Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.1% → 68.9%

Time: 19.1s

Alternatives: 11

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = 2.8362478185884508e+255
  2. w = 6.225005537905497e-286
  3. h = -8.120922703731018e-263
  4. D = -2.770461251395601e-305
  5. d = -6.304142093756634e+280
  6. M = 6.354553706980201e-18

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: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \frac{\frac{c0 \cdot \left(2 \cdot \left(d \cdot d\right)\right)}{D}}{\left(w \cdot h\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (((c0 * (2.0 * (d * d))) / D) / ((w * h) * D));
	} else {
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (((c0 * (2.0 * (d * d))) / D) / ((w * h) * D));
	} else {
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * (((c0 * (2.0 * (d * d))) / D) / ((w * h) * D))
	else:
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(Float64(Float64(c0 * Float64(2.0 * Float64(d * d))) / D) / Float64(Float64(w * h) * D)));
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(Float64(h * M) / d) * Float64(M / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * (((c0 * (2.0 * (d * d))) / D) / ((w * h) * D));
	else
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(N[(c0 * N[(2.0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.8%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 87.4%

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

   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

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 16.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 42.7%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.0%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 69.0%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{\color{blue}{d}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\frac{c0 \cdot \left(2 \cdot \left(d \cdot d\right)\right)}{D}}{\left(w \cdot h\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c0 \cdot \left(d \cdot d\right)\\ t_1 := \left(w \cdot h\right) \cdot \left(D \cdot D\right)\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{t\_0}{t\_1}\\ \mathbf{if}\;t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_2 \cdot \frac{2 \cdot t\_0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d)))
        (t_1 (* (* w h) (* D D)))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ t_0 t_1)))
   (if (<= (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))) INFINITY)
     (* t_2 (/ (* 2.0 t_0) t_1))
     (* 0.25 (* D (* D (* (/ (* h M) d) (/ M d))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = (w * h) * (D * D);
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_0 / t_1;
	double tmp;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_2 * ((2.0 * t_0) / t_1);
	} else {
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = (w * h) * (D * D);
	double t_2 = c0 / (2.0 * w);
	double t_3 = t_0 / t_1;
	double tmp;
	if ((t_2 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_2 * ((2.0 * t_0) / t_1);
	} else {
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 * (d * d)
	t_1 = (w * h) * (D * D)
	t_2 = c0 / (2.0 * w)
	t_3 = t_0 / t_1
	tmp = 0
	if (t_2 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))) <= math.inf:
		tmp = t_2 * ((2.0 * t_0) / t_1)
	else:
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(d * d))
	t_1 = Float64(Float64(w * h) * Float64(D * D))
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(t_0 / t_1)
	tmp = 0.0
	if (Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_2 * Float64(Float64(2.0 * t_0) / t_1));
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(Float64(h * M) / d) * Float64(M / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 * (d * d);
	t_1 = (w * h) * (D * D);
	t_2 = c0 / (2.0 * w);
	t_3 = t_0 / t_1;
	tmp = 0.0;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= Inf)
		tmp = t_2 * ((2.0 * t_0) / t_1);
	else
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 / t$95$1), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$2 * N[(N[(2.0 * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.8%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   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

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 16.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 42.7%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.0%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 69.0%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{\color{blue}{d}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(c0 \cdot 2\right) \cdot \frac{d \cdot d}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((c0 * 2.0) * ((d * d) / (D * ((w * h) * D))));
	} else {
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((c0 * 2.0) * ((d * d) / (D * ((w * h) * D))));
	} else {
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * ((c0 * 2.0) * ((d * d) / (D * ((w * h) * D))))
	else:
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(Float64(c0 * 2.0) * Float64(Float64(d * d) / Float64(D * Float64(Float64(w * h) * D)))));
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(Float64(h * M) / d) * Float64(M / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * ((c0 * 2.0) * ((d * d) / (D * ((w * h) * D))));
	else
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(c0 * 2.0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.8%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 81.9%

