Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.3% → 54.5%

Time: 30.2s

Alternatives: 5

Speedup: 21.6×

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: 25.3% 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

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: 54.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{D} \cdot \sqrt{\frac{\frac{c0}{w}}{h}}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(t\_0, -0.5 \cdot \left(\frac{D \cdot M}{d} \cdot \sqrt{\frac{w \cdot h}{c0}}\right) + \frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}, {t\_0}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d D) (sqrt (/ (/ c0 w) h))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (*
      t_1
      (fma
       t_0
       (+
        (* -0.5 (* (/ (* D M) d) (sqrt (/ (* w h) c0))))
        (* (/ d D) (sqrt (/ c0 (* w h)))))
       (pow t_0 2.0)))
     (* c0 (/ 0.0 (* 2.0 w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * sqrt(((c0 / w) / h));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * fma(t_0, ((-0.5 * (((D * M) / d) * sqrt(((w * h) / c0)))) + ((d / D) * sqrt((c0 / (w * h))))), pow(t_0, 2.0));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / D) * sqrt(Float64(Float64(c0 / w) / h)))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * fma(t_0, Float64(Float64(-0.5 * Float64(Float64(Float64(D * M) / d) * sqrt(Float64(Float64(w * h) / c0)))) + Float64(Float64(d / D) * sqrt(Float64(c0 / Float64(w * h))))), (t_0 ^ 2.0)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(t$95$0 * N[(N[(-0.5 * N[(N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(N[(w * h), $MachinePrecision] / c0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.1%

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

   3. Simplified 66.5%

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

   6. Applied egg-rr 68.2%

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

   7. Taylor expanded in c0 around inf 37.4%

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

      1. associate-/l/ 34.9%

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

   9. Simplified 34.9%

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

   10. Taylor expanded in M around 0 68.2%

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

       1. add-sqr-sqrt 68.2%

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

       2. pow2 68.2%

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

       3. *-commutative 68.2%

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

       4. sqrt-prod 68.2%

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

       5. sqrt-pow1 74.3%

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

       6. metadata-eval 74.3%

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

       7. pow1 74.3%

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

       8. associate-/r* 74.3%

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

   12. Applied egg-rr 74.3%

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

   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 24.9%

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

   5. Taylor expanded in c0 around -inf 1.9%

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

      1. distribute-lft-in 0.8%

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

      2. mul-1-neg 0.8%

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

      3. distribute-rgt-neg-in 0.8%

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

      4. associate-/l* 0.7%

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

      5. mul-1-neg 0.7%

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

      6. associate-/l* 0.1%

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

      7. distribute-lft1-in 0.1%

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

      8. metadata-eval 0.1%

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

      9. mul0-lft 37.7%

         \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

      10. metadata-eval 37.7%

          \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   7. Simplified 37.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{d}{D} \cdot \sqrt{\frac{\frac{c0}{w}}{h}}, -0.5 \cdot \left(\frac{D \cdot M}{d} \cdot \sqrt{\frac{w \cdot h}{c0}}\right) + \frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}, {\left(\frac{d}{D} \cdot \sqrt{\frac{\frac{c0}{w}}{h}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 38.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-111}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}, \frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 1.8e-111)
   (* c0 (/ 0.0 (* 2.0 w)))
   (*
    c0
    (/
     (fma
      c0
      (* d (/ d (* D (* h (* w D)))))
      (* (/ d D) (* (/ d D) (/ c0 (* w h)))))
     (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 1.8e-111) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * (fma(c0, (d * (d / (D * (h * (w * D))))), ((d / D) * ((d / D) * (c0 / (w * h))))) / (2.0 * w));
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 1.8e-111)
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(fma(c0, Float64(d * Float64(d / Float64(D * Float64(h * Float64(w * D))))), Float64(Float64(d / D) * Float64(Float64(d / D) * Float64(c0 / Float64(w * h))))) / Float64(2.0 * w)));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 1.8e-111], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(c0 * N[(d * N[(d / N[(D * N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(d / D), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.80000000000000005e-111

