Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.0% → 54.0%

Time: 30.1s

Alternatives: 6

Speedup: 21.6×

• could not determine a ground truth

  1. c0 = 1.913490257859518e+231
  2. w = -1.4851521532114767e-195
  3. h = 2.5360633981848977e-275
  4. D = -5.919197895986445e-248
  5. d = 1.851346411120628e+276
  6. M = 8.920373042916235e-100

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 54.0% accurate, 0.4× speedup

Math

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

FPCore

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

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 = c0 * ((2.0 * (((c0 * pow(d, 2.0)) / pow(D, 2.0)) / (w * h))) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	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 = c0 * ((2.0 * (((c0 * Math.pow(d, 2.0)) / Math.pow(D, 2.0)) / (w * h))) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	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 = c0 * ((2.0 * (((c0 * math.pow(d, 2.0)) / math.pow(D, 2.0)) / (w * h))) / (2.0 * w))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	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(c0 * Float64(Float64(2.0 * Float64(Float64(Float64(c0 * (d ^ 2.0)) / (D ^ 2.0)) / Float64(w * h))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	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 = c0 * ((2.0 * (((c0 * (d ^ 2.0)) / (D ^ 2.0)) / (w * h))) / (2.0 * w));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	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[(c0 * N[(N[(2.0 * N[(N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.3%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in c0 around inf 78.6%

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

      1. *-commutative 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-*r* 76.1%

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

      4. times-frac 74.9%

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

      5. *-commutative 74.9%

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

   7. Simplified 74.9%

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

      1. *-commutative 74.9%

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

      2. frac-times 76.1%

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

      3. *-commutative 76.1%

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

      4. associate-*r* 78.6%

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

      5. associate-/r* 82.0%

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

      6. *-commutative 82.0%

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

   9. Applied egg-rr 82.0%

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

   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%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in c0 around -inf 2.0%

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

      1. mul-1-neg 2.0%

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

      2. distribute-lft-in 0.6%

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

      3. mul-1-neg 0.6%

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

      4. distribute-rgt-neg-in 0.6%

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

      5. associate-/l* 0.0%

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

      6. mul-1-neg 0.0%

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

      7. associate-/l* 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. metadata-eval 0.0%

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

      10. mul0-lft 46.6%

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

      11. metadata-eval 46.6%

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

   7. Simplified 46.6%

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

4. Final simplification 57.9%

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

Alternative 2: 54.3% accurate, 0.4× speedup

Math

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

FPCore

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

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 = c0 * ((2.0 * ((c0 * pow(d, 2.0)) / ((w * h) * pow(D, 2.0)))) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	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 = c0 * ((2.0 * ((c0 * Math.pow(d, 2.0)) / ((w * h) * Math.pow(D, 2.0)))) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	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 = c0 * ((2.0 * ((c0 * math.pow(d, 2.0)) / ((w * h) * math.pow(D, 2.0)))) / (2.0 * w))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	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(c0 * Float64(Float64(2.0 * Float64(Float64(c0 * (d ^ 2.0)) / Float64(Float64(w * h) * (D ^ 2.0)))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	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 = c0 * ((2.0 * ((c0 * (d ^ 2.0)) / ((w * h) * (D ^ 2.0)))) / (2.0 * w));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	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[(c0 * N[(N[(2.0 * N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.3%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in c0 around inf 78.6%

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

   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%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in c0 around -inf 2.0%

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

      1. mul-1-neg 2.0%

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

      2. distribute-lft-in 0.6%

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

      3. mul-1-neg 0.6%

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

      4. distribute-rgt-neg-in 0.6%

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

      5. associate-/l* 0.0%

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

      6. mul-1-neg 0.0%

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

      7. associate-/l* 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. metadata-eval 0.0%

