 = (d_m * d_m) * c0;
	double t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_0 / D))) / (w * 2.0);
	} else {
		tmp = (c0 * (((pow((M * D), 2.0) * (h * w)) / t_0) * -0.5)) / (w * 2.0);
	}
	return tmp;
}

Java

d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D, double d_m, double M) {
	double t_0 = (d_m * d_m) * c0;
	double t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_0 / D))) / (w * 2.0);
	} else {
		tmp = (c0 * (((Math.pow((M * D), 2.0) * (h * w)) / t_0) * -0.5)) / (w * 2.0);
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(c0, w, h, D, d_m, M):
	t_0 = (d_m * d_m) * c0
	t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_0 / D))) / (w * 2.0)
	else:
		tmp = (c0 * (((math.pow((M * D), 2.0) * (h * w)) / t_0) * -0.5)) / (w * 2.0)
	return tmp

Julia

d_m = abs(d)
function code(c0, w, h, D, d_m, M)
	t_0 = Float64(Float64(d_m * d_m) * c0)
	t_1 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 / Float64(Float64(h * w) * D)) * Float64(t_0 / D))) / Float64(w * 2.0));
	else
		tmp = Float64(Float64(c0 * Float64(Float64(Float64((Float64(M * D) ^ 2.0) * Float64(h * w)) / t_0) * -0.5)) / Float64(w * 2.0));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(c0, w, h, D, d_m, M)
	t_0 = (d_m * d_m) * c0;
	t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_0 / D))) / (w * 2.0);
	else
		tmp = (c0 * (((((M * D) ^ 2.0) * (h * w)) / t_0) * -0.5)) / (w * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D_, d$95$m_, M_] := Block[{t$95$0 = N[(N[(d$95$m * d$95$m), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * N[(N[(2.0 / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[(N[(N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 72.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot w\right) \cdot \color{blue}{{D}^{2}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot {\color{blue}{D}}^{2}} \]

      10. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot \left(D \cdot \color{blue}{D}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      15. lower-*.f64 74.6

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

   5. Applied rewrites 74.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{2 \cdot w}} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{2 \cdot w}} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{2 \cdot w}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{2 \cdot w}} \]

   7. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{2}{\left(h \cdot w\right) \cdot D} \cdot \frac{\left(d \cdot d\right) \cdot c0}{D}\right)}{w \cdot 2}} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}}{w}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}}{\color{blue}{w}} \]

   5. Applied rewrites 6.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{h}, 2, \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{\left(d \cdot d\right) \cdot c0} \cdot -0.5\right)}{w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{{d}^{2} \cdot c0} \cdot \frac{-1}{2}\right) \]

      12. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      14. lift-*.f64 27.6

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot -0.5\right) \]

   8. Applied rewrites 27.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \color{blue}{-0.5}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{2 \cdot w}} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{2 \cdot w}} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

   10. Applied rewrites 33.5%

       \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot -0.5\right)}{w \cdot 2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 34.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)\\ t_1 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{\left(d\_m \cdot d\_m\right) \cdot c0}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \left(d\_m \cdot \frac{d\_m}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (c0 w h D d_m M)
 :precision binary64
 (let* ((t_0 (* (* (* w w) h) (* D D)))
        (t_1 (/ (* c0 (* d_m d_m)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
        INFINITY)
     (* c0 (/ (* (* d_m d_m) c0) t_0))
     (* (* c0 c0) (* d_m (/ d_m t_0))))))

C

d_m = fabs(d);
double code(double c0, double w, double h, double D, double d_m, double M) {
	double t_0 = ((w * w) * h) * (D * D);
	double t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * (((d_m * d_m) * c0) / t_0);
	} else {
		tmp = (c0 * c0) * (d_m * (d_m / t_0));
	}
	return tmp;
}

Java

d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D, double d_m, double M) {
	double t_0 = ((w * w) * h) * (D * D);
	double t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * (((d_m * d_m) * c0) / t_0);
	} else {
		tmp = (c0 * c0) * (d_m * (d_m / t_0));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(c0, w, h, D, d_m, M):
	t_0 = ((w * w) * h) * (D * D)
	t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = c0 * (((d_m * d_m) * c0) / t_0)
	else:
		tmp = (c0 * c0) * (d_m * (d_m / t_0))
	return tmp

Julia

d_m = abs(d)
function code(c0, w, h, D, d_m, M)
	t_0 = Float64(Float64(Float64(w * w) * h) * Float64(D * D))
	t_1 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(Float64(Float64(d_m * d_m) * c0) / t_0));
	else
		tmp = Float64(Float64(c0 * c0) * Float64(d_m * Float64(d_m / t_0)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(c0, w, h, D, d_m, M)
	t_0 = ((w * w) * h) * (D * D);
	t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = c0 * (((d_m * d_m) * c0) / t_0);
	else
		tmp = (c0 * c0) * (d_m * (d_m / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D_, d$95$m_, M_] := Block[{t$95$0 = N[(N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(N[(N[(d$95$m * d$95$m), $MachinePrecision] * c0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * c0), $MachinePrecision] * N[(d$95$m * N[(d$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 72.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h} \cdot {w}^{2}} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      12. lower-*.f64 61.5

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

   5. Applied rewrites 61.5%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right) \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot \color{blue}{{w}^{2}}\right)} \]

      13. associate-/l* N/A

          \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{{d}^{2}}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      18. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      20. associate-*r* N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

   7. Applied rewrites 53.9%

      \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{d \cdot d}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]

      3. associate-*l* N/A

         \[\leadsto c0 \cdot \color{blue}{\left(c0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c0 \cdot \left(c0 \cdot \frac{d \cdot d}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \left(w \cdot w\right)}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto c0 \cdot \left(c0 \cdot \frac{d \cdot d}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}}\right) \]

      6. pow2 N/A

         \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \left(w \cdot w\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot \color{blue}{w}\right)}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\color{blue}{w} \cdot w\right)}\right) \]

      11. pow2 N/A

          \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left({D}^{2} \cdot h\right) \cdot \left(w \cdot w\right)}\right) \]

      12. pow2 N/A

          \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left({D}^{2} \cdot h\right) \cdot {w}^{\color{blue}{2}}}\right) \]

      13. associate-*r* N/A

          \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \color{blue}{\left(h \cdot {w}^{2}\right)}}\right) \]

      14. associate-/l* N/A

          \[\leadsto c0 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

   9. Applied rewrites 63.4%

      \[\leadsto c0 \cdot \color{blue}{\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)}} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h} \cdot {w}^{2}} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      12. lower-*.f64 23.1