   1. Initial program 24.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in c0 around -inf 6.3%

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

      1. distribute-lft-in 5.2%

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

      2. mul-1-neg 5.2%

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

      3. distribute-rgt-neg-in 5.2%

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

      4. associate-/l* 6.2%

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

      5. mul-1-neg 6.2%

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

      6. associate-/l* 5.7%

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

      7. distribute-lft1-in 5.7%

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

      8. metadata-eval 5.7%

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

      9. mul0-lft 36.7%

         \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

      10. metadata-eval 36.7%

          \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   7. Simplified 36.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   if 1.80000000000000005e-111 < M

   1. Initial program 12.1%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in c0 around inf 31.7%

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

      1. *-commutative 31.7%

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

      2. *-commutative 31.7%

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

      3. frac-times 32.2%

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

      4. unpow2 32.2%

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

      5. unpow2 32.2%

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

      6. frac-times 42.8%

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

      7. associate-*r* 45.3%

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

   7. Applied egg-rr 45.3%

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

   8. Taylor expanded in w around 0 46.3%

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

      1. associate-*r* 45.3%

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

      2. *-commutative 45.3%

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

      3. associate-*l* 46.3%

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

   10. Simplified 46.3%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.8 \cdot 10^{-111}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\mathsf{fma}\left(c0, d \cdot \frac{d}{D \cdot \left(h \cdot \left(w \cdot D\right)\right)}, \frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)\right)}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 7.8 \cdot 10^{-258}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 7.8e-258)
   (* c0 (/ 0.0 (* 2.0 w)))
   (* c0 (/ (* 2.0 (* (/ (/ c0 w) h) (pow (/ d D) 2.0))) (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7.8e-258) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * ((2.0 * (((c0 / w) / h) * pow((d / D), 2.0))) / (2.0 * w));
	}
	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) :: tmp
    if (m <= 7.8d-258) then
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    else
        tmp = c0 * ((2.0d0 * (((c0 / w) / h) * ((d_1 / d) ** 2.0d0))) / (2.0d0 * w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7.8e-258) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * ((2.0 * (((c0 / w) / h) * Math.pow((d / D), 2.0))) / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 7.8e-258:
		tmp = c0 * (0.0 / (2.0 * w))
	else:
		tmp = c0 * ((2.0 * (((c0 / w) / h) * math.pow((d / D), 2.0))) / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 7.8e-258)
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(Float64(c0 / w) / h) * (Float64(d / D) ^ 2.0))) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 7.8e-258)
		tmp = c0 * (0.0 / (2.0 * w));
	else
		tmp = c0 * ((2.0 * (((c0 / w) / h) * ((d / D) ^ 2.0))) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 7.8e-258], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(2.0 * N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.80000000000000008e-258

   1. Initial program 23.2%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in c0 around -inf 6.6%

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

      1. distribute-lft-in 5.9%

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

      2. mul-1-neg 5.9%

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

      3. distribute-rgt-neg-in 5.9%

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

      4. associate-/l* 7.3%

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

      5. mul-1-neg 7.3%

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

      6. associate-/l* 6.6%

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

      7. distribute-lft1-in 6.6%

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

      8. metadata-eval 6.6%

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

      9. mul0-lft 33.3%

         \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

      10. metadata-eval 33.3%

          \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   7. Simplified 33.3%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   if 7.80000000000000008e-258 < M

