Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.7% → 54.9%

Time: 18.5s

Alternatives: 10

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = 2.770553989549313e+292
  2. w = -7.150871795117319e-306
  3. h = 7.810662470363612e-148
  4. D = -4.327667744619452e-261
  5. d = 8.696363325550854e+298
  6. M = 3.172258251254402e-288

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.7% 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: 54.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(d \cdot d\right) \cdot c0\\ t_1 := \frac{t\_0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\\ t_2 := \sqrt{t\_1 \cdot t\_1 - M \cdot M}\\ t_3 := \frac{c0}{w \cdot 2}\\ t_4 := \left(t\_2 + t\_1\right) \cdot t\_3\\ \mathbf{if}\;t\_4 \leq 10^{+180}:\\ \;\;\;\;\left(\frac{t\_0}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} + t\_2\right) \cdot t\_3\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{{\left(\frac{D}{c0}\right)}^{-2} \cdot d}{\frac{h \cdot w}{d} \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* d d) c0))
        (t_1 (/ t_0 (* (* h w) (* D D))))
        (t_2 (sqrt (- (* t_1 t_1) (* M M))))
        (t_3 (/ c0 (* w 2.0)))
        (t_4 (* (+ t_2 t_1) t_3)))
   (if (<= t_4 1e+180)
     (* (+ (/ t_0 (* (* (* h w) D) D)) t_2) t_3)
     (if (<= t_4 INFINITY)
       (/ (* (pow (/ D c0) -2.0) d) (* (/ (* h w) d) w))
       0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * d) * c0;
	double t_1 = t_0 / ((h * w) * (D * D));
	double t_2 = sqrt(((t_1 * t_1) - (M * M)));
	double t_3 = c0 / (w * 2.0);
	double t_4 = (t_2 + t_1) * t_3;
	double tmp;
	if (t_4 <= 1e+180) {
		tmp = ((t_0 / (((h * w) * D) * D)) + t_2) * t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (pow((D / c0), -2.0) * d) / (((h * w) / d) * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * d) * c0;
	double t_1 = t_0 / ((h * w) * (D * D));
	double t_2 = Math.sqrt(((t_1 * t_1) - (M * M)));
	double t_3 = c0 / (w * 2.0);
	double t_4 = (t_2 + t_1) * t_3;
	double tmp;
	if (t_4 <= 1e+180) {
		tmp = ((t_0 / (((h * w) * D) * D)) + t_2) * t_3;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.pow((D / c0), -2.0) * d) / (((h * w) / d) * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d * d) * c0
	t_1 = t_0 / ((h * w) * (D * D))
	t_2 = math.sqrt(((t_1 * t_1) - (M * M)))
	t_3 = c0 / (w * 2.0)
	t_4 = (t_2 + t_1) * t_3
	tmp = 0
	if t_4 <= 1e+180:
		tmp = ((t_0 / (((h * w) * D) * D)) + t_2) * t_3
	elif t_4 <= math.inf:
		tmp = (math.pow((D / c0), -2.0) * d) / (((h * w) / d) * w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * d) * c0)
	t_1 = Float64(t_0 / Float64(Float64(h * w) * Float64(D * D)))
	t_2 = sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))
	t_3 = Float64(c0 / Float64(w * 2.0))
	t_4 = Float64(Float64(t_2 + t_1) * t_3)
	tmp = 0.0
	if (t_4 <= 1e+180)
		tmp = Float64(Float64(Float64(t_0 / Float64(Float64(Float64(h * w) * D) * D)) + t_2) * t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64((Float64(D / c0) ^ -2.0) * d) / Float64(Float64(Float64(h * w) / d) * w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d * d) * c0;
	t_1 = t_0 / ((h * w) * (D * D));
	t_2 = sqrt(((t_1 * t_1) - (M * M)));
	t_3 = c0 / (w * 2.0);
	t_4 = (t_2 + t_1) * t_3;
	tmp = 0.0;
	if (t_4 <= 1e+180)
		tmp = ((t_0 / (((h * w) * D) * D)) + t_2) * t_3;
	elseif (t_4 <= Inf)
		tmp = (((D / c0) ^ -2.0) * d) / (((h * w) / d) * w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(h * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 + t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, 1e+180], N[(N[(N[(t$95$0 / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Power[N[(D / c0), $MachinePrecision], -2.0], $MachinePrecision] * d), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] / d), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]]

