Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.8% → 65.7%

Time: 18.3s

Alternatives: 11

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = -2.2529684445121697e+293
  2. w = -5.302119954033252e-288
  3. h = -2.853393189203833e-262
  4. D = 1.1210781283527635e-274
  5. d = 6.039953736322755e+268
  6. M = -8.139057275053306e+277

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: 24.8% 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: 65.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((h * w) * (D * D))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = ((d * c0) / (((D * w) * h) * (D * w))) * (d * c0)
	else:
		tmp = (((D / d) * (((M * M) * h) * D)) * 0.25) / d
	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 * c0) / Float64(Float64(Float64(D * w) * h) * Float64(D * w))) * Float64(d * c0));
	else
		tmp = Float64(Float64(Float64(Float64(D / d) * Float64(Float64(Float64(M * M) * h) * D)) * 0.25) / d);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.6%

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

   3. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 52.3%

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

   9. Applied rewrites 75.2%

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

   11. Applied rewrites 79.0%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 45.0%

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

   12. Applied rewrites 59.5%

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

4. Final simplification 65.2%

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

Alternative 2: 59.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\\ t_1 := \left(\sqrt{t\_0 \cdot t\_0 - M \cdot M} + t\_0\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-98}:\\ \;\;\;\;\frac{c0 \cdot c0}{\left(\left(h \cdot w\right) \cdot D\right) \cdot \left(D \cdot w\right)} \cdot \left(d \cdot d\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(\left(\frac{D}{d \cdot d} \cdot D\right) \cdot \left(M \cdot h\right)\right) \cdot M\right) \cdot 0.25\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{d \cdot c0}{\left(\left(D \cdot h\right) \cdot w\right) \cdot \left(D \cdot w\right)} \cdot c0\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{d \cdot d} \cdot D\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* (* d d) c0) (* (* h w) (* D D))))
        (t_1 (* (+ (sqrt (- (* t_0 t_0) (* M M))) t_0) (/ c0 (* w 2.0)))))
   (if (<= t_1 -2e-98)
     (* (/ (* c0 c0) (* (* (* h w) D) (* D w))) (* d d))
     (if (<= t_1 0.0)
       (* (* (* (* (/ D (* d d)) D) (* M h)) M) 0.25)
       (if (<= t_1 INFINITY)
         (* (* (/ (* d c0) (* (* (* D h) w) (* D w))) c0) d)
         (* (* (/ (* (* (* M M) h) D) (* d d)) D) 0.25))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) * c0) / ((h * w) * (D * D));
	double t_1 = (sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0));
	double tmp;
	if (t_1 <= -2e-98) {
		tmp = ((c0 * c0) / (((h * w) * D) * (D * w))) * (d * d);
	} else if (t_1 <= 0.0) {
		tmp = ((((D / (d * d)) * D) * (M * h)) * M) * 0.25;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((d * c0) / (((D * h) * w) * (D * w))) * c0) * d;
	} else {
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) * c0) / ((h * w) * (D * D));
	double t_1 = (Math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0));
	double tmp;
	if (t_1 <= -2e-98) {
		tmp = ((c0 * c0) / (((h * w) * D) * (D * w))) * (d * d);
	} else if (t_1 <= 0.0) {
		tmp = ((((D / (d * d)) * D) * (M * h)) * M) * 0.25;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (((d * c0) / (((D * h) * w) * (D * w))) * c0) * d;
	} else {
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((h * w) * (D * D))
	t_1 = (math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))
	tmp = 0
	if t_1 <= -2e-98:
		tmp = ((c0 * c0) / (((h * w) * D) * (D * w))) * (d * d)
	elif t_1 <= 0.0:
		tmp = ((((D / (d * d)) * D) * (M * h)) * M) * 0.25
	elif t_1 <= math.inf:
		tmp = (((d * c0) / (((D * h) * w) * (D * w))) * c0) * d
	else:
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d * d) * c0) / Float64(Float64(h * w) * Float64(D * D)))
	t_1 = Float64(Float64(sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))) + t_0) * Float64(c0 / Float64(w * 2.0)))
	tmp = 0.0
	if (t_1 <= -2e-98)
		tmp = Float64(Float64(Float64(c0 * c0) / Float64(Float64(Float64(h * w) * D) * Float64(D * w))) * Float64(d * d));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(D / Float64(d * d)) * D) * Float64(M * h)) * M) * 0.25);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(d * c0) / Float64(Float64(Float64(D * h) * w) * Float64(D * w))) * c0) * d);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) / Float64(d * d)) * D) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * d) * c0) / ((h * w) * (D * D));
	t_1 = (sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0));
	tmp = 0.0;
	if (t_1 <= -2e-98)
		tmp = ((c0 * c0) / (((h * w) * D) * (D * w))) * (d * d);
	elseif (t_1 <= 0.0)
		tmp = ((((D / (d * d)) * D) * (M * h)) * M) * 0.25;
	elseif (t_1 <= Inf)
		tmp = (((d * c0) / (((D * h) * w) * (D * w))) * c0) * d;
	else
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-98], N[(N[(N[(c0 * c0), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision] * 0.25), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(d * c0), $MachinePrecision] / N[(N[(N[(D * h), $MachinePrecision] * w), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * d), $MachinePrecision], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * 0.25), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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))))) < -1.99999999999999988e-98