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

      10. mul0-lft 46.6%

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

      11. metadata-eval 46.6%

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

   7. Simplified 46.6%

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

4. Final simplification 56.8%

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

Alternative 3: 54.3% accurate, 0.4× speedup

Math

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

FPCore

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

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 = c0 * ((2.0 * (((c0 * pow(d, 2.0)) / (w * h)) / pow(D, 2.0))) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	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 = c0 * ((2.0 * (((c0 * Math.pow(d, 2.0)) / (w * h)) / Math.pow(D, 2.0))) / (2.0 * w));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	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 = c0 * ((2.0 * (((c0 * math.pow(d, 2.0)) / (w * h)) / math.pow(D, 2.0))) / (2.0 * w))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	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(c0 * Float64(Float64(2.0 * Float64(Float64(Float64(c0 * (d ^ 2.0)) / Float64(w * h)) / (D ^ 2.0))) / Float64(2.0 * w)));
	else
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	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 = c0 * ((2.0 * (((c0 * (d ^ 2.0)) / (w * h)) / (D ^ 2.0))) / (2.0 * w));
	else
		tmp = c0 * (0.0 / (2.0 * w));
	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[(c0 * N[(N[(2.0 * N[(N[(N[(c0 * N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision] / N[Power[D, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.3%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in c0 around inf 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-/l/ 81.9%

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

      3. *-commutative 81.9%

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

   7. Simplified 81.9%

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

   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%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in c0 around -inf 2.0%

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

      1. mul-1-neg 2.0%

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

      2. distribute-lft-in 0.6%

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

      3. mul-1-neg 0.6%

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

      4. distribute-rgt-neg-in 0.6%

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

      5. associate-/l* 0.0%

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

      6. mul-1-neg 0.0%

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

      7. associate-/l* 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. metadata-eval 0.0%

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

      10. mul0-lft 46.6%

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

      11. metadata-eval 46.6%

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

   7. Simplified 46.6%

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

4. Final simplification 57.9%

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

Alternative 4: 53.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	} else {
		tmp = c0 * (0.0 / (2.0 * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D))
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	else:
		tmp = c0 * (0.0 / (2.0 * w))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.3%