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

   5. Applied rewrites 23.1%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right) \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot \color{blue}{{w}^{2}}\right)} \]

      13. associate-/l* N/A

          \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{{d}^{2}}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      18. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      20. associate-*r* N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

   7. Applied rewrites 11.1%

      \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \left(w \cdot w\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]

      3. associate-/l* N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \color{blue}{\frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \color{blue}{\frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot \color{blue}{w}\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\color{blue}{w} \cdot w\right)}\right) \]

      9. pow2 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left({D}^{2} \cdot h\right) \cdot \left(w \cdot w\right)}\right) \]

      10. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left({D}^{2} \cdot h\right) \cdot {w}^{\color{blue}{2}}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{{D}^{2} \cdot \color{blue}{\left(h \cdot {w}^{2}\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(h \cdot {w}^{2}\right) \cdot \color{blue}{{D}^{2}}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(h \cdot {w}^{2}\right) \cdot \color{blue}{{D}^{2}}}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left({w}^{2} \cdot h\right) \cdot {\color{blue}{D}}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left({w}^{2} \cdot h\right) \cdot {\color{blue}{D}}^{2}}\right) \]

      17. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot {D}^{2}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot {D}^{2}}\right) \]

      19. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot \color{blue}{D}\right)}\right) \]

      20. lift-*.f64 23.4

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot \color{blue}{D}\right)}\right) \]

   9. Applied rewrites 23.4%

      \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \color{blue}{\frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{{\left(d\_m \cdot c0\right)}^{2}}{\left(\left(D \cdot w\right) \cdot \left(D \cdot w\right)\right) \cdot h}\\ t_1 := \left(d\_m \cdot d\_m\right) \cdot c0\\ \mathbf{if}\;d\_m \leq 3.1 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d\_m \leq 1.85 \cdot 10^{+104}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{2}{\left(h \cdot w\right) \cdot D} \cdot \frac{t\_1}{D}\right)}{w \cdot 2}\\ \mathbf{elif}\;d\_m \leq 1.4 \cdot 10^{+277}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{t\_1}\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (c0 w h D d_m M)
 :precision binary64
 (let* ((t_0 (/ (pow (* d_m c0) 2.0) (* (* (* D w) (* D w)) h)))
        (t_1 (* (* d_m d_m) c0)))
   (if (<= d_m 3.1e-136)
     t_0
     (if (<= d_m 1.85e+104)
       (/ (* c0 (* (/ 2.0 (* (* h w) D)) (/ t_1 D))) (* w 2.0))
       (if (<= d_m 1.4e+277)
         t_0
         (*
          (/ c0 (* 2.0 w))
          (* (* (* (* M D) (* M D)) (/ (* h w) t_1)) -0.5)))))))

C

d_m = fabs(d);
double code(double c0, double w, double h, double D, double d_m, double M) {
	double t_0 = pow((d_m * c0), 2.0) / (((D * w) * (D * w)) * h);
	double t_1 = (d_m * d_m) * c0;
	double tmp;
	if (d_m <= 3.1e-136) {
		tmp = t_0;
	} else if (d_m <= 1.85e+104) {
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_1 / D))) / (w * 2.0);
	} else if (d_m <= 1.4e+277) {
		tmp = t_0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((((M * D) * (M * D)) * ((h * w) / t_1)) * -0.5);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(c0, w, h, d, d_m, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_m
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((d_m * c0) ** 2.0d0) / (((d * w) * (d * w)) * h)
    t_1 = (d_m * d_m) * c0
    if (d_m <= 3.1d-136) then
        tmp = t_0
    else if (d_m <= 1.85d+104) then
        tmp = (c0 * ((2.0d0 / ((h * w) * d)) * (t_1 / d))) / (w * 2.0d0)
    else if (d_m <= 1.4d+277) then
        tmp = t_0
    else
        tmp = (c0 / (2.0d0 * w)) * ((((m * d) * (m * d)) * ((h * w) / t_1)) * (-0.5d0))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D, double d_m, double M) {
	double t_0 = Math.pow((d_m * c0), 2.0) / (((D * w) * (D * w)) * h);
	double t_1 = (d_m * d_m) * c0;
	double tmp;
	if (d_m <= 3.1e-136) {
		tmp = t_0;
	} else if (d_m <= 1.85e+104) {
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_1 / D))) / (w * 2.0);
	} else if (d_m <= 1.4e+277) {
		tmp = t_0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((((M * D) * (M * D)) * ((h * w) / t_1)) * -0.5);
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(c0, w, h, D, d_m, M):
	t_0 = math.pow((d_m * c0), 2.0) / (((D * w) * (D * w)) * h)
	t_1 = (d_m * d_m) * c0
	tmp = 0
	if d_m <= 3.1e-136:
		tmp = t_0
	elif d_m <= 1.85e+104:
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_1 / D))) / (w * 2.0)
	elif d_m <= 1.4e+277:
		tmp = t_0
	else:
		tmp = (c0 / (2.0 * w)) * ((((M * D) * (M * D)) * ((h * w) / t_1)) * -0.5)
	return tmp

Julia

d_m = abs(d)
function code(c0, w, h, D, d_m, M)
	t_0 = Float64((Float64(d_m * c0) ^ 2.0) / Float64(Float64(Float64(D * w) * Float64(D * w)) * h))
	t_1 = Float64(Float64(d_m * d_m) * c0)
	tmp = 0.0
	if (d_m <= 3.1e-136)
		tmp = t_0;
	elseif (d_m <= 1.85e+104)
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 / Float64(Float64(h * w) * D)) * Float64(t_1 / D))) / Float64(w * 2.0));
	elseif (d_m <= 1.4e+277)
		tmp = t_0;
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(Float64(h * w) / t_1)) * -0.5));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(c0, w, h, D, d_m, M)
	t_0 = ((d_m * c0) ^ 2.0) / (((D * w) * (D * w)) * h);
	t_1 = (d_m * d_m) * c0;
	tmp = 0.0;
	if (d_m <= 3.1e-136)
		tmp = t_0;
	elseif (d_m <= 1.85e+104)
		tmp = (c0 * ((2.0 / ((h * w) * D)) * (t_1 / D))) / (w * 2.0);
	elseif (d_m <= 1.4e+277)
		tmp = t_0;
	else
		tmp = (c0 / (2.0 * w)) * ((((M * D) * (M * D)) * ((h * w) / t_1)) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D_, d$95$m_, M_] := Block[{t$95$0 = N[(N[Power[N[(d$95$m * c0), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(N[(D * w), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d$95$m * d$95$m), $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[d$95$m, 3.1e-136], t$95$0, If[LessEqual[d$95$m, 1.85e+104], N[(N[(c0 * N[(N[(2.0 / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d$95$m, 1.4e+277], t$95$0, N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[(h * w), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < 3.1e-136 or 1.8499999999999999e104 < d < 1.39999999999999993e277

