Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.1% → 55.2%

Time: 18.1s

Alternatives: 10

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = 1.9443657401579997e-78
  2. w = -1.4263470052306399e-275
  3. h = 4.551388675829924e-280
  4. D = 2.0075547623922242e-216
  5. d = 6.537380979908286e+243
  6. M = 2.625029893039813e+70

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: 25.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D * D), $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] * t$95$0), $MachinePrecision], Infinity], N[(N[(N[Sqrt[N[((-M) * M + N[Power[N[(N[(N[(N[(N[(h * w), $MachinePrecision] / d), $MachinePrecision] * D), $MachinePrecision] / c0), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(N[(d / D), $MachinePrecision] / N[(h * w), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] * t$95$0), $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 73.5%

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

   4. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 70.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 70.2

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

   6. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 72.5

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

   8. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 76.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 76.7

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

   10. Applied rewrites 76.7%

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

   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. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 42.6

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

   5. Applied rewrites 42.6%

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

4. Final simplification 54.6%

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

Alternative 2: 54.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(D * D) * w) * h)
	t_1 = Float64(c0 / Float64(w * 2.0))
	t_2 = Float64(Float64(Float64(d * d) * c0) / Float64(Float64(h * w) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))) + t_2) * t_1) <= Inf)
		tmp = Float64(Float64(sqrt(fma(Float64(-M), M, (Float64(t_0 / Float64(Float64(d * c0) * d)) ^ -2.0))) + Float64(Float64(Float64(d / t_0) * d) * c0)) * t_1);
	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] * w), $MachinePrecision] * h), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision], Infinity], N[(N[(N[Sqrt[N[((-M) * M + N[Power[N[(t$95$0 / N[(N[(d * c0), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(d / t$95$0), $MachinePrecision] * d), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] * t$95$1), $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 73.5%

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

   4. Applied rewrites 69.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. associate-*r* N/A

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

      25. lower-*.f64 N/A

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

      26. lower-*.f64 69.2

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

   6. Applied rewrites 69.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. clear-num N/A

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

      20. frac-2neg N/A

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

      21. lower-/.f64 N/A

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

   8. Applied rewrites 74.9%

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

   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. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 42.6

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

   5. Applied rewrites 42.6%

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

4. Final simplification 54.0%

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

Alternative 3: 55.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) * c0) / ((h * w) * (D * D));
	double t_1 = (Math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((h * w) * (D * D))
	t_1 = (math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * d) * c0) / ((h * w) * (D * D));
	t_1 = (sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.5%

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

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

   1. Initial program 0.0%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 42.6

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

   5. Applied rewrites 42.6%

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

4. Final simplification 53.5%

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

Alternative 4: 55.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((h * w) * (D * D))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = ((math.pow((d / D), 2.0) * (c0 / w)) / (h * w)) * 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(h * w) * Float64(D * D)))
	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 / D) ^ 2.0) * Float64(c0 / w)) / Float64(h * w)) * 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) / ((h * w) * (D * D));
	tmp = 0.0;
	if (((sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= Inf)
		tmp = ((((d / D) ^ 2.0) * (c0 / w)) / (h * w)) * 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[(h * w), $MachinePrecision] * N[(D * D), $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[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(c0 / w), $MachinePrecision]), $MachinePrecision] / N[(h * w), $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 73.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 59.6%

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

   9. Applied rewrites 72.1%

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

   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. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 42.6

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

   5. Applied rewrites 42.6%

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

4. Final simplification 53.0%

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

Alternative 5: 53.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((h * w) * (D * D))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = ((((d / w) / w) * (d * c0)) / ((D * D) * h)) * 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(h * w) * Float64(D * D)))
	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(Float64(d / w) / w) * Float64(d * c0)) / Float64(Float64(D * D) * h)) * 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) / ((h * w) * (D * D));
	tmp = 0.0;
	if (((sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= Inf)
		tmp = ((((d / w) / w) * (d * c0)) / ((D * D) * h)) * 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[(h * w), $MachinePrecision] * N[(D * D), $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 / w), $MachinePrecision] / w), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * h), $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 73.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 59.6%

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

   9. Applied rewrites 72.1%

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

   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. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 42.6

