Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 51.9%

Time: 12.8s

Alternatives: 15

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: 24.5% 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: 51.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot t\_1 \leq \infty:\\ \;\;\;\;\frac{c0}{w + w} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (+ t_0 (sqrt (- (* t_0 t_0) (* M M))))))
   (if (<= (* (/ c0 (* 2.0 w)) t_1) INFINITY)
     (* (/ c0 (+ w w)) t_1)
     (/ (* -0.25 (* (pow (* D M) 2.0) h)) (* d d)))))

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));
	double t_1 = t_0 + sqrt(((t_0 * t_0) - (M * M)));
	double tmp;
	if (((c0 / (2.0 * w)) * t_1) <= ((double) INFINITY)) {
		tmp = (c0 / (w + w)) * t_1;
	} else {
		tmp = (-0.25 * (pow((D * M), 2.0) * h)) / (d * d);
	}
	return tmp;
}

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));
	double t_1 = t_0 + Math.sqrt(((t_0 * t_0) - (M * M)));
	double tmp;
	if (((c0 / (2.0 * w)) * t_1) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 / (w + w)) * t_1;
	} else {
		tmp = (-0.25 * (Math.pow((D * M), 2.0) * h)) / (d * d);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = t_0 + math.sqrt(((t_0 * t_0) - (M * M)))
	tmp = 0
	if ((c0 / (2.0 * w)) * t_1) <= math.inf:
		tmp = (c0 / (w + w)) * t_1
	else:
		tmp = (-0.25 * (math.pow((D * M), 2.0) * h)) / (d * d)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_1 = Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * t_1) <= Inf)
		tmp = Float64(Float64(c0 / Float64(w + w)) * t_1);
	else
		tmp = Float64(Float64(-0.25 * Float64((Float64(D * M) ^ 2.0) * h)) / Float64(d * d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_1 = t_0 + sqrt(((t_0 * t_0) - (M * M)));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * t_1) <= Inf)
		tmp = (c0 / (w + w)) * t_1;
	else
		tmp = (-0.25 * (((D * M) ^ 2.0) * h)) / (d * d);
	end
	tmp_2 = tmp;
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]}, Block[{t$95$1 = N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], Infinity], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(-0.25 * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * d), $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 78.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 78.5

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

   4. Applied rewrites 78.5%

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

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 7.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{\color{blue}{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{\color{blue}{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2}} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2}} \]

      9. pow2 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d} \]

      10. lift-*.f64 37.2

          \[\leadsto \frac{-0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d} \]

   8. Applied rewrites 37.2%

      \[\leadsto \frac{-0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 51.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot \frac{2 \cdot \left(\left(c0 \cdot d\right) \cdot d\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* c0 (/ (* 2.0 (* (* c0 d) d)) (* (* (* D D) h) w))) (* w 2.0))
     (/ (* -0.25 (* (pow (* D M) 2.0) h)) (* d d)))))

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));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * ((2.0 * ((c0 * d) * d)) / (((D * D) * h) * w))) / (w * 2.0);
	} else {
		tmp = (-0.25 * (pow((D * M), 2.0) * h)) / (d * d);
	}
	return tmp;
}

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));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * ((2.0 * ((c0 * d) * d)) / (((D * D) * h) * w))) / (w * 2.0);
	} else {
		tmp = (-0.25 * (Math.pow((D * M), 2.0) * h)) / (d * d);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (c0 * ((2.0 * ((c0 * d) * d)) / (((D * D) * h) * w))) / (w * 2.0)
	else:
		tmp = (-0.25 * (math.pow((D * M), 2.0) * h)) / (d * d)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 * Float64(Float64(c0 * d) * d)) / Float64(Float64(Float64(D * D) * h) * w))) / Float64(w * 2.0));
	else
		tmp = Float64(Float64(-0.25 * Float64((Float64(D * M) ^ 2.0) * h)) / Float64(d * d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 * ((2.0 * ((c0 * d) * d)) / (((D * D) * h) * w))) / (w * 2.0);
	else
		tmp = (-0.25 * (((D * M) ^ 2.0) * h)) / (d * d);
	end
	tmp_2 = tmp;
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]}, If[LessEqual[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], Infinity], N[(N[(c0 * N[(N[(2.0 * N[(N[(c0 * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * d), $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 78.5%

