Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.3% → 52.0%

Time: 10.8s

Alternatives: 9

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: 24.3% 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: 52.0% 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 79.8%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   2. Add Preprocessing
   3. 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 79.8

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

   4. Applied rewrites 79.8%

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

   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 w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 6.7%

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

   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. lower-*.f64 N/A

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

      7. lift-pow.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 45.2

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

   8. Applied rewrites 45.2%

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

Alternative 2: 52.8% accurate, 0.5× speedup

Math

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

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 (+ w w)) (/ (* 2.0 (* d (* d c0))) (* (* (* h w) D) D)))
     (/ (* -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 / (w + w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	} 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 / (w + w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	} 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 / (w + w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D))
	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(w + w)) * Float64(Float64(2.0 * Float64(d * Float64(d * c0))) / Float64(Float64(Float64(h * w) * D) * D)));
	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 / (w + w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	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[(w + 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], 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 79.8%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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 75.6

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

   5. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      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 76.9

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

   7. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 76.9

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

   9. Applied rewrites 76.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 6.7%

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

   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. lower-*.f64 N/A

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

      7. lift-pow.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 45.2

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

   8. Applied rewrites 45.2%

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

Alternative 3: 42.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -7.5 \cdot 10^{+142}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \frac{d \cdot c0}{D}}{D}}{w \cdot \left(w \cdot h\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= w -7.5e+142)
   (* (* c0 c0) (* d (/ d (* (* (* (* D D) w) h) w))))
   (/ (/ (* (* d c0) (/ (* d c0) D)) D) (* w (* w h)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (w <= -7.5e+142) {
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)));
	} else {
		tmp = (((d * c0) * ((d * c0) / D)) / D) / (w * (w * h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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 (w <= (-7.5d+142)) then
        tmp = (c0 * c0) * (d_1 * (d_1 / ((((d * d) * w) * h) * w)))
    else
        tmp = (((d_1 * c0) * ((d_1 * c0) / d)) / d) / (w * (w * h))
    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 (w <= -7.5e+142) {
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)));
	} else {
		tmp = (((d * c0) * ((d * c0) / D)) / D) / (w * (w * h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if w <= -7.5e+142:
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)))
	else:
		tmp = (((d * c0) * ((d * c0) / D)) / D) / (w * (w * h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (w <= -7.5e+142)
		tmp = Float64(Float64(c0 * c0) * Float64(d * Float64(d / Float64(Float64(Float64(Float64(D * D) * w) * h) * w))));
	else
		tmp = Float64(Float64(Float64(Float64(d * c0) * Float64(Float64(d * c0) / D)) / D) / Float64(w * Float64(w * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (w <= -7.5e+142)
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)));
	else
		tmp = (((d * c0) * ((d * c0) / D)) / D) / (w * (w * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[w, -7.5e+142], N[(N[(c0 * c0), $MachinePrecision] * N[(d * N[(d / N[(N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(N[(d * c0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -7.5000000000000002e142

   1. Initial program 4.3%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 4.2

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

   5. Applied rewrites 4.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 4.2

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 12.4

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

   7. Applied rewrites 12.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/l/ N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   9. Applied rewrites 24.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. unpow2 N/A

