Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 55.0%

Time: 28.2s

Alternatives: 4

Speedup: 151.0×

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    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: 55.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 INFINITY) t_1 0.0)))

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 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.0;
	}
	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 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.0
	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(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.0;
	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 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.0;
	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[(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]}, If[LessEqual[t$95$1, Infinity], t$95$1, 0.0]]]

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

   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%

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

   3. Simplified 22.6%

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

   6. Applied egg-rr 14.9%

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

   7. Taylor expanded in c0 around -inf 0.1%

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

      1. mul-1-neg 0.1%

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

      2. associate-/r* 2.0%

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

      3. associate-/l* 2.0%

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

   9. Simplified 2.0%

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

       1. fma-define 2.0%

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

       2. associate-*l/ 1.4%

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

       3. distribute-rgt-neg-out 1.4%

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

       4. pow2 1.4%

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

       5. pow2 1.4%

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

       6. frac-times 1.4%

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

       7. pow2 1.4%

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

       8. *-commutative 1.4%

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

       9. fmm-undef 3.3%

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

   11. Applied egg-rr 3.9%

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

       1. +-inverses 34.3%

          \[\leadsto \color{blue}{0} \]

   13. Simplified 34.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 33.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := t\_0 \cdot \frac{d \cdot d}{D \cdot D}\\ \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-302}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-187} \lor \neg \left(M \cdot M \leq 4 \cdot 10^{+27}\right) \land M \cdot M \leq 4 \cdot 10^{+188}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{t\_1 \cdot t\_1 - M \cdot M} + t\_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (* t_0 (/ (* d d) (* D D)))))
   (if (<= (* M M) 2e-302)
     0.0
     (if (or (<= (* M M) 5e-187)
             (and (not (<= (* M M) 4e+27)) (<= (* M M) 4e+188)))
       (*
        (/ c0 (* 2.0 w))
        (+ (sqrt (- (* t_1 t_1) (* M M))) (* t_0 (* (/ d D) (/ d D)))))
       0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double tmp;
	if ((M * M) <= 2e-302) {
		tmp = 0.0;
	} else if (((M * M) <= 5e-187) || (!((M * M) <= 4e+27) && ((M * M) <= 4e+188))) {
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = t_0 * ((d_1 * d_1) / (d * d))
    if ((m * m) <= 2d-302) then
        tmp = 0.0d0
    else if (((m * m) <= 5d-187) .or. (.not. ((m * m) <= 4d+27)) .and. ((m * m) <= 4d+188)) then
        tmp = (c0 / (2.0d0 * w)) * (sqrt(((t_1 * t_1) - (m * m))) + (t_0 * ((d_1 / d) * (d_1 / d))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double tmp;
	if ((M * M) <= 2e-302) {
		tmp = 0.0;
	} else if (((M * M) <= 5e-187) || (!((M * M) <= 4e+27) && ((M * M) <= 4e+188))) {
		tmp = (c0 / (2.0 * w)) * (Math.sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = t_0 * ((d * d) / (D * D))
	tmp = 0
	if (M * M) <= 2e-302:
		tmp = 0.0
	elif ((M * M) <= 5e-187) or (not ((M * M) <= 4e+27) and ((M * M) <= 4e+188)):
		tmp = (c0 / (2.0 * w)) * (math.sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(t_0 * Float64(Float64(d * d) / Float64(D * D)))
	tmp = 0.0
	if (Float64(M * M) <= 2e-302)
		tmp = 0.0;
	elseif ((Float64(M * M) <= 5e-187) || (!(Float64(M * M) <= 4e+27) && (Float64(M * M) <= 4e+188)))
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))) + Float64(t_0 * Float64(Float64(d / D) * Float64(d / D)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = t_0 * ((d * d) / (D * D));
	tmp = 0.0;
	if ((M * M) <= 2e-302)
		tmp = 0.0;
	elseif (((M * M) <= 5e-187) || (~(((M * M) <= 4e+27)) && ((M * M) <= 4e+188)))
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * M), $MachinePrecision], 2e-302], 0.0, If[Or[LessEqual[N[(M * M), $MachinePrecision], 5e-187], And[N[Not[LessEqual[N[(M * M), $MachinePrecision], 4e+27]], $MachinePrecision], LessEqual[N[(M * M), $MachinePrecision], 4e+188]]], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 1.9999999999999999e-302 or 4.9999999999999996e-187 < (*.f64 M M) < 4.0000000000000001e27 or 4.0000000000000001e188 < (*.f64 M M)