   1. Initial program 96.0%

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

   3. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 87.5

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

   5. Applied rewrites 87.5%

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

   7. Applied rewrites 99.9%

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

   if -1.99999999999999988e-98 < (*.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))))) < -0.0

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 36.3%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 54.5%

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

   12. Applied rewrites 81.3%

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

   if -0.0 < (*.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))))) < +inf.0

   1. Initial program 64.1%

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

   3. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 36.8

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

   5. Applied rewrites 36.8%

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

   7. Applied rewrites 39.7%

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

   9. Applied rewrites 77.5%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 45.0%

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

   12. Applied rewrites 52.9%

       \[\leadsto 0.25 \cdot \left(D \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{\color{blue}{d \cdot d}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq -2 \cdot 10^{-98}:\\ \;\;\;\;\frac{c0 \cdot c0}{\left(\left(h \cdot w\right) \cdot D\right) \cdot \left(D \cdot w\right)} \cdot \left(d \cdot d\right)\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 0:\\ \;\;\;\;\left(\left(\left(\frac{D}{d \cdot d} \cdot D\right) \cdot \left(M \cdot h\right)\right) \cdot M\right) \cdot 0.25\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(h \cdot w\right) \cdot \left(D \cdot D\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\left(\frac{d \cdot c0}{\left(\left(D \cdot h\right) \cdot w\right) \cdot \left(D \cdot w\right)} \cdot c0\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{d \cdot d} \cdot D\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((h * w) * (D * D))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = ((d * c0) / (((D * w) * h) * (D * w))) * (d * c0)
	else:
		tmp = ((((D * h) * D) / d) * ((M * M) / d)) * 0.25
	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 * c0) / Float64(Float64(Float64(D * w) * h) * Float64(D * w))) * Float64(d * c0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(D * h) * D) / d) * Float64(Float64(M * M) / d)) * 0.25);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.6%

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

   3. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 52.3%

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

   9. Applied rewrites 75.2%

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

   11. Applied rewrites 79.0%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 45.0%

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

   12. Applied rewrites 54.8%

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

4. Final simplification 61.8%

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

Alternative 4: 60.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((h * w) * (D * D))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = ((d * c0) / (((D * w) * h) * (D * w))) * (d * c0)
	else:
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25
	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 * c0) / Float64(Float64(Float64(D * w) * h) * Float64(D * w))) * Float64(d * c0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) / Float64(d * d)) * D) * 0.25);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.6%

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

   3. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 52.3%

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

   9. Applied rewrites 75.2%

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

   11. Applied rewrites 79.0%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 45.0%

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

   12. Applied rewrites 52.9%

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

4. Final simplification 60.5%

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

Alternative 5: 59.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((h * w) * (D * D))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = ((d * c0) / (((D * h) * w) * (D * w))) * (d * c0)
	else:
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25
	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 * c0) / Float64(Float64(Float64(D * h) * w) * Float64(D * w))) * Float64(d * c0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) / Float64(d * d)) * D) * 0.25);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.6%

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

   3. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 52.3%

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

   9. Applied rewrites 75.2%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 45.0%

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

   12. Applied rewrites 52.9%

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

4. Final simplification 59.4%

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

Alternative 6: 59.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.6%

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

   3. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 52.3%

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

   9. Applied rewrites 75.2%

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

   11. Applied rewrites 74.0%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 45.0%

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

   12. Applied rewrites 52.9%

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

4. Final simplification 59.0%

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

Alternative 7: 59.4% accurate, 0.7× speedup

Math

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

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 (* (* (* (* D w) h) w) D)) c0) (* d c0))
     (* (* (/ (* (* (* M M) h) D) (* d d)) D) 0.25))))

