Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.6% → 58.0%

Time: 18.3s

Alternatives: 8

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = -2.2436746034681464e-65
  2. w = -7.681889493930691e-263
  3. h = -9.119653076995752e-163
  4. D = 2.410391886550218e-256
  5. d = -2.0031300562721206e+205
  6. M = -5.12459632158709e-241

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.6% 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: 58.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 70.5%

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

   9. Applied rewrites 82.6%

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

   11. Applied rewrites 86.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval N/A

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

      8. div0 45.6

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

   5. Applied rewrites 45.6%

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

4. Final simplification 58.4%

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

Alternative 2: 57.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 70.5%

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

   9. Applied rewrites 82.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval N/A

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

      8. div0 45.6

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

   5. Applied rewrites 45.6%

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

4. Final simplification 57.3%

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

Alternative 3: 57.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 70.5%

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

   9. Applied rewrites 82.6%

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

   11. Applied rewrites 81.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval N/A

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

      8. div0 45.6

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

   5. Applied rewrites 45.6%

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

4. Final simplification 57.0%

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

Alternative 4: 55.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 64.7%

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

   9. Applied rewrites 73.0%

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

   11. Applied rewrites 76.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval N/A

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

      8. div0 45.6

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

   5. Applied rewrites 45.6%

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

4. Final simplification 55.4%

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

Alternative 5: 53.3% accurate, 0.7× speedup

Math

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

FPCore

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

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((D * D) * (h * w))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = ((d * c0) / ((((w * w) * D) * 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(D * D) * Float64(h * w)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))) + t_0) * Float64(c0 / Float64(w * 2.0))) <= Inf)
		tmp = Float64(Float64(Float64(d * c0) / Float64(Float64(Float64(Float64(w * w) * D) * h) * D)) * Float64(d * c0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * d) * c0) / ((D * D) * (h * w));
	tmp = 0.0;
	if (((sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= Inf)
		tmp = ((d * c0) / ((((w * w) * D) * 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[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(d * c0), $MachinePrecision] / N[(N[(N[(N[(w * w), $MachinePrecision] * D), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 70.5%

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

   9. Applied rewrites 71.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval N/A

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

      8. div0 45.6

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

   5. Applied rewrites 45.6%

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

4. Final simplification 53.7%

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

Alternative 6: 52.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 55.9%

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

   9. Applied rewrites 64.7%

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

   11. Applied rewrites 64.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval N/A

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

      8. div0 45.6

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

   5. Applied rewrites 45.6%

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

4. Final simplification 51.7%

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

Alternative 7: 52.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 55.9%

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

   9. Applied rewrites 64.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval N/A

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

      8. div0 45.6

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

   5. Applied rewrites 45.6%

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

4. Final simplification 51.6%

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

Alternative 8: 33.9% 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 23.8%

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

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

   2. associate-*l/ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. mul0-lft N/A

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

   6. mul0-rgt N/A

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

   7. metadata-eval N/A

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

   8. div0 34.9

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

5. Applied rewrites 34.9%

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

Reproduce

herbie shell --seed 2024240 
(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))))))