   1. Initial program 16.7%

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

   3. Simplified 34.5%

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

   6. Applied egg-rr 27.9%

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

   7. Taylor expanded in c0 around -inf 1.8%

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

      1. mul-1-neg 1.8%

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

      2. associate-/r* 3.4%

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

      3. associate-/l* 3.5%

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

   9. Simplified 3.5%

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

       1. fma-define 3.0%

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

       2. associate-*l/ 2.5%

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

       3. distribute-rgt-neg-out 2.5%

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

       4. pow2 2.5%

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

       5. pow2 2.5%

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

       6. frac-times 2.6%

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

       7. pow2 2.6%

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

       8. *-commutative 2.6%

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

       9. fmm-undef 5.7%

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

   11. Applied egg-rr 6.6%

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

       1. +-inverses 33.0%

          \[\leadsto \color{blue}{0} \]

   13. Simplified 33.0%

       \[\leadsto \color{blue}{0} \]

   if 1.9999999999999999e-302 < (*.f64 M M) < 4.9999999999999996e-187 or 4.0000000000000001e27 < (*.f64 M M) < 4.0000000000000001e188

   1. Initial program 52.0%

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

   3. Simplified 52.0%

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

      1. times-frac 52.0%

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

   6. Applied egg-rr 52.0%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 2 \cdot 10^{-302}:\\ \;\;\;\;0\\ \mathbf{elif}\;M \cdot M \leq 5 \cdot 10^{-187} \lor \neg \left(M \cdot M \leq 4 \cdot 10^{+27}\right) \land M \cdot M \leq 4 \cdot 10^{+188}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) - M \cdot M} + \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 29.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\\ \mathbf{if}\;w \leq -4 \cdot 10^{-283}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (/ (* d d) (* D D)))))
   (if (<= w -4e-283)
     (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double tmp;
	if (w <= -4e-283) {
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c0 / (w * h)) * ((d_1 * d_1) / (d * d))
    if (w <= (-4d-283)) then
        tmp = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double tmp;
	if (w <= -4e-283) {
		tmp = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D))
	tmp = 0
	if w <= -4e-283:
		tmp = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D)))
	tmp = 0.0
	if (w <= -4e-283)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	tmp = 0.0;
	if (w <= -4e-283)
		tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -4e-283], 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], 0.0]]

Derivation

1. Split input into 2 regimes

2. if w < -3.99999999999999979e-283

   1. Initial program 30.4%

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

   3. Simplified 30.6%

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

   if -3.99999999999999979e-283 < w

   1. Initial program 20.2%

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

   3. Simplified 37.0%

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

   6. Applied egg-rr 27.7%

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

   7. Taylor expanded in c0 around -inf 1.0%

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

      1. mul-1-neg 1.0%

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

      2. associate-/r* 2.6%

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

      3. associate-/l* 1.8%

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

   9. Simplified 1.8%

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

       1. fma-define 1.2%

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

       2. associate-*l/ 1.1%

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

       3. distribute-rgt-neg-out 1.1%

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

       4. pow2 1.1%

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

       5. pow2 1.1%

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

       6. frac-times 1.1%

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

       7. pow2 1.1%

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

       8. *-commutative 1.1%

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

       9. fmm-undef 3.5%

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

   11. Applied egg-rr 3.4%

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

       1. +-inverses 35.3%

          \[\leadsto \color{blue}{0} \]

   13. Simplified 35.3%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 34.1% accurate, 151.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 0.0)

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    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 = 0.0d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Python

def code(c0, w, h, D, d, M):
	return 0.0

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.0;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 25.3%

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

3. Simplified 39.9%

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

6. Applied egg-rr 35.0%

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

7. Taylor expanded in c0 around -inf 1.4%

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

   1. mul-1-neg 1.4%

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

   2. associate-/r* 2.6%

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

   3. associate-/l* 2.7%

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

9. Simplified 2.7%

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

    1. fma-define 2.4%

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

    2. associate-*l/ 2.0%

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

    3. distribute-rgt-neg-out 2.0%

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

    4. pow2 2.0%

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

    5. pow2 2.0%

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

    6. frac-times 2.1%

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

    7. pow2 2.1%

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

    8. *-commutative 2.1%

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

    9. fmm-undef 4.4%

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

11. Applied egg-rr 5.1%

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

    1. +-inverses 27.6%

       \[\leadsto \color{blue}{0} \]

13. Simplified 27.6%

    \[\leadsto \color{blue}{0} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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))))))