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 77.3

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

   9. Applied rewrites 77.3%

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

   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 c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 7.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{\color{blue}{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{\color{blue}{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left(\left({D}^{2} \cdot {M}^{2}\right) \cdot h\right)}{{d}^{2}} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{{d}^{2}} \]

      9. pow2 N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d} \]

      10. lift-*.f64 37.2

          \[\leadsto \frac{-0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d} \]

   8. Applied rewrites 37.2%

      \[\leadsto \frac{-0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{\color{blue}{d \cdot d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0 \cdot \frac{2 \cdot \left(\left(c0 \cdot d\right) \cdot d\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left({\left(D \cdot M\right)}^{2} \cdot h\right)}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = t_0 / ((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 = (t_0 / ((((D * D) * h) * w) * w)) * c0;
	} else {
		tmp = (t_0 / ((D * (D * h)) * (w * w))) * c0;
	}
	return tmp;
}

Python

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(t_0 / 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(t_0 / Float64(Float64(Float64(Float64(D * D) * h) * w) * w)) * c0);
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(D * Float64(D * h)) * Float64(w * w))) * c0);
	end
	return tmp
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 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[(t$95$0 / N[(N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[(t$95$0 / N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $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 78.5%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 62.5%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 66.0

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

   10. Applied rewrites 66.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. pow2 N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 71.2

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

   12. Applied rewrites 71.2%

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

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 7.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 14.5%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 15.1

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

   10. Applied rewrites 15.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 19.2

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

   12. Applied rewrites 19.2%

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

Alternative 4: 37.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.3 \cdot 10^{-94}:\\ \;\;\;\;\left(\left(\frac{\frac{d}{D \cdot D}}{h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \frac{\frac{\frac{\frac{\left(\left(2 \cdot d\right) \cdot d\right) \cdot c0}{h}}{w}}{D}}{D}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 1.3e-94)
   (* (* (* (/ (/ d (* D D)) h) (/ d (* w w))) c0) c0)
   (/ (* c0 (/ (/ (/ (/ (* (* (* 2.0 d) d) c0) h) w) D) D)) (* 2.0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.3e-94) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 * (((((((2.0 * d) * d) * c0) / h) / w) / D) / D)) / (2.0 * w);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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) :: tmp
    if (d_1 <= 1.3d-94) then
        tmp = ((((d_1 / (d * d)) / h) * (d_1 / (w * w))) * c0) * c0
    else
        tmp = (c0 * (((((((2.0d0 * d_1) * d_1) * c0) / h) / w) / d) / d)) / (2.0d0 * w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.3e-94) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 * (((((((2.0 * d) * d) * c0) / h) / w) / D) / D)) / (2.0 * w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 1.3e-94:
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0
	else:
		tmp = (c0 * (((((((2.0 * d) * d) * c0) / h) / w) / D) / D)) / (2.0 * w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 1.3e-94)
		tmp = Float64(Float64(Float64(Float64(Float64(d / Float64(D * D)) / h) * Float64(d / Float64(w * w))) * c0) * c0);
	else
		tmp = Float64(Float64(c0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * d) * d) * c0) / h) / w) / D) / D)) / Float64(2.0 * w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 1.3e-94)
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	else
		tmp = (c0 * (((((((2.0 * d) * d) * c0) / h) / w) / D) / D)) / (2.0 * w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 1.3e-94], N[(N[(N[(N[(N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] * N[(d / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * c0), $MachinePrecision], N[(N[(c0 * N[(N[(N[(N[(N[(N[(N[(2.0 * d), $MachinePrecision] * d), $MachinePrecision] * c0), $MachinePrecision] / h), $MachinePrecision] / w), $MachinePrecision] / D), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.29999999999999997e-94

   1. Initial program 30.4%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 36.4%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 45.0%

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

   if 1.29999999999999997e-94 < d

   1. Initial program 28.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 38.2

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

   5. Applied rewrites 38.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   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 42.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 42.8

