Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.1% → 69.4%

Time: 16.6s

Alternatives: 13

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = 2.5607244280588838e+272
  2. w = 1.779454659562182e-294
  3. h = -8.922568902596605e-278
  4. D = -2.0314762490356662e-199
  5. d = 2.729016683910885e+300
  6. M = 6.424628494262528e+100

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 69.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ t_1 := \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_2 := \left(\sqrt{t\_1 \cdot t\_1 - M \cdot M} + t\_1\right) \cdot \frac{c0}{w \cdot 2}\\ t_3 := \left(t\_0 \cdot t\_0\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot h\right) \cdot D}}{w \cdot w}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-99}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\frac{{\left(\frac{d}{D} \cdot c0\right)}^{2}}{w}}{h \cdot w}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ M d) D))
        (t_1 (/ (* (* d d) c0) (* (* D D) (* h w))))
        (t_2 (* (+ (sqrt (- (* t_1 t_1) (* M M))) t_1) (/ c0 (* w 2.0))))
        (t_3 (* (* t_0 t_0) (* 0.25 h))))
   (if (<= t_2 -2e+125)
     (/ (/ (pow (* d c0) 2.0) (* (* D h) D)) (* w w))
     (if (<= t_2 2e-99)
       t_3
       (if (<= t_2 INFINITY)
         (/ (/ (pow (* (/ d D) c0) 2.0) w) (* h w))
         t_3)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (M / d) * D;
	double t_1 = ((d * d) * c0) / ((D * D) * (h * w));
	double t_2 = (sqrt(((t_1 * t_1) - (M * M))) + t_1) * (c0 / (w * 2.0));
	double t_3 = (t_0 * t_0) * (0.25 * h);
	double tmp;
	if (t_2 <= -2e+125) {
		tmp = (pow((d * c0), 2.0) / ((D * h) * D)) / (w * w);
	} else if (t_2 <= 2e-99) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (pow(((d / D) * c0), 2.0) / w) / (h * w);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (M / d) * D;
	double t_1 = ((d * d) * c0) / ((D * D) * (h * w));
	double t_2 = (Math.sqrt(((t_1 * t_1) - (M * M))) + t_1) * (c0 / (w * 2.0));
	double t_3 = (t_0 * t_0) * (0.25 * h);
	double tmp;
	if (t_2 <= -2e+125) {
		tmp = (Math.pow((d * c0), 2.0) / ((D * h) * D)) / (w * w);
	} else if (t_2 <= 2e-99) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.pow(((d / D) * c0), 2.0) / w) / (h * w);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (M / d) * D
	t_1 = ((d * d) * c0) / ((D * D) * (h * w))
	t_2 = (math.sqrt(((t_1 * t_1) - (M * M))) + t_1) * (c0 / (w * 2.0))
	t_3 = (t_0 * t_0) * (0.25 * h)
	tmp = 0
	if t_2 <= -2e+125:
		tmp = (math.pow((d * c0), 2.0) / ((D * h) * D)) / (w * w)
	elif t_2 <= 2e-99:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = (math.pow(((d / D) * c0), 2.0) / w) / (h * w)
	else:
		tmp = t_3
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(M / d) * D)
	t_1 = Float64(Float64(Float64(d * d) * c0) / Float64(Float64(D * D) * Float64(h * w)))
	t_2 = Float64(Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))) + t_1) * Float64(c0 / Float64(w * 2.0)))
	t_3 = Float64(Float64(t_0 * t_0) * Float64(0.25 * h))
	tmp = 0.0
	if (t_2 <= -2e+125)
		tmp = Float64(Float64((Float64(d * c0) ^ 2.0) / Float64(Float64(D * h) * D)) / Float64(w * w));
	elseif (t_2 <= 2e-99)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = Float64(Float64((Float64(Float64(d / D) * c0) ^ 2.0) / w) / Float64(h * w));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (M / d) * D;
	t_1 = ((d * d) * c0) / ((D * D) * (h * w));
	t_2 = (sqrt(((t_1 * t_1) - (M * M))) + t_1) * (c0 / (w * 2.0));
	t_3 = (t_0 * t_0) * (0.25 * h);
	tmp = 0.0;
	if (t_2 <= -2e+125)
		tmp = (((d * c0) ^ 2.0) / ((D * h) * D)) / (w * w);
	elseif (t_2 <= 2e-99)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = ((((d / D) * c0) ^ 2.0) / w) / (h * w);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+125], N[(N[(N[Power[N[(d * c0), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(D * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-99], t$95$3, If[LessEqual[t$95$2, Infinity], N[(N[(N[Power[N[(N[(d / D), $MachinePrecision] * c0), $MachinePrecision], 2.0], $MachinePrecision] / w), $MachinePrecision] / N[(h * w), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -1.9999999999999998e125