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

   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

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 16.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 42.7%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.0%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 69.0%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{\color{blue}{d}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(c0 \cdot 2\right) \cdot \frac{d \cdot d}{D \cdot \left(\left(w \cdot h\right) \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(w \cdot h\right) \cdot \left(D \cdot D\right)\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{t\_0}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{t\_0}}{w}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (w * h) * (D * D);
	double t_1 = (c0 * (d * d)) / t_0;
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (((d * d) * (c0 * c0)) / t_0) / w;
	} else {
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (w * h) * (D * D);
	double t_1 = (c0 * (d * d)) / t_0;
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (((d * d) * (c0 * c0)) / t_0) / w;
	} else {
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (w * h) * (D * D)
	t_1 = (c0 * (d * d)) / t_0
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = (((d * d) * (c0 * c0)) / t_0) / w
	else:
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(w * h) * Float64(D * D))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / t_0)
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(Float64(Float64(d * d) * Float64(c0 * c0)) / t_0) / w);
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(Float64(h * M) / d) * Float64(M / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (w * h) * (D * D);
	t_1 = (c0 * (d * d)) / t_0;
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = (((d * d) * (c0 * c0)) / t_0) / w;
	else
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(d * d), $MachinePrecision] * N[(c0 * c0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / w), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.8%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 82.4%

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

   8. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 75.3

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

   10. Applied rewrites 75.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 16.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 42.7%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.0%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 69.0%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{\color{blue}{d}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}}{w}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	} else {
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	} else {
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)))
	else:
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(Float64(d * d) * Float64(c0 * c0)) / Float64(Float64(D * D) * Float64(h * Float64(w * w))));
	else
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(Float64(h * M) / d) * Float64(M / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	else
		tmp = 0.25 * (D * (D * (((h * M) / d) * (M / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(d * d), $MachinePrecision] * N[(c0 * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(D * N[(D * N[(N[(N[(h * M), $MachinePrecision] / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.8%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

   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

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 16.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 42.7%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.0%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 69.0%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{\color{blue}{d}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{h \cdot M}{d} \cdot \frac{M}{d}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot M\right) \cdot \left(\frac{h}{d \cdot d} \cdot \left(D \cdot M\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* (* d d) (* c0 c0)) (* (* D D) (* h (* w w))))
     (* 0.25 (* (* D M) (* (/ h (* d d)) (* D 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));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	} else {
		tmp = 0.25 * ((D * M) * ((h / (d * d)) * (D * M)));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	} else {
		tmp = 0.25 * ((D * M) * ((h / (d * d)) * (D * M)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)))
	else:
		tmp = 0.25 * ((D * M) * ((h / (d * d)) * (D * M)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(Float64(d * d) * Float64(c0 * c0)) / Float64(Float64(D * D) * Float64(h * Float64(w * w))));
	else
		tmp = Float64(0.25 * Float64(Float64(D * M) * Float64(Float64(h / Float64(d * d)) * Float64(D * M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = ((d * d) * (c0 * c0)) / ((D * D) * (h * (w * w)));
	else
		tmp = 0.25 * ((D * M) * ((h / (d * d)) * (D * M)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(d * d), $MachinePrecision] * N[(c0 * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D * M), $MachinePrecision] * N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.8%