   1. Initial program 20.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h} \cdot {w}^{2}} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      12. lower-*.f64 34.6

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

   5. Applied rewrites 34.6%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right) \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot \color{blue}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot h} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2}}}{\color{blue}{h}} \]

   7. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{h}} \]
   8. Step-by-step derivation

      1. count-2-rev 42.5

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{\color{blue}{2}}}{{\left(D \cdot w\right)}^{2}}}{h} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{\color{blue}{h}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{h} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{h} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{h} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{h} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{h} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{h} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2}}}{h} \]

      10. associate-/l/ N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\color{blue}{{\left(D \cdot w\right)}^{2} \cdot h}} \]

   9. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2} \cdot h}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2} \cdot h} \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{{\left(D \cdot w\right)}^{2} \cdot h} \]

       3. unpow2 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot w\right) \cdot \left(D \cdot w\right)\right) \cdot h} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot w\right) \cdot \left(D \cdot w\right)\right) \cdot h} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot w\right) \cdot \left(D \cdot w\right)\right) \cdot h} \]

       6. lift-*.f64 42.9

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot w\right) \cdot \left(D \cdot w\right)\right) \cdot h} \]

   11. Applied rewrites 42.9%

       \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot w\right) \cdot \left(D \cdot w\right)\right) \cdot h} \]

   if 3.1e-136 < d < 1.8499999999999999e104

   1. Initial program 37.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot w\right) \cdot \color{blue}{{D}^{2}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot {\color{blue}{D}}^{2}} \]

      10. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot \left(D \cdot \color{blue}{D}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      15. lower-*.f64 53.5

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

   5. Applied rewrites 53.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{2 \cdot w}} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0}{2 \cdot w}} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{2 \cdot w}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c0 \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}}{2 \cdot w}} \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{2}{\left(h \cdot w\right) \cdot D} \cdot \frac{\left(d \cdot d\right) \cdot c0}{D}\right)}{w \cdot 2}} \]

   if 1.39999999999999993e277 < d

   1. Initial program 26.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}}{w}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}}{\color{blue}{w}} \]

   5. Applied rewrites 21.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{h}, 2, \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{\left(d \cdot d\right) \cdot c0} \cdot -0.5\right)}{w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{{d}^{2} \cdot c0} \cdot \frac{-1}{2}\right) \]

      12. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      14. lift-*.f64 47.8

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot -0.5\right) \]

   8. Applied rewrites 47.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \color{blue}{-0.5}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      6. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({M}^{2} \cdot {D}^{2}\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      11. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{{d}^{2} \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({M}^{2} \cdot {D}^{2}\right) \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      15. unpow-prod-down N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      20. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{{d}^{2} \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      21. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      23. lift-*.f64 47.9

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]

   10. Applied rewrites 47.9%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       3. unpow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       6. lift-*.f64 47.9

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]

   12. Applied rewrites 47.9%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 40.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;d\_m \leq 4.4 \cdot 10^{+271}:\\ \;\;\;\;t\_0 \cdot \frac{2 \cdot \left(\left(d\_m \cdot c0\right) \cdot d\_m\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d\_m \cdot d\_m\right) \cdot c0}\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (c0 w h D d_m M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))))
   (if (<= d_m 4.4e+271)
     (* t_0 (/ (* 2.0 (* (* d_m c0) d_m)) (* (* h (* w D)) D)))
     (* t_0 (* (* (* (* M D) (* M D)) (/ (* h w) (* (* d_m d_m) c0))) -0.5)))))

C

d_m = fabs(d);
double code(double c0, double w, double h, double D, double d_m, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if (d_m <= 4.4e+271) {
		tmp = t_0 * ((2.0 * ((d_m * c0) * d_m)) / ((h * (w * D)) * D));
	} else {
		tmp = t_0 * ((((M * D) * (M * D)) * ((h * w) / ((d_m * d_m) * c0))) * -0.5);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(c0, w, h, d, d_m, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_m
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (2.0d0 * w)
    if (d_m <= 4.4d+271) then
        tmp = t_0 * ((2.0d0 * ((d_m * c0) * d_m)) / ((h * (w * d)) * d))
    else
        tmp = t_0 * ((((m * d) * (m * d)) * ((h * w) / ((d_m * d_m) * c0))) * (-0.5d0))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D, double d_m, double M) {
	double t_0 = c0 / (2.0 * w);
	double tmp;
	if (d_m <= 4.4e+271) {
		tmp = t_0 * ((2.0 * ((d_m * c0) * d_m)) / ((h * (w * D)) * D));
	} else {
		tmp = t_0 * ((((M * D) * (M * D)) * ((h * w) / ((d_m * d_m) * c0))) * -0.5);
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(c0, w, h, D, d_m, M):
	t_0 = c0 / (2.0 * w)
	tmp = 0
	if d_m <= 4.4e+271:
		tmp = t_0 * ((2.0 * ((d_m * c0) * d_m)) / ((h * (w * D)) * D))
	else:
		tmp = t_0 * ((((M * D) * (M * D)) * ((h * w) / ((d_m * d_m) * c0))) * -0.5)
	return tmp

Julia

d_m = abs(d)
function code(c0, w, h, D, d_m, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (d_m <= 4.4e+271)
		tmp = Float64(t_0 * Float64(Float64(2.0 * Float64(Float64(d_m * c0) * d_m)) / Float64(Float64(h * Float64(w * D)) * D)));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(M * D) * Float64(M * D)) * Float64(Float64(h * w) / Float64(Float64(d_m * d_m) * c0))) * -0.5));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(c0, w, h, D, d_m, M)
	t_0 = c0 / (2.0 * w);
	tmp = 0.0;
	if (d_m <= 4.4e+271)
		tmp = t_0 * ((2.0 * ((d_m * c0) * d_m)) / ((h * (w * D)) * D));
	else
		tmp = t_0 * ((((M * D) * (M * D)) * ((h * w) / ((d_m * d_m) * c0))) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D_, d$95$m_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d$95$m, 4.4e+271], N[(t$95$0 * N[(N[(2.0 * N[(N[(d$95$m * c0), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(N[(h * w), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 4.40000000000000002e271

   1. Initial program 23.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot w\right) \cdot \color{blue}{{D}^{2}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot {\color{blue}{D}}^{2}} \]