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

   9. Applied rewrites 42.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 46.4

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

   11. Applied rewrites 46.4%

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

Alternative 5: 37.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;\left(\left(\frac{\frac{d}{D \cdot D}}{h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \frac{\frac{2 \cdot \left(\left(c0 \cdot d\right) \cdot d\right)}{D \cdot \left(h \cdot w\right)}}{D}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 1.2e-94)
   (* (* (* (/ (/ d (* D D)) h) (/ d (* w w))) c0) c0)
   (/ (* c0 (/ (/ (* 2.0 (* (* c0 d) d)) (* D (* h w))) D)) (* 2.0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.2e-94) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 * (((2.0 * ((c0 * d) * d)) / (D * (h * w))) / D)) / (2.0 * w);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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) :: tmp
    if (d_1 <= 1.2d-94) then
        tmp = ((((d_1 / (d * d)) / h) * (d_1 / (w * w))) * c0) * c0
    else
        tmp = (c0 * (((2.0d0 * ((c0 * d_1) * d_1)) / (d * (h * w))) / d)) / (2.0d0 * w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.2e-94) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 * (((2.0 * ((c0 * d) * d)) / (D * (h * w))) / D)) / (2.0 * w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 1.2e-94:
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0
	else:
		tmp = (c0 * (((2.0 * ((c0 * d) * d)) / (D * (h * w))) / D)) / (2.0 * w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 1.2e-94)
		tmp = Float64(Float64(Float64(Float64(Float64(d / Float64(D * D)) / h) * Float64(d / Float64(w * w))) * c0) * c0);
	else
		tmp = Float64(Float64(c0 * Float64(Float64(Float64(2.0 * Float64(Float64(c0 * d) * d)) / Float64(D * Float64(h * w))) / D)) / Float64(2.0 * w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 1.2e-94)
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	else
		tmp = (c0 * (((2.0 * ((c0 * d) * d)) / (D * (h * w))) / D)) / (2.0 * w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 1.2e-94], N[(N[(N[(N[(N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] * N[(d / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * c0), $MachinePrecision], N[(N[(c0 * N[(N[(N[(2.0 * N[(N[(c0 * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.2e-94

   1. Initial program 30.4%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 36.4%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 45.0%

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

   if 1.2e-94 < d

   1. Initial program 28.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 38.2

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

   5. Applied rewrites 38.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   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 42.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 42.8

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

   9. Applied rewrites 42.8%

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

   10. Taylor expanded in c0 around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 44.4

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

   12. Applied rewrites 44.4%

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

Alternative 6: 39.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\left(\left(\frac{\frac{d}{D \cdot D}}{h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(d \cdot \left(d \cdot c0\right)\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 5e-187)
   (* (* (* (/ (/ d (* D D)) h) (/ d (* w w))) c0) c0)
   (* (/ c0 (* 2.0 w)) (/ (* 2.0 (* d (* d c0))) (* (* (* h w) D) D)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 5e-187) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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) :: tmp
    if ((d_1 * d_1) <= 5d-187) then
        tmp = ((((d_1 / (d * d)) / h) * (d_1 / (w * w))) * c0) * c0
    else
        tmp = (c0 / (2.0d0 * w)) * ((2.0d0 * (d_1 * (d_1 * c0))) / (((h * w) * d) * d))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 5e-187) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 5e-187:
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0
	else:
		tmp = (c0 / (2.0 * w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 5e-187)
		tmp = Float64(Float64(Float64(Float64(Float64(d / Float64(D * D)) / h) * Float64(d / Float64(w * w))) * c0) * c0);
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(2.0 * Float64(d * Float64(d * c0))) / Float64(Float64(Float64(h * w) * D) * D)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 5e-187)
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	else
		tmp = (c0 / (2.0 * w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 5e-187], N[(N[(N[(N[(N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] * N[(d / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * c0), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(d * N[(d * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 d d) < 4.9999999999999996e-187

   1. Initial program 30.0%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 20.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 34.6%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 53.1%

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

   if 4.9999999999999996e-187 < (*.f64 d d)