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

   5. Applied rewrites 42.6%

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

4. Final simplification 53.0%

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

Alternative 6: 54.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* (* d d) c0) (* (* h w) (* D D)))))
   (if (<=
        (* (+ (sqrt (- (* t_0 t_0) (* M M))) t_0) (/ c0 (* w 2.0)))
        INFINITY)
     (* (* (/ (/ d w) (* h w)) (* (/ 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) / ((h * w) * (D * D));
	double tmp;
	if (((sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= ((double) INFINITY)) {
		tmp = (((d / w) / (h * w)) * ((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) / ((h * w) * (D * D));
	double tmp;
	if (((Math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= Double.POSITIVE_INFINITY) {
		tmp = (((d / w) / (h * w)) * ((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) / ((h * w) * (D * D))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = (((d / w) / (h * w)) * ((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(h * w) * Float64(D * D)))
	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 / w) / Float64(h * w)) * 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) / ((h * w) * (D * D));
	tmp = 0.0;
	if (((sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= Inf)
		tmp = (((d / w) / (h * w)) * ((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[(h * w), $MachinePrecision] * N[(D * D), $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 / w), $MachinePrecision] / N[(h * w), $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 73.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 59.6%

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

   9. Applied rewrites 69.9%

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

   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. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 42.6

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

   5. Applied rewrites 42.6%

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

4. Final simplification 52.2%

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

Alternative 7: 53.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* (* d d) c0) (* (* h w) (* D D)))))
   (if (<=
        (* (+ (sqrt (- (* t_0 t_0) (* M M))) t_0) (/ c0 (* w 2.0)))
        INFINITY)
     (* (* (/ (/ d w) (* h w)) 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) / ((h * w) * (D * D));
	double tmp;
	if (((sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= ((double) INFINITY)) {
		tmp = (((d / w) / (h * w)) * 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) / ((h * w) * (D * D));
	double tmp;
	if (((Math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= Double.POSITIVE_INFINITY) {
		tmp = (((d / w) / (h * w)) * 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) / ((h * w) * (D * D))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = (((d / w) / (h * w)) * 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(h * w) * Float64(D * D)))
	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 / w) / Float64(h * w)) * 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) / ((h * w) * (D * D));
	tmp = 0.0;
	if (((sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= Inf)
		tmp = (((d / w) / (h * w)) * 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[(h * w), $MachinePrecision] * N[(D * D), $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 / w), $MachinePrecision] / N[(h * w), $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 73.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 68.6%

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

   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. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 42.6

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

   5. Applied rewrites 42.6%

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

4. Final simplification 51.8%

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

Alternative 8: 53.6% accurate, 0.7× speedup

Math

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

FPCore

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

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((h * w) * (D * D))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = (((d * c0) * d) / (((h * w) * w) * (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(h * w) * Float64(D * D)))
	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) * d) / Float64(Float64(Float64(h * w) * w) * 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) / ((h * w) * (D * D));
	tmp = 0.0;
	if (((sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= Inf)
		tmp = (((d * c0) * d) / (((h * w) * w) * (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[(h * w), $MachinePrecision] * N[(D * D), $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] * d), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $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 73.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 59.6%

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

   9. Applied rewrites 66.7%

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

   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. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 42.6

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

   5. Applied rewrites 42.6%

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

4. Final simplification 51.1%

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

Alternative 9: 52.5% accurate, 0.7× speedup

Math

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

FPCore

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

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((h * w) * (D * D))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = (((d * c0) * d) / (((w * w) * h) * (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(h * w) * Float64(D * D)))
	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) * d) / Float64(Float64(Float64(w * w) * h) * 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) / ((h * w) * (D * D));
	tmp = 0.0;
	if (((sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= Inf)
		tmp = (((d * c0) * d) / (((w * w) * h) * (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[(h * w), $MachinePrecision] * N[(D * D), $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] * d), $MachinePrecision] / N[(N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $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 73.5%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 59.6%

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

   9. Applied rewrites 58.9%

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

   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. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 42.6

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

   5. Applied rewrites 42.6%

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

4. Final simplification 48.3%

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

Alternative 10: 33.6% 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 25.8%

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

3. Taylor expanded in c0 around -inf

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

   1. associate-/l* N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

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

   5. div0 N/A

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

   6. mul0-rgt N/A

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

   7. metadata-eval 31.2

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

5. Applied rewrites 31.2%

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

Reproduce

herbie shell --seed 2024251 
(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))))))