      10. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot \left(D \cdot \color{blue}{D}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      15. lower-*.f64 37.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

   5. Applied rewrites 37.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      3. associate-*l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D} \]

      5. lower-*.f64 37.4

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D} \]

   7. Applied rewrites 37.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \color{blue}{\left(w \cdot D\right)}\right) \cdot D} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(\color{blue}{w} \cdot D\right)\right) \cdot D} \]

      3. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{\left(h \cdot \left(\color{blue}{w} \cdot D\right)\right) \cdot D} \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\left(h \cdot \color{blue}{\left(w \cdot D\right)}\right) \cdot D} \]

      5. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(h \cdot \left(w \cdot \color{blue}{D}\right)\right) \cdot D} \]

      6. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(c0 \cdot d\right) \cdot d\right)}{\left(h \cdot \color{blue}{\left(w \cdot D\right)}\right) \cdot D} \]

      7. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(h \cdot \left(\color{blue}{w} \cdot D\right)\right) \cdot D} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(h \cdot \color{blue}{\left(w \cdot D\right)}\right) \cdot D} \]

      9. lift-*.f64 43.0

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(h \cdot \left(\color{blue}{w} \cdot D\right)\right) \cdot D} \]

   9. Applied rewrites 43.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(h \cdot \color{blue}{\left(w \cdot D\right)}\right) \cdot D} \]

   if 4.40000000000000002e271 < d

   1. Initial program 26.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}}{w}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot {w}^{2}\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot h}}{\color{blue}{w}} \]

   5. Applied rewrites 21.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{c0}{D \cdot D} \cdot \frac{d \cdot d}{h}, 2, \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{\left(d \cdot d\right) \cdot c0} \cdot -0.5\right)}{w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}}}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{c0 \cdot {d}^{2}} \cdot \frac{-1}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{{d}^{2} \cdot c0} \cdot \frac{-1}{2}\right) \]

      12. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      14. lift-*.f64 47.8

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot -0.5\right) \]

   8. Applied rewrites 47.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \color{blue}{-0.5}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{{\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w\right)}{\left(d \cdot d\right) \cdot c0} \cdot \frac{-1}{2}\right) \]

      6. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({M}^{2} \cdot {D}^{2}\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      11. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{{d}^{2} \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({D}^{2} \cdot {M}^{2}\right) \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left({M}^{2} \cdot {D}^{2}\right) \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      15. unpow-prod-down N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      16. lift-pow.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{c0 \cdot {d}^{2}}\right) \cdot \frac{-1}{2}\right) \]

      20. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{{d}^{2} \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      21. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

      23. lift-*.f64 47.9

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]

   10. Applied rewrites 47.9%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left({\left(M \cdot D\right)}^{2} \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       3. unpow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot \frac{-1}{2}\right) \]

       6. lift-*.f64 47.9

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]

   12. Applied rewrites 47.9%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{h \cdot w}{\left(d \cdot d\right) \cdot c0}\right) \cdot -0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 39.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 1.4 \cdot 10^{-161} \lor \neg \left(d\_m \leq 1.35 \cdot 10^{+177}\right):\\ \;\;\;\;\frac{\left(d\_m \cdot c0\right) \cdot \left(d\_m \cdot c0\right)}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{2 \cdot \left(\left(d\_m \cdot d\_m\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (c0 w h D d_m M)
 :precision binary64
 (if (or (<= d_m 1.4e-161) (not (<= d_m 1.35e+177)))
   (/ (* (* d_m c0) (* d_m c0)) (* (* D (* D h)) (* w w)))
   (* (/ c0 (+ w w)) (/ (* 2.0 (* (* d_m d_m) c0)) (* (* (* h w) D) D)))))

C

d_m = fabs(d);
double code(double c0, double w, double h, double D, double d_m, double M) {
	double tmp;
	if ((d_m <= 1.4e-161) || !(d_m <= 1.35e+177)) {
		tmp = ((d_m * c0) * (d_m * c0)) / ((D * (D * h)) * (w * w));
	} else {
		tmp = (c0 / (w + w)) * ((2.0 * ((d_m * d_m) * c0)) / (((h * w) * D) * D));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(c0, w, h, d, d_m, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_m
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d_m <= 1.4d-161) .or. (.not. (d_m <= 1.35d+177))) then
        tmp = ((d_m * c0) * (d_m * c0)) / ((d * (d * h)) * (w * w))
    else
        tmp = (c0 / (w + w)) * ((2.0d0 * ((d_m * d_m) * c0)) / (((h * w) * d) * d))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D, double d_m, double M) {
	double tmp;
	if ((d_m <= 1.4e-161) || !(d_m <= 1.35e+177)) {
		tmp = ((d_m * c0) * (d_m * c0)) / ((D * (D * h)) * (w * w));
	} else {
		tmp = (c0 / (w + w)) * ((2.0 * ((d_m * d_m) * c0)) / (((h * w) * D) * D));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(c0, w, h, D, d_m, M):
	tmp = 0
	if (d_m <= 1.4e-161) or not (d_m <= 1.35e+177):
		tmp = ((d_m * c0) * (d_m * c0)) / ((D * (D * h)) * (w * w))
	else:
		tmp = (c0 / (w + w)) * ((2.0 * ((d_m * d_m) * c0)) / (((h * w) * D) * D))
	return tmp

Julia

d_m = abs(d)
function code(c0, w, h, D, d_m, M)
	tmp = 0.0
	if ((d_m <= 1.4e-161) || !(d_m <= 1.35e+177))
		tmp = Float64(Float64(Float64(d_m * c0) * Float64(d_m * c0)) / Float64(Float64(D * Float64(D * h)) * Float64(w * w)));
	else
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(Float64(2.0 * Float64(Float64(d_m * d_m) * c0)) / Float64(Float64(Float64(h * w) * D) * D)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(c0, w, h, D, d_m, M)
	tmp = 0.0;
	if ((d_m <= 1.4e-161) || ~((d_m <= 1.35e+177)))
		tmp = ((d_m * c0) * (d_m * c0)) / ((D * (D * h)) * (w * w));
	else
		tmp = (c0 / (w + w)) * ((2.0 * ((d_m * d_m) * c0)) / (((h * w) * D) * D));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D_, d$95$m_, M_] := If[Or[LessEqual[d$95$m, 1.4e-161], N[Not[LessEqual[d$95$m, 1.35e+177]], $MachinePrecision]], N[(N[(N[(d$95$m * c0), $MachinePrecision] * N[(d$95$m * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(N[(d$95$m * d$95$m), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.39999999999999996e-161 or 1.34999999999999995e177 < d