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 / ((((D * w) * h) * w) * D)) * c0) * (d * c0);
	} else {
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	}
	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 / ((((D * w) * h) * w) * D)) * c0) * (d * c0);
	} else {
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	}
	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 / ((((D * w) * h) * w) * D)) * c0) * (d * c0)
	else:
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25
	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(Float64(Float64(D * w) * h) * w) * D)) * c0) * Float64(d * c0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) / Float64(d * d)) * D) * 0.25);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.6%

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

   3. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 52.3%

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

   9. Applied rewrites 75.2%

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

   11. Applied rewrites 73.9%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 45.0%

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

   12. Applied rewrites 52.9%

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

4. Final simplification 59.0%

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

Alternative 8: 58.8% accurate, 0.7× speedup

Math

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

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 (* (* (* D h) w) (* D w))) (* d c0)) c0)
     (* (* (/ (* (* (* M M) h) D) (* d d)) D) 0.25))))

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 / (((D * h) * w) * (D * w))) * (d * c0)) * c0;
	} else {
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	}
	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 / (((D * h) * w) * (D * w))) * (d * c0)) * c0;
	} else {
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	}
	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 / (((D * h) * w) * (D * w))) * (d * c0)) * c0
	else:
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25
	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(Float64(D * h) * w) * Float64(D * w))) * Float64(d * c0)) * c0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) / Float64(d * d)) * D) * 0.25);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.6%

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

   3. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 61.9%

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

   9. Applied rewrites 71.5%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 45.0%

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

   12. Applied rewrites 52.9%

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

4. Final simplification 58.3%

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

Alternative 9: 57.9% accurate, 0.7× speedup

Math

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

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 d) (* (* (* D h) w) (* D w))) c0) c0)
     (* (* (/ (* (* (* M M) h) D) (* d d)) D) 0.25))))

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 * d) / (((D * h) * w) * (D * w))) * c0) * c0;
	} else {
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	}
	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 * d) / (((D * h) * w) * (D * w))) * c0) * c0;
	} else {
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	}
	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 * d) / (((D * h) * w) * (D * w))) * c0) * c0
	else:
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25
	return tmp

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.6%

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

   3. Taylor expanded in w around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 61.9%

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

   9. Applied rewrites 67.6%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 27.7%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 45.0%

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

   12. Applied rewrites 52.9%

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

4. Final simplification 57.2%

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

Alternative 10: 36.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 6.8 \cdot 10^{-279}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{d \cdot d} \cdot D\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 6.8e-279) 0.0 (* (* (/ (* (* (* M M) h) D) (* d d)) D) 0.25)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 6.8e-279) {
		tmp = 0.0;
	} else {
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 6.8d-279) then
        tmp = 0.0d0
    else
        tmp = (((((m * m) * h) * d) / (d_1 * d_1)) * d) * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 6.8e-279) {
		tmp = 0.0;
	} else {
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 6.8e-279:
		tmp = 0.0
	else:
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 6.8e-279)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * D) / Float64(d * d)) * D) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 6.8e-279)
		tmp = 0.0;
	else
		tmp = (((((M * M) * h) * D) / (d * d)) * D) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 6.8e-279], 0.0, N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] * 0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 6.8000000000000003e-279

   1. Initial program 21.9%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval N/A

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

      8. div0 37.0

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

   5. Applied rewrites 37.0%

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

   if 6.8000000000000003e-279 < D

   1. Initial program 20.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 19.2%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 29.6%

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

   8. Taylor expanded in c0 around 0

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

   10. Applied rewrites 45.9%

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

   12. Applied rewrites 52.0%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 6.8 \cdot 10^{-279}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot D}{d \cdot d} \cdot D\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.8% 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 21.0%

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

3. Taylor expanded in c0 around -inf

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. mul0-lft N/A

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

   6. mul0-rgt N/A

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

   7. metadata-eval N/A

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

   8. div0 38.3

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

5. Applied rewrites 38.3%

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

Reproduce

herbie shell --seed 2024237 
(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))))))