Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.0% → 53.4%

Time: 30.4s

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: 25.0% 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: 53.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (*
      c0
      (/ (* 2.0 (* c0 (* (pow D -2.0) (/ (pow d 2.0) w)))) (* h (* 2.0 w))))
     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 tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = c0 * ((2.0 * (c0 * (pow(D, -2.0) * (pow(d, 2.0) / w)))) / (h * (2.0 * w)));
	} 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 tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = c0 * ((2.0 * (c0 * (Math.pow(D, -2.0) * (Math.pow(d, 2.0) / w)))) / (h * (2.0 * w)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = c0 * ((2.0 * (c0 * (math.pow(D, -2.0) * (math.pow(d, 2.0) / w)))) / (h * (2.0 * w)))
	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)))
	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(c0 * Float64(Float64(2.0 * Float64(c0 * Float64((D ^ -2.0) * Float64((d ^ 2.0) / w)))) / Float64(h * Float64(2.0 * w))));
	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));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = c0 * ((2.0 * (c0 * ((D ^ -2.0) * ((d ^ 2.0) / w)))) / (h * (2.0 * w)));
	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]}, 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[(c0 * N[(N[(2.0 * N[(c0 * N[(N[Power[D, -2.0], $MachinePrecision] * N[(N[Power[d, 2.0], $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h * N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 72.5%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in w around 0 69.3%

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

   6. Taylor expanded in h around 0 19.9%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{0.5 \cdot \frac{{D}^{2} \cdot \left(h \cdot \left(w \cdot \left(-1 \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w} + \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{{D}^{2} \cdot w}\right)\right)\right)}{c0 \cdot {d}^{2}} + 2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}}{h}}}{2 \cdot w} \]

   8. Simplified 70.3%

      \[\leadsto c0 \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{{D}^{2}}{c0} \cdot \frac{0}{{d}^{2}}, 2 \cdot \left(\frac{c0}{{D}^{2}} \cdot \frac{{d}^{2}}{w}\right)\right)}{h}}}{2 \cdot w} \]
   9. Step-by-step derivation

      1. pow1 70.3%

         \[\leadsto \color{blue}{{\left(c0 \cdot \frac{\frac{\mathsf{fma}\left(0.5, \frac{{D}^{2}}{c0} \cdot \frac{0}{{d}^{2}}, 2 \cdot \left(\frac{c0}{{D}^{2}} \cdot \frac{{d}^{2}}{w}\right)\right)}{h}}{2 \cdot w}\right)}^{1}} \]

   10. Applied egg-rr 72.7%

       \[\leadsto \color{blue}{{\left(c0 \cdot \frac{0 + 2 \cdot \left(\left(c0 \cdot {D}^{-2}\right) \cdot \frac{{d}^{2}}{w}\right)}{\left(w \cdot 2\right) \cdot h}\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 72.7%

          \[\leadsto \color{blue}{c0 \cdot \frac{0 + 2 \cdot \left(\left(c0 \cdot {D}^{-2}\right) \cdot \frac{{d}^{2}}{w}\right)}{\left(w \cdot 2\right) \cdot h}} \]

       2. +-lft-identity 72.7%

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

       3. associate-*l* 76.4%

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

       4. *-commutative 76.4%

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

   12. Simplified 76.4%

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

   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%

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

   3. Simplified 23.0%

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

   5. Taylor expanded in c0 around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

      2. neg-mul-1 0.2%

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

      3. distribute-lft-neg-in 0.2%

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

   7. Simplified 0.2%

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

      1. fma-undefine 0.1%

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

      2. associate-*r/ 0.1%

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

      3. pow2 0.1%

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

      4. associate-*r* 0.1%

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

      5. associate-/l* 0.2%

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

      6. pow2 0.2%

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

      7. associate-*r* 0.3%

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

      8. pow2 0.3%

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

   9. Applied egg-rr 0.3%

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

   10. Taylor expanded in c0 around 0 0.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
   11. Step-by-step derivation

       1. associate-*r/ 0.0%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \left({c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]

   12. Simplified 48.4%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(c0 \cdot \left({D}^{-2} \cdot \frac{{d}^{2}}{w}\right)\right)}{h \cdot \left(2 \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.3% accurate, 0.5× speedup

Math

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

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 72.5%

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

   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 23.0%

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

   5. Taylor expanded in c0 around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

      2. neg-mul-1 0.2%

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

      3. distribute-lft-neg-in 0.2%

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

   7. Simplified 0.2%

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

      1. fma-undefine 0.1%

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

      2. associate-*r/ 0.1%

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

      3. pow2 0.1%

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

      4. associate-*r* 0.1%

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

      5. associate-/l* 0.2%

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

      6. pow2 0.2%

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

      7. associate-*r* 0.3%

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

      8. pow2 0.3%

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

   9. Applied egg-rr 0.3%

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

   10. Taylor expanded in c0 around 0 0.0%

       \[\leadsto \color{blue}{0.5 \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
   11. Step-by-step derivation