   1. Initial program 29.7%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 43.9

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

   5. Applied rewrites 43.9%

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

      1. lift-*.f64 N/A

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

      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

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 45.3

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

   7. Applied rewrites 45.3%

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

Alternative 7: 36.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;\left(\left(\frac{\frac{d}{D \cdot D}}{h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \frac{\frac{\left(2 \cdot \left(d \cdot d\right)\right) \cdot c0}{\left(h \cdot w\right) \cdot D}}{D}}{w + w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 1.2e-94)
   (* (* (* (/ (/ d (* D D)) h) (/ d (* w w))) c0) c0)
   (/ (* c0 (/ (/ (* (* 2.0 (* d d)) c0) (* (* h w) D)) D)) (+ w w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.2e-94) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 * ((((2.0 * (d * d)) * c0) / ((h * w) * D)) / D)) / (w + w);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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) :: tmp
    if (d_1 <= 1.2d-94) then
        tmp = ((((d_1 / (d * d)) / h) * (d_1 / (w * w))) * c0) * c0
    else
        tmp = (c0 * ((((2.0d0 * (d_1 * d_1)) * c0) / ((h * w) * d)) / d)) / (w + w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 1.2e-94) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 * ((((2.0 * (d * d)) * c0) / ((h * w) * D)) / D)) / (w + w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 1.2e-94:
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0
	else:
		tmp = (c0 * ((((2.0 * (d * d)) * c0) / ((h * w) * D)) / D)) / (w + w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 1.2e-94)
		tmp = Float64(Float64(Float64(Float64(Float64(d / Float64(D * D)) / h) * Float64(d / Float64(w * w))) * c0) * c0);
	else
		tmp = Float64(Float64(c0 * Float64(Float64(Float64(Float64(2.0 * Float64(d * d)) * c0) / Float64(Float64(h * w) * D)) / D)) / Float64(w + w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 1.2e-94)
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	else
		tmp = (c0 * ((((2.0 * (d * d)) * c0) / ((h * w) * D)) / D)) / (w + w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 1.2e-94], N[(N[(N[(N[(N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] * N[(d / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * c0), $MachinePrecision], N[(N[(c0 * N[(N[(N[(N[(2.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.2e-94

   1. Initial program 30.4%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 36.4%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 45.0%

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

   if 1.2e-94 < d

   1. Initial program 28.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 38.2

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

   5. Applied rewrites 38.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   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 42.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 42.8

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

   9. Applied rewrites 42.8%

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

Alternative 8: 38.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 2 \cdot 10^{-188}:\\ \;\;\;\;\left(\left(\frac{\frac{d}{D \cdot D}}{h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 2e-188)
   (* (* (* (/ (/ d (* D D)) h) (/ d (* w w))) c0) c0)
   (* (/ c0 (+ w w)) (/ (* 2.0 (* (* d d) c0)) (* (* (* h w) D) D)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 2e-188) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 / (w + w)) * ((2.0 * ((d * d) * c0)) / (((h * w) * D) * D));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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) :: tmp
    if ((d_1 * d_1) <= 2d-188) then
        tmp = ((((d_1 / (d * d)) / h) * (d_1 / (w * w))) * c0) * c0
    else
        tmp = (c0 / (w + w)) * ((2.0d0 * ((d_1 * d_1) * c0)) / (((h * w) * d) * d))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 2e-188) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 / (w + w)) * ((2.0 * ((d * d) * c0)) / (((h * w) * D) * D));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 2e-188:
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0
	else:
		tmp = (c0 / (w + w)) * ((2.0 * ((d * d) * c0)) / (((h * w) * D) * D))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 2e-188)
		tmp = Float64(Float64(Float64(Float64(Float64(d / Float64(D * D)) / h) * Float64(d / Float64(w * w))) * c0) * c0);
	else
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(Float64(2.0 * Float64(Float64(d * d) * c0)) / Float64(Float64(Float64(h * w) * D) * D)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 2e-188)
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	else
		tmp = (c0 / (w + w)) * ((2.0 * ((d * d) * c0)) / (((h * w) * D) * D));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 2e-188], N[(N[(N[(N[(N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] * N[(d / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * c0), $MachinePrecision], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(N[(d * d), $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 (*.f64 d d) < 1.9999999999999999e-188

   1. Initial program 30.6%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 21.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 35.3%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 54.3%

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

   if 1.9999999999999999e-188 < (*.f64 d d)

   1. Initial program 29.6%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 43.7

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

   5. Applied rewrites 43.7%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   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 43.7