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

   3. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

   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

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 16.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 42.7%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.4%

       \[\leadsto 0.25 \cdot \left(\left(\frac{h}{d \cdot d} \cdot \left(D \cdot M\right)\right) \cdot \left(D \cdot \color{blue}{M}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot \left(c0 \cdot c0\right)}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot M\right) \cdot \left(\frac{h}{d \cdot d} \cdot \left(D \cdot M\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq 2.1 \cdot 10^{+261}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) 2.1e+261) (* 0.25 (* D (* D (* (* M M) (/ h (* d d)))))) 0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 2.1e+261) {
		tmp = 0.25 * (D * (D * ((M * M) * (h / (d * d)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d_1 * d_1) <= 2.1d+261) then
        tmp = 0.25d0 * (d * (d * ((m * m) * (h / (d_1 * d_1)))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= 2.1e+261) {
		tmp = 0.25 * (D * (D * ((M * M) * (h / (d * d)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= 2.1e+261:
		tmp = 0.25 * (D * (D * ((M * M) * (h / (d * d)))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= 2.1e+261)
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(Float64(M * M) * Float64(h / Float64(d * d))))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= 2.1e+261)
		tmp = 0.25 * (D * (D * ((M * M) * (h / (d * d)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], 2.1e+261], N[(0.25 * N[(D * N[(D * N[(N[(M * M), $MachinePrecision] * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 d d) < 2.1000000000000001e261

   1. Initial program 29.8%

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

   3. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 13.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 34.9%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.9%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 44.2%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{\color{blue}{d \cdot d}}\right)\right)\right) \]

   if 2.1000000000000001e261 < (*.f64 d d)

   1. Initial program 26.4%

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

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. associate-/l* N/A

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

      6. mul0-lft N/A

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

      7. metadata-eval 41.2

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

   5. Applied rewrites 41.2%

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

Alternative 8: 42.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-165}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \frac{M \cdot M}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot M\right) \cdot \left(\frac{h}{d \cdot d} \cdot \left(D \cdot M\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h -2e-165)
   (* 0.25 (* D (* D (* h (/ (* M M) (* d d))))))
   (* 0.25 (* (* D M) (* (/ h (* d d)) (* D M))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -2e-165) {
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))));
	} else {
		tmp = 0.25 * ((D * M) * ((h / (d * d)) * (D * M)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (h <= (-2d-165)) then
        tmp = 0.25d0 * (d * (d * (h * ((m * m) / (d_1 * d_1)))))
    else
        tmp = 0.25d0 * ((d * m) * ((h / (d_1 * d_1)) * (d * m)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -2e-165) {
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))));
	} else {
		tmp = 0.25 * ((D * M) * ((h / (d * d)) * (D * M)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= -2e-165:
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))))
	else:
		tmp = 0.25 * ((D * M) * ((h / (d * d)) * (D * M)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= -2e-165)
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(h * Float64(Float64(M * M) / Float64(d * d))))));
	else
		tmp = Float64(0.25 * Float64(Float64(D * M) * Float64(Float64(h / Float64(d * d)) * Float64(D * M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= -2e-165)
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))));
	else
		tmp = 0.25 * ((D * M) * ((h / (d * d)) * (D * M)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, -2e-165], N[(0.25 * N[(D * N[(D * N[(h * N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D * M), $MachinePrecision] * N[(N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -2e-165

   1. Initial program 24.5%

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

   3. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 18.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 35.4%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.1%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 52.1%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{M \cdot M}{d \cdot d} \cdot h\right)\right)\right) \]

   if -2e-165 < h

   1. Initial program 29.8%

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

   3. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 11.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 31.3%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 43.3%