       1. associate-*r/ 0.0%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \left({c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]

   12. Simplified 48.4%

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

Alternative 3: 33.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := t\_0 \cdot \frac{d \cdot d}{D \cdot D}\\ \mathbf{if}\;D \leq 3.1 \cdot 10^{+16}:\\ \;\;\;\;0\\ \mathbf{elif}\;D \leq 5.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (* t_0 (/ (* d d) (* D D)))))
   (if (<= D 3.1e+16)
     0.0
     (if (<= D 5.5e+152)
       (*
        (/ c0 (* 2.0 w))
        (+ (* t_0 (* (/ d D) (/ d D))) (sqrt (- (* t_1 t_1) (* 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);
	double t_1 = t_0 * ((d * d) / (D * D));
	double tmp;
	if (D <= 3.1e+16) {
		tmp = 0.0;
	} else if (D <= 5.5e+152) {
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d / D) * (d / D))) + sqrt(((t_1 * t_1) - (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) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = t_0 * ((d_1 * d_1) / (d * d))
    if (d <= 3.1d+16) then
        tmp = 0.0d0
    else if (d <= 5.5d+152) then
        tmp = (c0 / (2.0d0 * w)) * ((t_0 * ((d_1 / d) * (d_1 / d))) + sqrt(((t_1 * t_1) - (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);
	double t_1 = t_0 * ((d * d) / (D * D));
	double tmp;
	if (D <= 3.1e+16) {
		tmp = 0.0;
	} else if (D <= 5.5e+152) {
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d / D) * (d / D))) + Math.sqrt(((t_1 * t_1) - (M * M))));
	} 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 D <= 3.1e+16:
		tmp = 0.0
	elif D <= 5.5e+152:
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d / D) * (d / D))) + math.sqrt(((t_1 * t_1) - (M * M))))
	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 (D <= 3.1e+16)
		tmp = 0.0;
	elseif (D <= 5.5e+152)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(t_0 * Float64(Float64(d / D) * Float64(d / D))) + sqrt(Float64(Float64(t_1 * t_1) - 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);
	t_1 = t_0 * ((d * d) / (D * D));
	tmp = 0.0;
	if (D <= 3.1e+16)
		tmp = 0.0;
	elseif (D <= 5.5e+152)
		tmp = (c0 / (2.0 * w)) * ((t_0 * ((d / D) * (d / D))) + sqrt(((t_1 * t_1) - (M * M))));
	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[D, 3.1e+16], 0.0, If[LessEqual[D, 5.5e+152], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 2 regimes

2. if D < 3.1e16 or 5.4999999999999999e152 < D

   1. Initial program 22.5%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in c0 around -inf 2.4%

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

      1. associate-*r/ 2.4%

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

      2. neg-mul-1 2.4%

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

      3. distribute-lft-neg-in 2.4%

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

   7. Simplified 2.4%

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

      1. fma-undefine 2.8%

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

      2. associate-*r/ 3.2%

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

      3. pow2 3.2%

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

      4. associate-*r* 3.1%

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

      5. associate-/l* 4.0%

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

      6. pow2 4.0%

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

      7. associate-*r* 3.6%

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

      8. pow2 3.6%

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

   9. Applied egg-rr 3.6%

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

   10. Taylor expanded in c0 around 0 2.9%

       \[\leadsto \color{blue}{0.5 \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
   11. Step-by-step derivation

       1. associate-*r/ 2.9%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \left({c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]

   12. Simplified 39.0%

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

   if 3.1e16 < D < 5.4999999999999999e152

   1. Initial program 57.5%

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

   3. Simplified 63.1%

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

      1. times-frac 63.1%

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

   6. Applied egg-rr 63.1%

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

Alternative 4: 32.9% 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 24.6%

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

3. Simplified 38.9%

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

5. Taylor expanded in c0 around -inf 2.2%

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

   1. associate-*r/ 2.2%

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

   2. neg-mul-1 2.2%

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

   3. distribute-lft-neg-in 2.2%

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

7. Simplified 2.2%

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

   1. fma-undefine 2.6%

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

   2. associate-*r/ 3.0%

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

   3. pow2 3.0%

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

   4. associate-*r* 3.0%

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

   5. associate-/l* 3.7%

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

   6. pow2 3.7%

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

   7. associate-*r* 3.4%

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

   8. pow2 3.4%

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

9. Applied egg-rr 3.4%

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

10. Taylor expanded in c0 around 0 2.7%

    \[\leadsto \color{blue}{0.5 \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
11. Step-by-step derivation

    1. associate-*r/ 2.7%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left({c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]

12. Simplified 37.1%

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

Reproduce

herbie shell --seed 2024154 
(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))))))