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

   7. Applied rewrites 43.7%

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

Alternative 9: 36.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 2.6 \cdot 10^{-94}:\\ \;\;\;\;\left(\left(\frac{\frac{d}{D \cdot D}}{h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 2.6e-94)
   (* (* (* (/ (/ d (* D D)) h) (/ d (* w w))) c0) c0)
   (* (/ c0 (* 2.0 w)) (/ (* 2.0 (* (* d d) c0)) (* (* h (* w D)) D)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 2.6e-94) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((2.0 * ((d * d) * c0)) / ((h * (w * D)) * D));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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) :: tmp
    if (d_1 <= 2.6d-94) then
        tmp = ((((d_1 / (d * d)) / h) * (d_1 / (w * w))) * c0) * c0
    else
        tmp = (c0 / (2.0d0 * w)) * ((2.0d0 * ((d_1 * d_1) * c0)) / ((h * (w * d)) * d))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 2.6e-94) {
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	} else {
		tmp = (c0 / (2.0 * w)) * ((2.0 * ((d * d) * c0)) / ((h * (w * D)) * D));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 2.6e-94:
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0
	else:
		tmp = (c0 / (2.0 * w)) * ((2.0 * ((d * d) * c0)) / ((h * (w * D)) * D))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 2.6e-94)
		tmp = Float64(Float64(Float64(Float64(Float64(d / Float64(D * D)) / h) * Float64(d / Float64(w * w))) * c0) * c0);
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(2.0 * Float64(Float64(d * d) * c0)) / Float64(Float64(h * Float64(w * D)) * D)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 2.6e-94)
		tmp = ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
	else
		tmp = (c0 / (2.0 * w)) * ((2.0 * ((d * d) * c0)) / ((h * (w * D)) * D));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 2.6e-94], N[(N[(N[(N[(N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] * N[(d / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * c0), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.59999999999999994e-94

   1. Initial program 30.4%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 36.4%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 45.0%

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

   if 2.59999999999999994e-94 < d

   1. Initial program 28.9%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 38.2

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

   5. Applied rewrites 38.2%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   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 40.1

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

   7. Applied rewrites 40.1%

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

Alternative 10: 32.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ \mathbf{if}\;D \leq 9 \cdot 10^{-23}:\\ \;\;\;\;\frac{t\_0}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot w\right) \cdot w} \cdot c0\\ \mathbf{elif}\;D \leq 3.9 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\left(c0 \cdot d\right) \cdot d}{D \cdot D}}{h \cdot \left(w \cdot w\right)} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(D \cdot \left(D \cdot h\right)\right) \cdot \left(w \cdot w\right)} \cdot c0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d))))
   (if (<= D 9e-23)
     (* (/ t_0 (* (* (* (* D D) h) w) w)) c0)
     (if (<= D 3.9e+130)
       (* (/ (/ (* (* c0 d) d) (* D D)) (* h (* w w))) c0)
       (* (/ t_0 (* (* D (* D h)) (* w w))) c0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double tmp;
	if (D <= 9e-23) {
		tmp = (t_0 / ((((D * D) * h) * w) * w)) * c0;
	} else if (D <= 3.9e+130) {
		tmp = ((((c0 * d) * d) / (D * D)) / (h * (w * w))) * c0;
	} else {
		tmp = (t_0 / ((D * (D * h)) * (w * w))) * c0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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
    real(8) :: tmp
    t_0 = c0 * (d_1 * d_1)
    if (d <= 9d-23) then
        tmp = (t_0 / ((((d * d) * h) * w) * w)) * c0
    else if (d <= 3.9d+130) then
        tmp = ((((c0 * d_1) * d_1) / (d * d)) / (h * (w * w))) * c0
    else
        tmp = (t_0 / ((d * (d * h)) * (w * w))) * c0
    end if
    code = tmp
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);
	double tmp;
	if (D <= 9e-23) {
		tmp = (t_0 / ((((D * D) * h) * w) * w)) * c0;
	} else if (D <= 3.9e+130) {
		tmp = ((((c0 * d) * d) / (D * D)) / (h * (w * w))) * c0;
	} else {
		tmp = (t_0 / ((D * (D * h)) * (w * w))) * c0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 * (d * d)
	tmp = 0
	if D <= 9e-23:
		tmp = (t_0 / ((((D * D) * h) * w) * w)) * c0
	elif D <= 3.9e+130:
		tmp = ((((c0 * d) * d) / (D * D)) / (h * (w * w))) * c0
	else:
		tmp = (t_0 / ((D * (D * h)) * (w * w))) * c0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	tmp = 0.0
	if (D <= 9e-23)
		tmp = Float64(Float64(t_0 / Float64(Float64(Float64(Float64(D * D) * h) * w) * w)) * c0);
	elseif (D <= 3.9e+130)
		tmp = Float64(Float64(Float64(Float64(Float64(c0 * d) * d) / Float64(D * D)) / Float64(h * Float64(w * w))) * c0);
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(D * Float64(D * h)) * Float64(w * w))) * c0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 * (d * d);
	tmp = 0.0;
	if (D <= 9e-23)
		tmp = (t_0 / ((((D * D) * h) * w) * w)) * c0;
	elseif (D <= 3.9e+130)
		tmp = ((((c0 * d) * d) / (D * D)) / (h * (w * w))) * c0;
	else
		tmp = (t_0 / ((D * (D * h)) * (w * w))) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 9e-23], N[(N[(t$95$0 / N[(N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[D, 3.9e+130], N[(N[(N[(N[(N[(c0 * d), $MachinePrecision] * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], N[(N[(t$95$0 / N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if D < 8.9999999999999995e-23