   1. Initial program 20.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h} \cdot {w}^{2}} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      12. lower-*.f64 33.4

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

   5. Applied rewrites 33.4%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right) \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot \color{blue}{{w}^{2}}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      14. unpow-prod-down N/A

          \[\leadsto \frac{{\left(c0 \cdot d\right)}^{2}}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{{\left(c0 \cdot d\right)}^{2}}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{{\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{{\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      18. associate-*r* N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left({D}^{2} \cdot h\right) \cdot {\color{blue}{w}}^{2}} \]

      21. pow2 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot {w}^{2}} \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot {w}^{2}} \]

      23. pow2 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot \color{blue}{w}\right)} \]

   7. Applied rewrites 32.8%

      \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\color{blue}{w} \cdot w\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(\color{blue}{w} \cdot w\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(\color{blue}{w} \cdot w\right)} \]

      5. lower-*.f64 35.4

         \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)} \]

   9. Applied rewrites 35.4%

      \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(\color{blue}{w} \cdot w\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\color{blue}{D} \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)} \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot \left(w \cdot w\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot \left(w \cdot w\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot \left(w \cdot w\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(\color{blue}{D} \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)} \]

       6. lift-*.f64 35.4

          \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(D \cdot \color{blue}{\left(D \cdot h\right)}\right) \cdot \left(w \cdot w\right)} \]

   11. Applied rewrites 35.4%

       \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot \left(w \cdot w\right)} \]

   if 1.39999999999999996e-161 < d < 1.34999999999999995e177

   1. Initial program 35.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot w\right) \cdot \color{blue}{{D}^{2}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot {\color{blue}{D}}^{2}} \]

      10. pow2 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot \left(D \cdot \color{blue}{D}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

      13. *-commutative N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      15. lower-*.f64 51.9

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

   5. Applied rewrites 51.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c0}{\color{blue}{2 \cdot w}} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      2. count-2-rev N/A

         \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

      3. lower-+.f64 51.9

         \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

   7. Applied rewrites 51.9%

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.4 \cdot 10^{-161} \lor \neg \left(d \leq 1.35 \cdot 10^{+177}\right):\\ \;\;\;\;\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d\_m \cdot c0\right) \cdot d\_m\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (c0 w h D d_m M)
 :precision binary64
 (* (/ c0 (* 2.0 w)) (/ (* 2.0 (* (* d_m c0) d_m)) (* (* h (* w D)) D))))

C

d_m = fabs(d);
double code(double c0, double w, double h, double D, double d_m, double M) {
	return (c0 / (2.0 * w)) * ((2.0 * ((d_m * c0) * d_m)) / ((h * (w * D)) * D));
}

Fortran

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

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

real(8) function code(c0, w, h, d, d_m, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_m
    real(8), intent (in) :: m
    code = (c0 / (2.0d0 * w)) * ((2.0d0 * ((d_m * c0) * d_m)) / ((h * (w * d)) * d))
end function

Java

d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D, double d_m, double M) {
	return (c0 / (2.0 * w)) * ((2.0 * ((d_m * c0) * d_m)) / ((h * (w * D)) * D));
}

Python

d_m = math.fabs(d)
def code(c0, w, h, D, d_m, M):
	return (c0 / (2.0 * w)) * ((2.0 * ((d_m * c0) * d_m)) / ((h * (w * D)) * D))

Julia

d_m = abs(d)
function code(c0, w, h, D, d_m, M)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(2.0 * Float64(Float64(d_m * c0) * d_m)) / Float64(Float64(h * Float64(w * D)) * D)))
end

MATLAB

d_m = abs(d);
function tmp = code(c0, w, h, D, d_m, M)
	tmp = (c0 / (2.0 * w)) * ((2.0 * ((d_m * c0) * d_m)) / ((h * (w * D)) * D));
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D_, d$95$m_, M_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(N[(d$95$m * c0), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.7%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Add Preprocessing

3. Taylor expanded in c0 around inf

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2} \cdot \left(h \cdot w\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\color{blue}{{D}^{2}} \cdot \left(h \cdot w\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{{D}^{\color{blue}{2}} \cdot \left(h \cdot w\right)} \]

   6. pow2 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{{D}^{2} \cdot \left(h \cdot w\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot w\right) \cdot \color{blue}{{D}^{2}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot {\color{blue}{D}}^{2}} \]

   10. pow2 N/A

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(w \cdot h\right) \cdot \left(D \cdot \color{blue}{D}\right)} \]

   11. associate-*r* N/A

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(w \cdot h\right) \cdot D\right) \cdot \color{blue}{D}} \]

   13. *-commutative N/A

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

   15. lower-*.f64 36.6

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

5. Applied rewrites 36.6%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

   3. associate-*l* N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D} \]

   5. lower-*.f64 36.6

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D} \]

7. Applied rewrites 36.6%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \color{blue}{\left(w \cdot D\right)}\right) \cdot D} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(\color{blue}{w} \cdot D\right)\right) \cdot D} \]

   3. pow2 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left({d}^{2} \cdot c0\right)}{\left(h \cdot \left(\color{blue}{w} \cdot D\right)\right) \cdot D} \]

   4. *-commutative N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot {d}^{2}\right)}{\left(h \cdot \color{blue}{\left(w \cdot D\right)}\right) \cdot D} \]

   5. pow2 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(h \cdot \left(w \cdot \color{blue}{D}\right)\right) \cdot D} \]

   6. associate-*r* N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(c0 \cdot d\right) \cdot d\right)}{\left(h \cdot \color{blue}{\left(w \cdot D\right)}\right) \cdot D} \]

   7. *-commutative N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(h \cdot \left(\color{blue}{w} \cdot D\right)\right) \cdot D} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(h \cdot \color{blue}{\left(w \cdot D\right)}\right) \cdot D} \]

   9. lift-*.f64 42.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(h \cdot \left(\color{blue}{w} \cdot D\right)\right) \cdot D} \]

9. Applied rewrites 42.2%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(h \cdot \color{blue}{\left(w \cdot D\right)}\right) \cdot D} \]
10. Add Preprocessing

Alternative 8: 36.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 1.7 \cdot 10^{-158} \lor \neg \left(d\_m \leq 3.4 \cdot 10^{+122}\right):\\ \;\;\;\;\frac{\left(d\_m \cdot c0\right) \cdot \left(d\_m \cdot c0\right)}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d\_m \cdot d\_m}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (c0 w h D d_m M)
 :precision binary64
 (if (or (<= d_m 1.7e-158) (not (<= d_m 3.4e+122)))
   (/ (* (* d_m c0) (* d_m c0)) (* (* D (* D h)) (* w w)))
   (* (* c0 c0) (/ (* d_m d_m) (* (* (* (* h w) D) D) w)))))