Derivation

1. Split input into 3 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))))) < 1e180

   1. Initial program 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 80.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 80.9

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

   4. Applied rewrites 80.9%

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

   if 1e180 < (*.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 71.8%

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 67.6

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

   7. Applied rewrites 67.6%

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

   9. Applied rewrites 74.9%

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

   11. Applied rewrites 87.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 10^{+180}:\\ \;\;\;\;\left(\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} + \sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\frac{{\left(\frac{D}{c0}\right)}^{-2} \cdot d}{\frac{h \cdot w}{d} \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\\ t_1 := \sqrt{t\_0 \cdot t\_0 - M \cdot M} + t\_0\\ t_2 := t\_1 \cdot \frac{c0}{w \cdot 2}\\ \mathbf{if}\;t\_2 \leq 10^{+180}:\\ \;\;\;\;\left(\frac{0.5}{w} \cdot c0\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{{\left(\frac{D}{c0}\right)}^{-2} \cdot d}{\frac{h \cdot w}{d} \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* (* d d) c0) (* (* h w) (* D D))))
        (t_1 (+ (sqrt (- (* t_0 t_0) (* M M))) t_0))
        (t_2 (* t_1 (/ c0 (* w 2.0)))))
   (if (<= t_2 1e+180)
     (* (* (/ 0.5 w) c0) t_1)
     (if (<= t_2 INFINITY)
       (/ (* (pow (/ D c0) -2.0) d) (* (/ (* h w) d) w))
       0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) * c0) / ((h * w) * (D * D));
	double t_1 = sqrt(((t_0 * t_0) - (M * M))) + t_0;
	double t_2 = t_1 * (c0 / (w * 2.0));
	double tmp;
	if (t_2 <= 1e+180) {
		tmp = ((0.5 / w) * c0) * t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (pow((D / c0), -2.0) * d) / (((h * w) / d) * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) * c0) / ((h * w) * (D * D));
	double t_1 = Math.sqrt(((t_0 * t_0) - (M * M))) + t_0;
	double t_2 = t_1 * (c0 / (w * 2.0));
	double tmp;
	if (t_2 <= 1e+180) {
		tmp = ((0.5 / w) * c0) * t_1;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.pow((D / c0), -2.0) * d) / (((h * w) / d) * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((h * w) * (D * D))
	t_1 = math.sqrt(((t_0 * t_0) - (M * M))) + t_0
	t_2 = t_1 * (c0 / (w * 2.0))
	tmp = 0
	if t_2 <= 1e+180:
		tmp = ((0.5 / w) * c0) * t_1
	elif t_2 <= math.inf:
		tmp = (math.pow((D / c0), -2.0) * d) / (((h * w) / d) * w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d * d) * c0) / Float64(Float64(h * w) * Float64(D * D)))
	t_1 = Float64(sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))) + t_0)
	t_2 = Float64(t_1 * Float64(c0 / Float64(w * 2.0)))
	tmp = 0.0
	if (t_2 <= 1e+180)
		tmp = Float64(Float64(Float64(0.5 / w) * c0) * t_1);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64((Float64(D / c0) ^ -2.0) * d) / Float64(Float64(Float64(h * w) / d) * w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * d) * c0) / ((h * w) * (D * D));
	t_1 = sqrt(((t_0 * t_0) - (M * M))) + t_0;
	t_2 = t_1 * (c0 / (w * 2.0));
	tmp = 0.0;
	if (t_2 <= 1e+180)
		tmp = ((0.5 / w) * c0) * t_1;
	elseif (t_2 <= Inf)
		tmp = (((D / c0) ^ -2.0) * d) / (((h * w) / d) * w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e+180], N[(N[(N[(0.5 / w), $MachinePrecision] * c0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Power[N[(D / c0), $MachinePrecision], -2.0], $MachinePrecision] * d), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] / d), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 3 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))))) < 1e180