   1. Initial program 30.2%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 25.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 34.0%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 35.6

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

   10. Applied rewrites 35.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. pow2 N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 39.0

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

   12. Applied rewrites 39.0%

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

   if 8.9999999999999995e-23 < D < 3.9000000000000002e130

   1. Initial program 32.7%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 16.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 33.4%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 33.7

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

   10. Applied rewrites 33.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. pow2 N/A

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

       11. associate-*r* N/A

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

       12. associate-/r* N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. pow2 N/A

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

       16. associate-*r* N/A

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

       17. lower-*.f64 N/A

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

       18. lift-*.f64 N/A

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

       19. pow2 N/A

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

       20. lift-*.f64 N/A

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

       21. lower-*.f64 N/A

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

   12. Applied rewrites 45.2%

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

   if 3.9000000000000002e130 < D

   1. Initial program 18.8%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 12.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 14.1%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 18.8

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

   10. Applied rewrites 18.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 43.9

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

   12. Applied rewrites 43.9%

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

Alternative 11: 33.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(D \cdot D\right) \cdot h\\ \mathbf{if}\;d \cdot d \leq 10^{-255}:\\ \;\;\;\;\left(\frac{d}{t\_0} \cdot \frac{d}{w \cdot w}\right) \cdot \left(c0 \cdot c0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(d \cdot d\right)}{\left(t\_0 \cdot w\right) \cdot w} \cdot c0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* D D) h)))
   (if (<= (* d d) 1e-255)
     (* (* (/ d t_0) (/ d (* w w))) (* c0 c0))
     (* (/ (* c0 (* d d)) (* (* t_0 w) w)) c0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * D) * h;
	double tmp;
	if ((d * d) <= 1e-255) {
		tmp = ((d / t_0) * (d / (w * w))) * (c0 * c0);
	} else {
		tmp = ((c0 * (d * d)) / ((t_0 * w) * w)) * c0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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
    real(8) :: tmp
    t_0 = (d * d) * h
    if ((d_1 * d_1) <= 1d-255) then
        tmp = ((d_1 / t_0) * (d_1 / (w * w))) * (c0 * c0)
    else
        tmp = ((c0 * (d_1 * d_1)) / ((t_0 * w) * w)) * c0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * D) * h;
	double tmp;
	if ((d * d) <= 1e-255) {
		tmp = ((d / t_0) * (d / (w * w))) * (c0 * c0);
	} else {
		tmp = ((c0 * (d * d)) / ((t_0 * w) * w)) * c0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (D * D) * h
	tmp = 0
	if (d * d) <= 1e-255:
		tmp = ((d / t_0) * (d / (w * w))) * (c0 * c0)
	else:
		tmp = ((c0 * (d * d)) / ((t_0 * w) * w)) * c0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(D * D) * h)
	tmp = 0.0
	if (Float64(d * d) <= 1e-255)
		tmp = Float64(Float64(Float64(d / t_0) * Float64(d / Float64(w * w))) * Float64(c0 * c0));
	else
		tmp = Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(t_0 * w) * w)) * c0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (D * D) * h;
	tmp = 0.0;
	if ((d * d) <= 1e-255)
		tmp = ((d / t_0) * (d / (w * w))) * (c0 * c0);
	else
		tmp = ((c0 * (d * d)) / ((t_0 * w) * w)) * c0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision]}, If[LessEqual[N[(d * d), $MachinePrecision], 1e-255], N[(N[(N[(d / t$95$0), $MachinePrecision] * N[(d / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c0 * c0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 d d) < 1e-255