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

   3. Simplified 76.7%

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

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

   1. Initial program 0.0%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in c0 around -inf 2.0%

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

      1. mul-1-neg 2.0%

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

      2. distribute-lft-in 0.6%

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

      3. mul-1-neg 0.6%

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

      4. distribute-rgt-neg-in 0.6%

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

      5. associate-/l* 0.0%

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

      6. mul-1-neg 0.0%

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

      7. associate-/l* 0.0%

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

      8. distribute-lft1-in 0.0%

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

      9. metadata-eval 0.0%

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

      10. mul0-lft 46.6%

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

      11. metadata-eval 46.6%

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

   7. Simplified 46.6%

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

4. Final simplification 56.2%

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

Alternative 5: 30.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := t\_0 \cdot \frac{d \cdot d}{D \cdot D}\\ \mathbf{if}\;c0 \leq -3.2 \cdot 10^{+102} \lor \neg \left(c0 \leq -5.6 \cdot 10^{-198}\right) \land c0 \leq 10^{+155}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{t\_1 \cdot t\_1 - M \cdot M} + t\_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h))) (t_1 (* t_0 (/ (* d d) (* D D)))))
   (if (or (<= c0 -3.2e+102) (and (not (<= c0 -5.6e-198)) (<= c0 1e+155)))
     (* c0 (/ 0.0 (* 2.0 w)))
     (*
      (/ c0 (* 2.0 w))
      (+ (sqrt (- (* t_1 t_1) (* M M))) (* t_0 (* (/ d D) (/ d D))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double tmp;
	if ((c0 <= -3.2e+102) || (!(c0 <= -5.6e-198) && (c0 <= 1e+155))) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c0 / (w * h)
    t_1 = t_0 * ((d_1 * d_1) / (d * d))
    if ((c0 <= (-3.2d+102)) .or. (.not. (c0 <= (-5.6d-198))) .and. (c0 <= 1d+155)) then
        tmp = c0 * (0.0d0 / (2.0d0 * w))
    else
        tmp = (c0 / (2.0d0 * w)) * (sqrt(((t_1 * t_1) - (m * m))) + (t_0 * ((d_1 / d) * (d_1 / d))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double tmp;
	if ((c0 <= -3.2e+102) || (!(c0 <= -5.6e-198) && (c0 <= 1e+155))) {
		tmp = c0 * (0.0 / (2.0 * w));
	} else {
		tmp = (c0 / (2.0 * w)) * (Math.sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * h)
	t_1 = t_0 * ((d * d) / (D * D))
	tmp = 0
	if (c0 <= -3.2e+102) or (not (c0 <= -5.6e-198) and (c0 <= 1e+155)):
		tmp = c0 * (0.0 / (2.0 * w))
	else:
		tmp = (c0 / (2.0 * w)) * (math.sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(t_0 * Float64(Float64(d * d) / Float64(D * D)))
	tmp = 0.0
	if ((c0 <= -3.2e+102) || (!(c0 <= -5.6e-198) && (c0 <= 1e+155)))
		tmp = Float64(c0 * Float64(0.0 / Float64(2.0 * w)));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))) + Float64(t_0 * Float64(Float64(d / D) * Float64(d / D)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * h);
	t_1 = t_0 * ((d * d) / (D * D));
	tmp = 0.0;
	if ((c0 <= -3.2e+102) || (~((c0 <= -5.6e-198)) && (c0 <= 1e+155)))
		tmp = c0 * (0.0 / (2.0 * w));
	else
		tmp = (c0 / (2.0 * w)) * (sqrt(((t_1 * t_1) - (M * M))) + (t_0 * ((d / D) * (d / D))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[c0, -3.2e+102], And[N[Not[LessEqual[c0, -5.6e-198]], $MachinePrecision], LessEqual[c0, 1e+155]]], N[(c0 * N[(0.0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c0 < -3.1999999999999999e102 or -5.5999999999999998e-198 < c0 < 1.00000000000000001e155

   1. Initial program 19.1%

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

   3. Simplified 34.6%

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

   5. Taylor expanded in c0 around -inf 3.5%

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

      1. mul-1-neg 3.5%

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

      2. distribute-lft-in 2.1%

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

      3. mul-1-neg 2.1%

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

      4. distribute-rgt-neg-in 2.1%

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

      5. associate-/l* 2.1%

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

      6. mul-1-neg 2.1%

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

      7. associate-/l* 1.5%

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

      8. distribute-lft1-in 1.5%

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

      9. metadata-eval 1.5%

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

      10. mul0-lft 44.2%

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

      11. metadata-eval 44.2%

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

   7. Simplified 44.2%

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

   if -3.1999999999999999e102 < c0 < -5.5999999999999998e-198 or 1.00000000000000001e155 < c0

   1. Initial program 32.9%

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

   3. Simplified 34.9%

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

      1. times-frac 34.9%

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

   6. Applied egg-rr 34.9%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -3.2 \cdot 10^{+102} \lor \neg \left(c0 \leq -5.6 \cdot 10^{-198}\right) \land c0 \leq 10^{+155}:\\ \;\;\;\;c0 \cdot \frac{0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) - M \cdot M} + \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.2% accurate, 21.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = c0 * (0.0d0 / (2.0d0 * w))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

3. Simplified 38.0%

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

5. Taylor expanded in c0 around -inf 3.7%

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

   1. mul-1-neg 3.7%

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

   2. distribute-lft-in 2.8%

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

   3. mul-1-neg 2.8%

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

   4. distribute-rgt-neg-in 2.8%

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

   5. associate-/l* 2.7%

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

   6. mul-1-neg 2.7%

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

   7. associate-/l* 2.4%

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

   8. distribute-lft1-in 2.4%

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

   9. metadata-eval 2.4%

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

   10. mul0-lft 34.4%

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

   11. metadata-eval 34.4%

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

7. Simplified 34.4%

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

8. Final simplification 34.4%

   \[\leadsto c0 \cdot \frac{0}{2 \cdot w} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(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))))))