Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.0% → 53.5%

Time: 16.6s

Alternatives: 8

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = -2.400667647537937e+239
  2. w = 3.330952724569262e-221
  3. h = 1.5804719854676109e-263
  4. D = 1.1216470049276549e-307
  5. d = 3.0560750983155815e+287
  6. M = 9.113577325122019e+243

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: 53.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(h / N[(d / D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(0.5 * c0), $MachinePrecision] * N[(N[(d / N[(t$95$0 * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0 + N[Sqrt[N[((-M) * M + N[Power[N[(N[(N[(w / c0), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $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 75.5%

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 75.6

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

   6. Applied rewrites 75.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 78.0

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

   8. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.0

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

   10. Applied rewrites 78.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

   5. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

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

Alternative 2: 53.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(N[(d / D), $MachinePrecision] / h), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] * c0 + N[Sqrt[N[((-M) * M + N[Power[N[(N[(N[(D * w), $MachinePrecision] / N[(d * c0), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 * c0), $MachinePrecision]), $MachinePrecision] / w), $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 75.5%

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 75.6

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

   6. Applied rewrites 75.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 78.0

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

   8. Applied rewrites 78.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 77.9

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

   10. Applied rewrites 77.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

   5. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

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

Alternative 3: 53.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(d / N[(N[(D * h), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * c0 + N[Sqrt[N[((-M) * M + N[Power[N[(N[(N[(w / c0), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 * c0), $MachinePrecision]), $MachinePrecision] / w), $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 75.5%

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 75.6

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

   6. Applied rewrites 75.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 78.0

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

   8. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. associate-*r* N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. associate-*r* N/A

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

   10. Applied rewrites 77.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

   5. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

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

Alternative 4: 53.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] / w), $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 75.5%

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 77.7

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

   7. Applied rewrites 77.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

   5. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

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

Alternative 5: 50.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(c0 * c0), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] / w), $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 75.5%

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

   4. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 75.6

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

   6. Applied rewrites 75.6%

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

   7. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 68.2

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

   9. Applied rewrites 68.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

   5. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.6%

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

Alternative 6: 51.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(d / N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * c0), $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 75.5%

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

   3. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 61.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

   5. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.5%

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

Alternative 7: 50.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(d / N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * N[(N[(c0 / N[(D * D), $MachinePrecision]), $MachinePrecision] * c0), $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 75.5%

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

   3. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

   5. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

   6. Taylor expanded in w around 0

      \[\leadsto 0 \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

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

Alternative 8: 33.1% accurate, 156.0× 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 24.5%

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

3. Taylor expanded in c0 around -inf

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

   1. *-commutative N/A

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

5. Applied rewrites 26.4%

   \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]

6. Taylor expanded in w around 0

   \[\leadsto 0 \]
7. Step-by-step derivation

8. Applied rewrites 34.4%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(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))))))