   1. Initial program 91.6%

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

   4. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.5

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

   7. Applied rewrites 82.5%

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

   9. Applied rewrites 82.6%

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

   11. Applied rewrites 95.5%

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

   if -1.9999999999999998e125 < (*.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))))) < 2e-99 or +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 7.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 63.6%

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

   if 2e-99 < (*.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 85.5%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   9. Applied rewrites 79.3%

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

   11. Applied rewrites 91.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot h\right) \cdot D}}{w \cdot w}\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 2 \cdot 10^{-99}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\frac{\frac{{\left(\frac{d}{D} \cdot c0\right)}^{2}}{w}}{h \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(d \cdot c0\right)}^{2}\\ t_1 := \frac{M}{d} \cdot D\\ t_2 := \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_3 := \left(\sqrt{t\_2 \cdot t\_2 - M \cdot M} + t\_2\right) \cdot \frac{c0}{w \cdot 2}\\ t_4 := \left(t\_1 \cdot t\_1\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{t\_0}{\left(D \cdot h\right) \cdot D}}{w \cdot w}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-99}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\frac{t\_0}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}}{w}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (* d c0) 2.0))
        (t_1 (* (/ M d) D))
        (t_2 (/ (* (* d d) c0) (* (* D D) (* h w))))
        (t_3 (* (+ (sqrt (- (* t_2 t_2) (* M M))) t_2) (/ c0 (* w 2.0))))
        (t_4 (* (* t_1 t_1) (* 0.25 h))))
   (if (<= t_3 -2e+125)
     (/ (/ t_0 (* (* D h) D)) (* w w))
     (if (<= t_3 2e-99)
       t_4
       (if (<= t_3 INFINITY) (/ (/ t_0 (* (* (* D D) w) h)) w) t_4)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((d * c0), 2.0);
	double t_1 = (M / d) * D;
	double t_2 = ((d * d) * c0) / ((D * D) * (h * w));
	double t_3 = (sqrt(((t_2 * t_2) - (M * M))) + t_2) * (c0 / (w * 2.0));
	double t_4 = (t_1 * t_1) * (0.25 * h);
	double tmp;
	if (t_3 <= -2e+125) {
		tmp = (t_0 / ((D * h) * D)) / (w * w);
	} else if (t_3 <= 2e-99) {
		tmp = t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_0 / (((D * D) * w) * h)) / w;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow((d * c0), 2.0);
	double t_1 = (M / d) * D;
	double t_2 = ((d * d) * c0) / ((D * D) * (h * w));
	double t_3 = (Math.sqrt(((t_2 * t_2) - (M * M))) + t_2) * (c0 / (w * 2.0));
	double t_4 = (t_1 * t_1) * (0.25 * h);
	double tmp;
	if (t_3 <= -2e+125) {
		tmp = (t_0 / ((D * h) * D)) / (w * w);
	} else if (t_3 <= 2e-99) {
		tmp = t_4;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / (((D * D) * w) * h)) / w;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((d * c0), 2.0)
	t_1 = (M / d) * D
	t_2 = ((d * d) * c0) / ((D * D) * (h * w))
	t_3 = (math.sqrt(((t_2 * t_2) - (M * M))) + t_2) * (c0 / (w * 2.0))
	t_4 = (t_1 * t_1) * (0.25 * h)
	tmp = 0
	if t_3 <= -2e+125:
		tmp = (t_0 / ((D * h) * D)) / (w * w)
	elif t_3 <= 2e-99:
		tmp = t_4
	elif t_3 <= math.inf:
		tmp = (t_0 / (((D * D) * w) * h)) / w
	else:
		tmp = t_4
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(d * c0) ^ 2.0
	t_1 = Float64(Float64(M / d) * D)
	t_2 = Float64(Float64(Float64(d * d) * c0) / Float64(Float64(D * D) * Float64(h * w)))
	t_3 = Float64(Float64(sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))) + t_2) * Float64(c0 / Float64(w * 2.0)))
	t_4 = Float64(Float64(t_1 * t_1) * Float64(0.25 * h))
	tmp = 0.0
	if (t_3 <= -2e+125)
		tmp = Float64(Float64(t_0 / Float64(Float64(D * h) * D)) / Float64(w * w));
	elseif (t_3 <= 2e-99)
		tmp = t_4;
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_0 / Float64(Float64(Float64(D * D) * w) * h)) / w);
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d * c0) ^ 2.0;
	t_1 = (M / d) * D;
	t_2 = ((d * d) * c0) / ((D * D) * (h * w));
	t_3 = (sqrt(((t_2 * t_2) - (M * M))) + t_2) * (c0 / (w * 2.0));
	t_4 = (t_1 * t_1) * (0.25 * h);
	tmp = 0.0;
	if (t_3 <= -2e+125)
		tmp = (t_0 / ((D * h) * D)) / (w * w);
	elseif (t_3 <= 2e-99)
		tmp = t_4;
	elseif (t_3 <= Inf)
		tmp = (t_0 / (((D * D) * w) * h)) / w;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(d * c0), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+125], N[(N[(t$95$0 / N[(N[(D * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-99], t$95$4, If[LessEqual[t$95$3, Infinity], N[(N[(t$95$0 / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision], t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -1.9999999999999998e125