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

       14. *-commutative N/A

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

       15. lower-/.f64 N/A

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

       16. *-commutative N/A

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

       17. unpow2 N/A

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

       18. associate-*r* N/A

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

       19. *-commutative N/A

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

       20. associate-*l* N/A

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

   11. Applied rewrites 32.0%

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

   if -7.5000000000000002e142 < w

   1. Initial program 24.8%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 39.3

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 41.9

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

   7. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 41.9

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

   9. Applied rewrites 41.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 42.0

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

   11. Applied rewrites 42.0%

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

Alternative 4: 35.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 3.5 \cdot 10^{+174}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c0 \cdot c0\right) \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{w \cdot \left(w \cdot h\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 3.5e+174)
   (* (* c0 c0) (* d (/ d (* (* (* (* D D) w) h) w))))
   (/ (* (* c0 c0) (* (/ d D) (/ d D))) (* w (* w h)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 3.5e+174) {
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)));
	} else {
		tmp = ((c0 * c0) * ((d / D) * (d / D))) / (w * (w * h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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 <= 3.5d+174) then
        tmp = (c0 * c0) * (d_1 * (d_1 / ((((d * d) * w) * h) * w)))
    else
        tmp = ((c0 * c0) * ((d_1 / d) * (d_1 / d))) / (w * (w * h))
    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 <= 3.5e+174) {
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)));
	} else {
		tmp = ((c0 * c0) * ((d / D) * (d / D))) / (w * (w * h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 3.5e+174:
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)))
	else:
		tmp = ((c0 * c0) * ((d / D) * (d / D))) / (w * (w * h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 3.5e+174)
		tmp = Float64(Float64(c0 * c0) * Float64(d * Float64(d / Float64(Float64(Float64(Float64(D * D) * w) * h) * w))));
	else
		tmp = Float64(Float64(Float64(c0 * c0) * Float64(Float64(d / D) * Float64(d / D))) / Float64(w * Float64(w * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 3.5e+174)
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)));
	else
		tmp = ((c0 * c0) * ((d / D) * (d / D))) / (w * (w * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 3.5e+174], N[(N[(c0 * c0), $MachinePrecision] * N[(d * N[(d / N[(N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c0 * c0), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.5000000000000001e174

   1. Initial program 23.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 29.9

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

   5. Applied rewrites 29.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 35.5

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 38.8

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

   7. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/l/ N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   9. Applied rewrites 28.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. unpow2 N/A

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

       14. *-commutative N/A

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

       15. lower-/.f64 N/A

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

       16. *-commutative N/A

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

       17. unpow2 N/A

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

       18. associate-*r* N/A

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

       19. *-commutative N/A

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

       20. associate-*l* N/A

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

   11. Applied rewrites 34.1%

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

   if 3.5000000000000001e174 < D

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 41.0

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 41.2

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

   7. Applied rewrites 41.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 41.2

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

   9. Applied rewrites 41.2%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-/l/ N/A

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

       7. pow2 N/A

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

       8. *-commutative N/A

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

       9. pow-prod-down N/A

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

       10. pow2 N/A

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

       11. associate-/l* N/A

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

       12. lower-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 N/A

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

       15. pow2 N/A

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

       16. pow2 N/A

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

       17. times-frac N/A

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

       18. lower-*.f64 N/A

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

       19. lower-/.f64 N/A

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

       20. lower-/.f64 50.8

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

   11. Applied rewrites 50.8%

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

Alternative 5: 35.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 3.5 \cdot 10^{+174}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\left(w \cdot w\right) \cdot h}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 3.5e+174)
   (* (* c0 c0) (* d (/ d (* (* (* (* D D) w) h) w))))
   (* (* c0 c0) (/ (* (/ d D) (/ d D)) (* (* w w) h)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 3.5e+174) {
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)));
	} else {
		tmp = (c0 * c0) * (((d / D) * (d / D)) / ((w * w) * h));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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 <= 3.5d+174) then
        tmp = (c0 * c0) * (d_1 * (d_1 / ((((d * d) * w) * h) * w)))
    else
        tmp = (c0 * c0) * (((d_1 / d) * (d_1 / d)) / ((w * w) * h))
    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 <= 3.5e+174) {
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)));
	} else {
		tmp = (c0 * c0) * (((d / D) * (d / D)) / ((w * w) * h));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 3.5e+174:
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)))
	else:
		tmp = (c0 * c0) * (((d / D) * (d / D)) / ((w * w) * h))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 3.5e+174)
		tmp = Float64(Float64(c0 * c0) * Float64(d * Float64(d / Float64(Float64(Float64(Float64(D * D) * w) * h) * w))));
	else
		tmp = Float64(Float64(c0 * c0) * Float64(Float64(Float64(d / D) * Float64(d / D)) / Float64(Float64(w * w) * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 3.5e+174)
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)));
	else
		tmp = (c0 * c0) * (((d / D) * (d / D)) / ((w * w) * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 3.5e+174], N[(N[(c0 * c0), $MachinePrecision] * N[(d * N[(d / N[(N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * c0), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.5000000000000001e174

   1. Initial program 23.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 29.9

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

   5. Applied rewrites 29.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 35.5

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 38.8

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

   7. Applied rewrites 38.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/l/ N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   9. Applied rewrites 28.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. unpow2 N/A

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

       14. *-commutative N/A

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

       15. lower-/.f64 N/A

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

       16. *-commutative N/A

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

       17. unpow2 N/A

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

       18. associate-*r* N/A

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

       19. *-commutative N/A

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

       20. associate-*l* N/A

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

   11. Applied rewrites 34.1%

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

   if 3.5000000000000001e174 < D

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 41.0

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 41.2

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

   7. Applied rewrites 41.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/l/ N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   9. Applied rewrites 0.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. unpow2 N/A