   1. Initial program 24.2%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   6. Taylor expanded in c0 around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 48.8

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

   8. Applied rewrites 48.8%

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

   if 1e-255 < (*.f64 d d)

   1. Initial program 30.6%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 25.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

   7. Applied rewrites 33.0%

      \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

   8. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 35.8

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

   10. Applied rewrites 35.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. pow2 N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 39.1

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

   12. Applied rewrites 39.1%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq 10^{-255}:\\ \;\;\;\;\left(\frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot \left(c0 \cdot c0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(d \cdot d\right)}{\left(\left(\left(D \cdot D\right) \cdot h\right) \cdot w\right) \cdot w} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.9% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (* (* (/ (/ d (* D D)) h) (/ d (* w w))) c0) c0))

C

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

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
    code = ((((d_1 / (d * d)) / h) * (d_1 / (w * w))) * c0) * c0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0;
}

Python

def code(c0, w, h, D, d, M):
	return ((((d / (D * D)) / h) * (d / (w * w))) * c0) * c0

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(N[(N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] * N[(d / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * c0), $MachinePrecision]

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in c0 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 24.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

7. Applied rewrites 32.7%

   \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

8. Taylor expanded in c0 around inf

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

   1. pow2 N/A

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

   2. associate-*r* N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. frac-times N/A

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

   7. lift-/.f64 N/A

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

   8. pow2 N/A

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

   9. lower-*.f64 N/A

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

10. Applied rewrites 41.3%

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

Alternative 13: 30.0% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ (* c0 (* d d)) (* (* D (* D h)) (* w w))) c0))

C

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

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
    code = ((c0 * (d_1 * d_1)) / ((d * (d * h)) * (w * w))) * c0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return ((c0 * (d * d)) / ((D * (D * h)) * (w * w))) * c0;
}

Python

def code(c0, w, h, D, d, M):
	return ((c0 * (d * d)) / ((D * (D * h)) * (w * w))) * c0

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in c0 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 24.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

7. Applied rewrites 32.7%

   \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

8. Taylor expanded in c0 around inf

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 34.4

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

10. Applied rewrites 34.4%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 36.6

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

12. Applied rewrites 36.6%

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

Alternative 14: 29.0% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ (* c0 (* d d)) (* (* D D) (* h (* w w)))) c0))

C

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

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
    code = ((c0 * (d_1 * d_1)) / ((d * d) * (h * (w * w)))) * c0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return ((c0 * (d * d)) / ((D * D) * (h * (w * w)))) * c0;
}

Python

def code(c0, w, h, D, d, M):
	return ((c0 * (d * d)) / ((D * D) * (h * (w * w)))) * c0

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in c0 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 24.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\right) \cdot \left(c0 \cdot c0\right)} \]

7. Applied rewrites 32.7%

   \[\leadsto \left(\mathsf{fma}\left({\left(D \cdot M\right)}^{2} \cdot \frac{h}{{\left(c0 \cdot d\right)}^{2}}, -0.25, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w \cdot w}\right) \cdot c0\right) \cdot \color{blue}{c0} \]

8. Taylor expanded in c0 around inf

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 34.4

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

10. Applied rewrites 34.4%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. pow2 N/A

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

    5. lift-*.f64 N/A

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

    6. pow2 N/A

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

    7. associate-*r* N/A

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

    8. lower-*.f64 N/A

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

    9. pow2 N/A

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

    10. lift-*.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. pow2 N/A

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

    13. lift-*.f64 33.0

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

12. Applied rewrites 33.0%

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

Alternative 15: 0.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{c0}{w + w} \cdot \left(\sqrt{-1} \cdot M\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ c0 (+ w w)) (* (sqrt -1.0) M)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (w + w)) * (sqrt(-1.0) * 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
    code = (c0 / (w + w)) * (sqrt((-1.0d0)) * m)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (w + w)) * (Math.sqrt(-1.0) * M);
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (w + w)) * (math.sqrt(-1.0) * M)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(w + w)) * Float64(sqrt(-1.0) * M))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 / (w + w)) * (sqrt(-1.0) * M);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[-1.0], $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.8%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
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 2025061 
(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))))))