   1. Initial program 18.3%

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

   3. Simplified 32.7%

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

   5. Taylor expanded in c0 around inf 29.6%

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

      1. *-commutative 29.6%

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

      2. *-commutative 29.6%

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

      3. frac-times 29.9%

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

      4. unpow2 29.9%

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

      5. unpow2 29.9%

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

      6. frac-times 38.4%

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

      7. associate-*r* 40.1%

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

   7. Applied egg-rr 40.1%

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

   8. Taylor expanded in c0 around 0 30.3%

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

      1. *-commutative 30.3%

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

      2. *-commutative 30.3%

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

      3. times-frac 30.7%

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

      4. associate-/r* 31.5%

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

      5. unpow2 31.5%

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

      6. unpow2 31.5%

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

      7. times-frac 44.1%

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

      8. unpow2 44.1%

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

   10. Simplified 44.1%

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

Alternative 4: 38.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 7.4 \cdot 10^{-112}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= M 7.4e-112)
   (* c0 (/ 0.0 (* 2.0 w)))
   (* c0 (/ (* 2.0 (* (/ c0 (* w h)) (pow (/ d D) 2.0))) (* 2.0 w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7.4e-112) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * ((2.0 * ((c0 / (w * h)) * pow((d / D), 2.0))) / (2.0 * w));
	}
	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) :: tmp
    if (m <= 7.4d-112) then
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    else
        tmp = c0 * ((2.0d0 * ((c0 / (w * h)) * ((d_1 / d) ** 2.0d0))) / (2.0d0 * w))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (M <= 7.4e-112) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = c0 * ((2.0 * ((c0 / (w * h)) * Math.pow((d / D), 2.0))) / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if M <= 7.4e-112:
		tmp = c0 * (0.0 / (2.0 * w))
	else:
		tmp = c0 * ((2.0 * ((c0 / (w * h)) * math.pow((d / D), 2.0))) / (2.0 * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (M <= 7.4e-112)
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(Float64(2.0 * Float64(Float64(c0 / Float64(w * h)) * (Float64(d / D) ^ 2.0))) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (M <= 7.4e-112)
		tmp = c0 * (0.0 / (2.0 * w));
	else
		tmp = c0 * ((2.0 * ((c0 / (w * h)) * ((d / D) ^ 2.0))) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[M, 7.4e-112], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(2.0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.3999999999999996e-112

   1. Initial program 24.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in c0 around -inf 6.3%

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

      1. distribute-lft-in 5.2%

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

      2. mul-1-neg 5.2%

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

      3. distribute-rgt-neg-in 5.2%

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

      4. associate-/l* 6.2%

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

      5. mul-1-neg 6.2%

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

      6. associate-/l* 5.7%

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

      7. distribute-lft1-in 5.7%

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

      8. metadata-eval 5.7%

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

      9. mul0-lft 36.7%

         \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

      10. metadata-eval 36.7%

          \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   7. Simplified 36.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

   if 7.3999999999999996e-112 < M

   1. Initial program 12.1%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in c0 around inf 31.7%

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

      1. *-commutative 31.7%

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

      2. *-commutative 31.7%

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

      3. frac-times 32.2%

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

      4. unpow2 32.2%

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

      5. unpow2 32.2%

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

      6. frac-times 42.8%

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

      7. associate-*r* 45.3%

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

   7. Applied egg-rr 45.3%

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

   8. Taylor expanded in c0 around 0 31.7%

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

      1. *-commutative 31.7%

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

      2. *-commutative 31.7%

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

      3. times-frac 32.2%

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

      4. *-commutative 32.2%

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

      5. unpow2 32.2%

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

      6. unpow2 32.2%

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

      7. times-frac 45.6%

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

      8. unpow2 45.6%

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

   10. Simplified 45.6%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.4 \cdot 10^{-112}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{2 \cdot \left(\frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{2}\right)}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \frac{0}{2 \cdot w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* c0 (/ 0.0 (* 2.0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

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 = c0 * (0.0d0 / (2.0d0 * w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

Python

def code(c0, w, h, D, d, M):
	return c0 * (0.0 / (2.0 * w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(c0 * Float64(0.0 / Float64(2.0 * w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = c0 * (0.0 / (2.0 * w));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.1%

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

3. Simplified 36.8%

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

5. Taylor expanded in c0 around -inf 4.7%

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

   1. distribute-lft-in 3.9%

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

   2. mul-1-neg 3.9%

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

   3. distribute-rgt-neg-in 3.9%

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

   4. associate-/l* 4.6%

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

   5. mul-1-neg 4.6%

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

   6. associate-/l* 4.2%

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

   7. distribute-lft1-in 4.2%

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

   8. metadata-eval 4.2%

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

   9. mul0-lft 30.7%

      \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

   10. metadata-eval 30.7%

       \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

7. Simplified 30.7%

   \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024106 
(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))))))