       \[\leadsto 0.25 \cdot \left(\left(\frac{h}{d \cdot d} \cdot \left(D \cdot M\right)\right) \cdot \left(D \cdot \color{blue}{M}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2 \cdot 10^{-165}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \frac{M \cdot M}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot M\right) \cdot \left(\frac{h}{d \cdot d} \cdot \left(D \cdot M\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 7.2 \cdot 10^{+131}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \frac{M \cdot M}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 7.2e+131)
   (* 0.25 (* D (* D (* h (/ (* M M) (* d d))))))
   (* 0.25 (* (* D (* D M)) (* M (/ h (* d d)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 7.2e+131) {
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))));
	} else {
		tmp = 0.25 * ((D * (D * M)) * (M * (h / (d * d))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 7.2d+131) then
        tmp = 0.25d0 * (d * (d * (h * ((m * m) / (d_1 * d_1)))))
    else
        tmp = 0.25d0 * ((d * (d * m)) * (m * (h / (d_1 * d_1))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 7.2e+131) {
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))));
	} else {
		tmp = 0.25 * ((D * (D * M)) * (M * (h / (d * d))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 7.2e+131:
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))))
	else:
		tmp = 0.25 * ((D * (D * M)) * (M * (h / (d * d))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 7.2e+131)
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(h * Float64(Float64(M * M) / Float64(d * d))))));
	else
		tmp = Float64(0.25 * Float64(Float64(D * Float64(D * M)) * Float64(M * Float64(h / Float64(d * d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 7.2e+131)
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))));
	else
		tmp = 0.25 * ((D * (D * M)) * (M * (h / (d * d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 7.2e+131], N[(0.25 * N[(D * N[(D * N[(h * N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D * N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(M * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 7.20000000000000063e131

   1. Initial program 29.7%

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

   3. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 14.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 35.9%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.6%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 45.6%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{M \cdot M}{d \cdot d} \cdot h\right)\right)\right) \]

   if 7.20000000000000063e131 < d

   1. Initial program 23.1%

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

   3. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 10.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 22.1%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 34.8%

       \[\leadsto 0.25 \cdot \left(\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(M \cdot \color{blue}{\frac{h}{d \cdot d}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 7.2 \cdot 10^{+131}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \frac{M \cdot M}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(D \cdot \left(D \cdot M\right)\right) \cdot \left(M \cdot \frac{h}{d \cdot d}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 9.2 \cdot 10^{+131}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \frac{M \cdot M}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 9.2e+131) (* 0.25 (* D (* D (* h (/ (* M M) (* d d)))))) 0.0))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 9.2e+131) {
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 9.2d+131) then
        tmp = 0.25d0 * (d * (d * (h * ((m * m) / (d_1 * d_1)))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 9.2e+131) {
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 9.2e+131:
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 9.2e+131)
		tmp = Float64(0.25 * Float64(D * Float64(D * Float64(h * Float64(Float64(M * M) / Float64(d * d))))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 9.2e+131)
		tmp = 0.25 * (D * (D * (h * ((M * M) / (d * d)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 9.2e+131], N[(0.25 * N[(D * N[(D * N[(h * N[(N[(M * M), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if d < 9.19999999999999966e131

   1. Initial program 29.7%

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

   3. Taylor expanded in c0 around -inf

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}\right) \]

      5. mul0-lft N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\frac{-1}{2} \cdot \color{blue}{0}}{w}\right) \]

      6. metadata-eval N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{\color{blue}{0}}{w}\right) \]

      7. div0 N/A

         \[\leadsto {c0}^{2} \cdot \left(\frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}} + \color{blue}{0}\right) \]

   5. Applied rewrites 14.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}, 0\right)} \]

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 35.9%

      \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{h}{d \cdot d}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 44.6%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \color{blue}{\frac{h \cdot \left(M \cdot M\right)}{d \cdot d}}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 45.6%

       \[\leadsto 0.25 \cdot \left(D \cdot \left(D \cdot \left(\frac{M \cdot M}{d \cdot d} \cdot h\right)\right)\right) \]

   if 9.19999999999999966e131 < d

   1. Initial program 23.1%

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

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. associate-/l* N/A

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

      6. mul0-lft N/A

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

      7. metadata-eval 33.7

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

   5. Applied rewrites 33.7%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 9.2 \cdot 10^{+131}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \frac{M \cdot M}{d \cdot d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.2% 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 28.1%

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

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

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

   5. associate-/l* N/A

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

   6. mul0-lft N/A

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

   7. metadata-eval 35.3

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

5. Applied rewrites 35.3%

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

Reproduce

herbie shell --seed 2024225 
(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))))))