C

d_m = fabs(d);
double code(double c0, double w, double h, double D, double d_m, double M) {
	double tmp;
	if ((d_m <= 1.7e-158) || !(d_m <= 3.4e+122)) {
		tmp = ((d_m * c0) * (d_m * c0)) / ((D * (D * h)) * (w * w));
	} else {
		tmp = (c0 * c0) * ((d_m * d_m) / ((((h * w) * D) * D) * w));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(c0, w, h, d, d_m, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_m
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d_m <= 1.7d-158) .or. (.not. (d_m <= 3.4d+122))) then
        tmp = ((d_m * c0) * (d_m * c0)) / ((d * (d * h)) * (w * w))
    else
        tmp = (c0 * c0) * ((d_m * d_m) / ((((h * w) * d) * d) * w))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D, double d_m, double M) {
	double tmp;
	if ((d_m <= 1.7e-158) || !(d_m <= 3.4e+122)) {
		tmp = ((d_m * c0) * (d_m * c0)) / ((D * (D * h)) * (w * w));
	} else {
		tmp = (c0 * c0) * ((d_m * d_m) / ((((h * w) * D) * D) * w));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(c0, w, h, D, d_m, M):
	tmp = 0
	if (d_m <= 1.7e-158) or not (d_m <= 3.4e+122):
		tmp = ((d_m * c0) * (d_m * c0)) / ((D * (D * h)) * (w * w))
	else:
		tmp = (c0 * c0) * ((d_m * d_m) / ((((h * w) * D) * D) * w))
	return tmp

Julia

d_m = abs(d)
function code(c0, w, h, D, d_m, M)
	tmp = 0.0
	if ((d_m <= 1.7e-158) || !(d_m <= 3.4e+122))
		tmp = Float64(Float64(Float64(d_m * c0) * Float64(d_m * c0)) / Float64(Float64(D * Float64(D * h)) * Float64(w * w)));
	else
		tmp = Float64(Float64(c0 * c0) * Float64(Float64(d_m * d_m) / Float64(Float64(Float64(Float64(h * w) * D) * D) * w)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(c0, w, h, D, d_m, M)
	tmp = 0.0;
	if ((d_m <= 1.7e-158) || ~((d_m <= 3.4e+122)))
		tmp = ((d_m * c0) * (d_m * c0)) / ((D * (D * h)) * (w * w));
	else
		tmp = (c0 * c0) * ((d_m * d_m) / ((((h * w) * D) * D) * w));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D_, d$95$m_, M_] := If[Or[LessEqual[d$95$m, 1.7e-158], N[Not[LessEqual[d$95$m, 3.4e+122]], $MachinePrecision]], N[(N[(N[(d$95$m * c0), $MachinePrecision] * N[(d$95$m * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * c0), $MachinePrecision] * N[(N[(d$95$m * d$95$m), $MachinePrecision] / N[(N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.7e-158 or 3.4e122 < d

   1. Initial program 21.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h} \cdot {w}^{2}} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      12. lower-*.f64 34.3

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

   5. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right) \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot \color{blue}{{w}^{2}}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      14. unpow-prod-down N/A

          \[\leadsto \frac{{\left(c0 \cdot d\right)}^{2}}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{{\left(c0 \cdot d\right)}^{2}}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{{\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{{\color{blue}{D}}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      18. associate-*r* N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left({D}^{2} \cdot h\right) \cdot {\color{blue}{w}}^{2}} \]

      21. pow2 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot {w}^{2}} \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot {w}^{2}} \]

      23. pow2 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot \color{blue}{w}\right)} \]

   7. Applied rewrites 34.1%

      \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\color{blue}{w} \cdot w\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(\color{blue}{w} \cdot w\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(\color{blue}{w} \cdot w\right)} \]

      5. lower-*.f64 36.6

         \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)} \]

   9. Applied rewrites 36.6%

      \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(\color{blue}{w} \cdot w\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\left(\color{blue}{D} \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)} \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{{\left(d \cdot c0\right)}^{2}}{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot \left(w \cdot w\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot \left(w \cdot w\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot \left(w \cdot w\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(\color{blue}{D} \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)} \]

       6. lift-*.f64 36.6

          \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(D \cdot \color{blue}{\left(D \cdot h\right)}\right) \cdot \left(w \cdot w\right)} \]

   11. Applied rewrites 36.6%

       \[\leadsto \frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\color{blue}{\left(D \cdot \left(D \cdot h\right)\right)} \cdot \left(w \cdot w\right)} \]

   if 1.7e-158 < d < 3.4e122

   1. Initial program 34.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h} \cdot {w}^{2}} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      12. lower-*.f64 41.6

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

   5. Applied rewrites 41.6%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right) \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot \color{blue}{{w}^{2}}\right)} \]

      13. associate-/l* N/A

          \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{{d}^{2}}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      18. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      20. associate-*r* N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

   7. Applied rewrites 34.2%

      \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot \color{blue}{w}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot w\right) \cdot \color{blue}{w}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot w\right) \cdot w} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot w\right) \cdot w} \]

      6. pow2 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left({D}^{2} \cdot h\right) \cdot w\right) \cdot w} \]

      7. associate-*r* N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot \left(h \cdot w\right)\right) \cdot w} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot \left(h \cdot w\right)\right) \cdot \color{blue}{w}} \]

      9. *-commutative N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot \left(w \cdot h\right)\right) \cdot w} \]

      10. *-commutative N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot {D}^{2}\right) \cdot w} \]

      11. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot \left(D \cdot D\right)\right) \cdot w} \]

      12. associate-*r* N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(w \cdot h\right) \cdot D\right) \cdot D\right) \cdot w} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(w \cdot h\right) \cdot D\right) \cdot D\right) \cdot w} \]

      14. *-commutative N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w} \]

      16. lift-*.f64 46.4

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w} \]

   9. Applied rewrites 46.4%

      \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot \color{blue}{w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.7 \cdot 10^{-158} \lor \neg \left(d \leq 3.4 \cdot 10^{+122}\right):\\ \;\;\;\;\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 5 \cdot 10^{-136} \lor \neg \left(d\_m \leq 4 \cdot 10^{+171}\right):\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \left(d\_m \cdot \frac{d\_m}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d\_m \cdot d\_m}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (c0 w h D d_m M)
 :precision binary64
 (if (or (<= d_m 5e-136) (not (<= d_m 4e+171)))
   (* (* c0 c0) (* d_m (/ d_m (* (* (* w w) h) (* D D)))))
   (* (* c0 c0) (/ (* d_m d_m) (* (* (* (* h w) D) D) w)))))