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

       13. *-commutative N/A

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

       14. associate-/r* N/A

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

       15. lower-/.f64 N/A

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

       16. pow2 N/A

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

       17. pow2 N/A

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

       18. times-frac N/A

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

       19. lower-*.f64 N/A

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

       20. lower-/.f64 N/A

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

       21. lower-/.f64 N/A

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

   11. Applied rewrites 50.6%

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

Alternative 6: 33.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -2.15 \cdot 10^{-244}:\\ \;\;\;\;\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(w \cdot \left(w \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h -2.15e-244)
   (/ (* (* d c0) (* d c0)) (* (* D D) (* w (* w h))))
   (* (* c0 c0) (* d (/ d (* (* (* (* D D) w) h) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -2.15e-244) {
		tmp = ((d * c0) * (d * c0)) / ((D * D) * (w * (w * h)));
	} else {
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * 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 (h <= (-2.15d-244)) then
        tmp = ((d_1 * c0) * (d_1 * c0)) / ((d * d) * (w * (w * h)))
    else
        tmp = (c0 * c0) * (d_1 * (d_1 / ((((d * d) * w) * h) * 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 (h <= -2.15e-244) {
		tmp = ((d * c0) * (d * c0)) / ((D * D) * (w * (w * h)));
	} else {
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= -2.15e-244:
		tmp = ((d * c0) * (d * c0)) / ((D * D) * (w * (w * h)))
	else:
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= -2.15e-244)
		tmp = Float64(Float64(Float64(d * c0) * Float64(d * c0)) / Float64(Float64(D * D) * Float64(w * Float64(w * h))));
	else
		tmp = Float64(Float64(c0 * c0) * Float64(d * Float64(d / Float64(Float64(Float64(Float64(D * D) * w) * h) * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= -2.15e-244)
		tmp = ((d * c0) * (d * c0)) / ((D * D) * (w * (w * h)));
	else
		tmp = (c0 * c0) * (d * (d / ((((D * D) * w) * h) * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, -2.15e-244], N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * c0), $MachinePrecision] * N[(d * N[(d / N[(N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -2.14999999999999993e-244

   1. Initial program 23.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.6

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

   5. Applied rewrites 33.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 34.4

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 37.2

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

   7. Applied rewrites 37.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 37.2

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

   9. Applied rewrites 37.2%

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

   if -2.14999999999999993e-244 < h

   1. Initial program 22.1%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 24.8

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

   5. Applied rewrites 24.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 32.6

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 35.5

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

   7. Applied rewrites 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/l/ N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

   9. Applied rewrites 27.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. unpow2 N/A

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

       14. *-commutative N/A

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

       15. lower-/.f64 N/A

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

       16. *-commutative N/A

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

       17. unpow2 N/A

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

       18. associate-*r* N/A

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

       19. *-commutative N/A

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

       20. associate-*l* N/A

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

   11. Applied rewrites 33.9%

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

Alternative 7: 34.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in c0 around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 28.7

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

5. Applied rewrites 28.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow-prod-down N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. lift-pow.f64 N/A

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

   11. lift-*.f64 35.7

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 38.9

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

7. Applied rewrites 38.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unpow-prod-down N/A

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

   9. associate-/l/ N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. associate-/r* N/A

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

   15. associate-/l* N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-/.f64 N/A

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

9. Applied rewrites 27.3%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-/l* N/A

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. pow2 N/A

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

    10. associate-*l* N/A

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

    11. *-commutative N/A

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

    12. associate-*r* N/A

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

    13. unpow2 N/A

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

    14. *-commutative N/A

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

    15. lower-/.f64 N/A

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

    16. *-commutative N/A

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

    17. unpow2 N/A

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

    18. associate-*r* N/A

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

    19. *-commutative N/A

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

    20. associate-*l* N/A

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

11. Applied rewrites 33.1%

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

Alternative 8: 32.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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) * c0) / ((((d * d) * w) * h) * w))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in c0 around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 28.7

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

5. Applied rewrites 28.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow-prod-down N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow-prod-down N/A

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

   10. lift-pow.f64 N/A

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

   11. lift-*.f64 35.7

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 38.9

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

7. Applied rewrites 38.9%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 38.9

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

9. Applied rewrites 38.9%

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

11. Applied rewrites 29.2%

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

Alternative 9: 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 22.8%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
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 2025032 
(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))))))