   1. Initial program 80.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 80.5

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

   4. Applied rewrites 80.5%

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

   if 1e180 < (*.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 71.8%

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 67.6

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

   7. Applied rewrites 67.6%

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

   9. Applied rewrites 74.9%

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

   11. Applied rewrites 87.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 10^{+180}:\\ \;\;\;\;\left(\frac{0.5}{w} \cdot c0\right) \cdot \left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right)\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\frac{{\left(\frac{D}{c0}\right)}^{-2} \cdot d}{\frac{h \cdot w}{d} \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot 2}\\ t_1 := \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\\ t_2 := \left(\sqrt{t\_1 \cdot t\_1 - M \cdot M} + t\_1\right) \cdot t\_0\\ \mathbf{if}\;t\_2 \leq 10^{+180}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D}, \frac{d \cdot c0}{D}, \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w} \cdot \left(d \cdot d\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{{\left(\frac{D}{c0}\right)}^{-2} \cdot d}{\frac{h \cdot w}{d} \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w 2.0)))
        (t_1 (/ (* (* d d) c0) (* (* h w) (* D D))))
        (t_2 (* (+ (sqrt (- (* t_1 t_1) (* M M))) t_1) t_0)))
   (if (<= t_2 1e+180)
     (*
      (fma
       (/ d (* (* h w) D))
       (/ (* d c0) D)
       (* (/ c0 (* (* (* D D) h) w)) (* d d)))
      t_0)
     (if (<= t_2 INFINITY)
       (/ (* (pow (/ D c0) -2.0) d) (* (/ (* h w) d) w))
       0.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * 2.0);
	double t_1 = ((d * d) * c0) / ((h * w) * (D * D));
	double t_2 = (sqrt(((t_1 * t_1) - (M * M))) + t_1) * t_0;
	double tmp;
	if (t_2 <= 1e+180) {
		tmp = fma((d / ((h * w) * D)), ((d * c0) / D), ((c0 / (((D * D) * h) * w)) * (d * d))) * t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (pow((D / c0), -2.0) * d) / (((h * w) / d) * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * 2.0))
	t_1 = Float64(Float64(Float64(d * d) * c0) / Float64(Float64(h * w) * Float64(D * D)))
	t_2 = Float64(Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))) + t_1) * t_0)
	tmp = 0.0
	if (t_2 <= 1e+180)
		tmp = Float64(fma(Float64(d / Float64(Float64(h * w) * D)), Float64(Float64(d * c0) / D), Float64(Float64(c0 / Float64(Float64(Float64(D * D) * h) * w)) * Float64(d * d))) * t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64((Float64(D / c0) ^ -2.0) * d) / Float64(Float64(Float64(h * w) / d) * w));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, 1e+180], N[(N[(N[(d / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * c0), $MachinePrecision] / D), $MachinePrecision] + N[(N[(c0 / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Power[N[(D / c0), $MachinePrecision], -2.0], $MachinePrecision] * d), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] / d), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 3 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))))) < 1e180

   1. Initial program 80.5%

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

   4. Applied rewrites 70.4%

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

   6. Applied rewrites 64.5%

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

   7. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 76.9

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

   9. Applied rewrites 76.9%

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

   if 1e180 < (*.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 71.8%

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 67.6

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

   7. Applied rewrites 67.6%

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

   9. Applied rewrites 74.9%

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

   11. Applied rewrites 87.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 10^{+180}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{\left(h \cdot w\right) \cdot D}, \frac{d \cdot c0}{D}, \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w} \cdot \left(d \cdot d\right)\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\frac{{\left(\frac{D}{c0}\right)}^{-2} \cdot d}{\frac{h \cdot w}{d} \cdot w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.7%

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

   4. Applied rewrites 71.9%

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

   6. Applied rewrites 65.5%

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

   7. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 75.2

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

   9. Applied rewrites 75.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 56.2%

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

Alternative 5: 53.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.7%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   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

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 56.0%

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

Alternative 6: 53.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.7%

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

   3. Taylor expanded in w around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   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%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 55.9%

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

Alternative 7: 52.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.7%

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 57.7

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

   7. Applied rewrites 57.7%

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

   9. Applied rewrites 63.5%

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

   11. Applied rewrites 70.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 54.6%

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

Alternative 8: 51.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.7%

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 57.7

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

   7. Applied rewrites 57.7%

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

   9. Applied rewrites 63.5%

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

   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

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 52.3%

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

Alternative 9: 50.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.7%

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

   3. Taylor expanded in w around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 57.7

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

   5. Applied rewrites 57.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. div0 N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 50.4%

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

Alternative 10: 33.7% accurate, 156.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 25.8%

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

3. Taylor expanded in c0 around -inf

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

   1. associate-/l* N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

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

   5. div0 N/A

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

   6. mul0-rgt N/A

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

   7. metadata-eval 35.0

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

5. Applied rewrites 35.0%

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

Reproduce

herbie shell --seed 2024263 
(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))))))