C

d_m = fabs(d);
double code(double c0, double w, double h, double D, double d_m, double M) {
	double tmp;
	if ((d_m <= 5e-136) || !(d_m <= 4e+171)) {
		tmp = (c0 * c0) * (d_m * (d_m / (((w * w) * h) * (D * D))));
	} else {
		tmp = (c0 * c0) * ((d_m * d_m) / ((((h * w) * D) * D) * w));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(c0, w, h, d, d_m, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_m
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d_m <= 5d-136) .or. (.not. (d_m <= 4d+171))) then
        tmp = (c0 * c0) * (d_m * (d_m / (((w * w) * h) * (d * d))))
    else
        tmp = (c0 * c0) * ((d_m * d_m) / ((((h * w) * d) * d) * w))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D, double d_m, double M) {
	double tmp;
	if ((d_m <= 5e-136) || !(d_m <= 4e+171)) {
		tmp = (c0 * c0) * (d_m * (d_m / (((w * w) * h) * (D * D))));
	} else {
		tmp = (c0 * c0) * ((d_m * d_m) / ((((h * w) * D) * D) * w));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
def code(c0, w, h, D, d_m, M):
	tmp = 0
	if (d_m <= 5e-136) or not (d_m <= 4e+171):
		tmp = (c0 * c0) * (d_m * (d_m / (((w * w) * h) * (D * D))))
	else:
		tmp = (c0 * c0) * ((d_m * d_m) / ((((h * w) * D) * D) * w))
	return tmp

Julia

d_m = abs(d)
function code(c0, w, h, D, d_m, M)
	tmp = 0.0
	if ((d_m <= 5e-136) || !(d_m <= 4e+171))
		tmp = Float64(Float64(c0 * c0) * Float64(d_m * Float64(d_m / Float64(Float64(Float64(w * w) * h) * Float64(D * D)))));
	else
		tmp = Float64(Float64(c0 * c0) * Float64(Float64(d_m * d_m) / Float64(Float64(Float64(Float64(h * w) * D) * D) * w)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
function tmp_2 = code(c0, w, h, D, d_m, M)
	tmp = 0.0;
	if ((d_m <= 5e-136) || ~((d_m <= 4e+171)))
		tmp = (c0 * c0) * (d_m * (d_m / (((w * w) * h) * (D * D))));
	else
		tmp = (c0 * c0) * ((d_m * d_m) / ((((h * w) * D) * D) * w));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D_, d$95$m_, M_] := If[Or[LessEqual[d$95$m, 5e-136], N[Not[LessEqual[d$95$m, 4e+171]], $MachinePrecision]], N[(N[(c0 * c0), $MachinePrecision] * N[(d$95$m * N[(d$95$m / N[(N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * c0), $MachinePrecision] * N[(N[(d$95$m * d$95$m), $MachinePrecision] / N[(N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.0000000000000002e-136 or 3.99999999999999982e171 < d

   1. Initial program 20.1%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h} \cdot {w}^{2}} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      12. lower-*.f64 33.8

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

   5. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right) \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot \color{blue}{{w}^{2}}\right)} \]

      13. associate-/l* N/A

          \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{{d}^{2}}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      18. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      20. associate-*r* N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

   7. Applied rewrites 22.4%

      \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \left(w \cdot w\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]

      3. associate-/l* N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \color{blue}{\frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \color{blue}{\frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot \color{blue}{w}\right)}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\color{blue}{w} \cdot w\right)}\right) \]

      9. pow2 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left({D}^{2} \cdot h\right) \cdot \left(w \cdot w\right)}\right) \]

      10. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left({D}^{2} \cdot h\right) \cdot {w}^{\color{blue}{2}}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{{D}^{2} \cdot \color{blue}{\left(h \cdot {w}^{2}\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(h \cdot {w}^{2}\right) \cdot \color{blue}{{D}^{2}}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(h \cdot {w}^{2}\right) \cdot \color{blue}{{D}^{2}}}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left({w}^{2} \cdot h\right) \cdot {\color{blue}{D}}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left({w}^{2} \cdot h\right) \cdot {\color{blue}{D}}^{2}}\right) \]

      17. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot {D}^{2}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot {D}^{2}}\right) \]

      19. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot \color{blue}{D}\right)}\right) \]

      20. lift-*.f64 33.7

          \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot \color{blue}{D}\right)}\right) \]

   9. Applied rewrites 33.7%

      \[\leadsto \left(c0 \cdot c0\right) \cdot \left(d \cdot \color{blue}{\frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)}}\right) \]

   if 5.0000000000000002e-136 < d < 3.99999999999999982e171

   1. Initial program 35.3%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h} \cdot {w}^{2}} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      12. lower-*.f64 41.7

          \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

   5. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right) \cdot h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

      8. associate-/l/ N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot \color{blue}{{w}^{2}}\right)} \]

      13. associate-/l* N/A

          \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{{d}^{2}}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

      18. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

      20. associate-*r* N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

   7. Applied rewrites 34.0%

      \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot \color{blue}{w}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot w\right) \cdot \color{blue}{w}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot w\right) \cdot w} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot w\right) \cdot w} \]

      6. pow2 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left({D}^{2} \cdot h\right) \cdot w\right) \cdot w} \]

      7. associate-*r* N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot \left(h \cdot w\right)\right) \cdot w} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot \left(h \cdot w\right)\right) \cdot \color{blue}{w}} \]

      9. *-commutative N/A

         \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot \left(w \cdot h\right)\right) \cdot w} \]

      10. *-commutative N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot {D}^{2}\right) \cdot w} \]

      11. pow2 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot \left(D \cdot D\right)\right) \cdot w} \]

      12. associate-*r* N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(w \cdot h\right) \cdot D\right) \cdot D\right) \cdot w} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(w \cdot h\right) \cdot D\right) \cdot D\right) \cdot w} \]

      14. *-commutative N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w} \]

      16. lift-*.f64 43.9

          \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w} \]

   9. Applied rewrites 43.9%

      \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot \color{blue}{w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5 \cdot 10^{-136} \lor \neg \left(d \leq 4 \cdot 10^{+171}\right):\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ c0 \cdot \frac{\left(d\_m \cdot d\_m\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)} \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (c0 w h D d_m M)
 :precision binary64
 (* c0 (/ (* (* d_m d_m) c0) (* (* (* w w) h) (* D D)))))

C

d_m = fabs(d);
double code(double c0, double w, double h, double D, double d_m, double M) {
	return c0 * (((d_m * d_m) * c0) / (((w * w) * h) * (D * D)));
}

Fortran

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

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

real(8) function code(c0, w, h, d, d_m, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_m
    real(8), intent (in) :: m
    code = c0 * (((d_m * d_m) * c0) / (((w * w) * h) * (d * d)))
end function

Java

d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D, double d_m, double M) {
	return c0 * (((d_m * d_m) * c0) / (((w * w) * h) * (D * D)));
}

Python

d_m = math.fabs(d)
def code(c0, w, h, D, d_m, M):
	return c0 * (((d_m * d_m) * c0) / (((w * w) * h) * (D * D)))

Julia

d_m = abs(d)
function code(c0, w, h, D, d_m, M)
	return Float64(c0 * Float64(Float64(Float64(d_m * d_m) * c0) / Float64(Float64(Float64(w * w) * h) * Float64(D * D))))
end

MATLAB

d_m = abs(d);
function tmp = code(c0, w, h, D, d_m, M)
	tmp = c0 * (((d_m * d_m) * c0) / (((w * w) * h) * (D * D)));
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D_, d$95$m_, M_] := N[(c0 * N[(N[(N[(d$95$m * d$95$m), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.7%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Add Preprocessing

3. Taylor expanded in c0 around inf

   \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h \cdot {w}^{2}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\color{blue}{h} \cdot {w}^{2}} \]

   4. pow-prod-down N/A

      \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{{D}^{2}}}{h \cdot {w}^{2}} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{h \cdot {w}^{2}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{{w}^{2} \cdot \color{blue}{h}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

   12. lower-*.f64 35.7

       \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

5. Applied rewrites 35.7%

   \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right) \cdot h}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\color{blue}{\left(w \cdot w\right)} \cdot h} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(w \cdot w\right) \cdot h} \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{{\left(c0 \cdot d\right)}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

   6. unpow-prod-down N/A

      \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{D \cdot D}}{\left(\color{blue}{w} \cdot w\right) \cdot h} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2}}}{\left(w \cdot \color{blue}{w}\right) \cdot h} \]

   8. associate-/l/ N/A

      \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{h}\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)} \]

   11. pow2 N/A

       \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left({w}^{2} \cdot h\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot \color{blue}{{w}^{2}}\right)} \]

   13. associate-/l* N/A

       \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto {c0}^{2} \cdot \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

   15. unpow2 N/A

       \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{{d}^{2}}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \]

   17. lower-/.f64 N/A

       \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{{d}^{2}}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

   18. pow2 N/A

       \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

   19. lift-*.f64 N/A

       \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\color{blue}{{D}^{2}} \cdot \left(h \cdot {w}^{2}\right)} \]

   20. associate-*r* N/A

       \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

   21. lower-*.f64 N/A

       \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left({D}^{2} \cdot h\right) \cdot \color{blue}{{w}^{2}}} \]

7. Applied rewrites 25.2%

   \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(c0 \cdot c0\right) \cdot \frac{\color{blue}{d \cdot d}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(c0 \cdot c0\right) \cdot \color{blue}{\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}} \]

   3. associate-*l* N/A

      \[\leadsto c0 \cdot \color{blue}{\left(c0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto c0 \cdot \left(c0 \cdot \frac{d \cdot d}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \left(w \cdot w\right)}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto c0 \cdot \left(c0 \cdot \frac{d \cdot d}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}}\right) \]

   6. pow2 N/A

      \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\color{blue}{\left(\left(D \cdot D\right) \cdot h\right)} \cdot \left(w \cdot w\right)}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot \color{blue}{w}\right)}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \color{blue}{\left(w \cdot w\right)}}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(\color{blue}{w} \cdot w\right)}\right) \]

   11. pow2 N/A

       \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left({D}^{2} \cdot h\right) \cdot \left(w \cdot w\right)}\right) \]

   12. pow2 N/A

       \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{\left({D}^{2} \cdot h\right) \cdot {w}^{\color{blue}{2}}}\right) \]

   13. associate-*r* N/A

       \[\leadsto c0 \cdot \left(c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \color{blue}{\left(h \cdot {w}^{2}\right)}}\right) \]

   14. associate-/l* N/A

       \[\leadsto c0 \cdot \frac{c0 \cdot {d}^{2}}{\color{blue}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]

9. Applied rewrites 30.3%

   \[\leadsto c0 \cdot \color{blue}{\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)}} \]

10. Final simplification 30.3%

    \[\leadsto c0 \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)} \]
11. Add Preprocessing

Alternative 11: 0.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ \frac{c0}{w + w} \cdot \left(\sqrt{-1} \cdot M\right) \end{array} \]

FPCore

d_m = (fabs.f64 d)
(FPCore (c0 w h D d_m M)
 :precision binary64
 (* (/ c0 (+ w w)) (* (sqrt -1.0) M)))

C

d_m = fabs(d);
double code(double c0, double w, double h, double D, double d_m, double M) {
	return (c0 / (w + w)) * (sqrt(-1.0) * M);
}

Fortran

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

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

real(8) function code(c0, w, h, d, d_m, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_m
    real(8), intent (in) :: m
    code = (c0 / (w + w)) * (sqrt((-1.0d0)) * m)
end function

Java

d_m = Math.abs(d);
public static double code(double c0, double w, double h, double D, double d_m, double M) {
	return (c0 / (w + w)) * (Math.sqrt(-1.0) * M);
}

Python

d_m = math.fabs(d)
def code(c0, w, h, D, d_m, M):
	return (c0 / (w + w)) * (math.sqrt(-1.0) * M)

Julia

d_m = abs(d)
function code(c0, w, h, D, d_m, M)
	return Float64(Float64(c0 / Float64(w + w)) * Float64(sqrt(-1.0) * M))
end

MATLAB

d_m = abs(d);
function tmp = code(c0, w, h, D, d_m, M)
	tmp = (c0 / (w + w)) * (sqrt(-1.0) * M);
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
code[c0_, w_, h_, D_, d$95$m_, M_] := N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[-1.0], $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.7%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Add Preprocessing

3. Taylor expanded in c0 around 0

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot \color{blue}{M}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot \color{blue}{M}\right) \]

   3. lower-sqrt.f64 0.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot M\right) \]

5. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{-1} \cdot M\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{c0}{\color{blue}{2 \cdot w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

   2. count-2-rev N/A

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

   3. lower-+.f64 0.0

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

7. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025040 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))