   1. Initial program 91.6%

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

   4. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.5

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

   7. Applied rewrites 82.5%

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

   9. Applied rewrites 82.6%

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

   11. Applied rewrites 95.5%

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

   if -1.9999999999999998e125 < (*.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))))) < 2e-99 or +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 7.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 63.6%

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

   if 2e-99 < (*.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 85.5%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   9. Applied rewrites 85.4%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot h\right) \cdot D}}{w \cdot w}\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 2 \cdot 10^{-99}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}}{w}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(d \cdot d\right) \cdot c0\\ t_1 := \frac{t\_0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_2 := \frac{c0}{w \cdot 2}\\ t_3 := \left(\sqrt{t\_1 \cdot t\_1 - M \cdot M} + t\_1\right) \cdot t\_2\\ t_4 := \frac{M}{d} \cdot D\\ t_5 := \left(t\_4 \cdot t\_4\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot h\right) \cdot D}}{w \cdot w}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-99}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{w}{t\_0} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \left(D \cdot D\right), -0.5, \left(\frac{2}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot \left(d \cdot d\right)\right) \cdot c0\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* d d) c0))
        (t_1 (/ t_0 (* (* D D) (* h w))))
        (t_2 (/ c0 (* w 2.0)))
        (t_3 (* (+ (sqrt (- (* t_1 t_1) (* M M))) t_1) t_2))
        (t_4 (* (/ M d) D))
        (t_5 (* (* t_4 t_4) (* 0.25 h))))
   (if (<= t_3 -2e+125)
     (/ (/ (pow (* d c0) 2.0) (* (* D h) D)) (* w w))
     (if (<= t_3 2e-99)
       t_5
       (if (<= t_3 INFINITY)
         (*
          (fma
           (* (* (/ w t_0) (* (* M M) h)) (* D D))
           -0.5
           (* (* (/ 2.0 (* (* (* D D) w) h)) (* d d)) c0))
          t_2)
         t_5)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d * d) * c0;
	double t_1 = t_0 / ((D * D) * (h * w));
	double t_2 = c0 / (w * 2.0);
	double t_3 = (sqrt(((t_1 * t_1) - (M * M))) + t_1) * t_2;
	double t_4 = (M / d) * D;
	double t_5 = (t_4 * t_4) * (0.25 * h);
	double tmp;
	if (t_3 <= -2e+125) {
		tmp = (pow((d * c0), 2.0) / ((D * h) * D)) / (w * w);
	} else if (t_3 <= 2e-99) {
		tmp = t_5;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma((((w / t_0) * ((M * M) * h)) * (D * D)), -0.5, (((2.0 / (((D * D) * w) * h)) * (d * d)) * c0)) * t_2;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d * d) * c0)
	t_1 = Float64(t_0 / Float64(Float64(D * D) * Float64(h * w)))
	t_2 = Float64(c0 / Float64(w * 2.0))
	t_3 = Float64(Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))) + t_1) * t_2)
	t_4 = Float64(Float64(M / d) * D)
	t_5 = Float64(Float64(t_4 * t_4) * Float64(0.25 * h))
	tmp = 0.0
	if (t_3 <= -2e+125)
		tmp = Float64(Float64((Float64(d * c0) ^ 2.0) / Float64(Float64(D * h) * D)) / Float64(w * w));
	elseif (t_3 <= 2e-99)
		tmp = t_5;
	elseif (t_3 <= Inf)
		tmp = Float64(fma(Float64(Float64(Float64(w / t_0) * Float64(Float64(M * M) * h)) * Float64(D * D)), -0.5, Float64(Float64(Float64(2.0 / Float64(Float64(Float64(D * D) * w) * h)) * Float64(d * d)) * c0)) * t_2);
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+125], N[(N[(N[Power[N[(d * c0), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(D * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-99], t$95$5, If[LessEqual[t$95$3, Infinity], N[(N[(N[(N[(N[(w / t$95$0), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(N[(N[(2.0 / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], t$95$5]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -1.9999999999999998e125

   1. Initial program 91.6%

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

   4. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.5

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

   7. Applied rewrites 82.5%

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

   9. Applied rewrites 82.6%

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

   11. Applied rewrites 95.5%

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

   if -1.9999999999999998e125 < (*.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))))) < 2e-99 or +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 7.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 63.6%

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

   if 2e-99 < (*.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 85.5%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 81.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{w}{\left(d \cdot d\right) \cdot c0}\right), -0.5, \frac{\left(\left(d \cdot d\right) \cdot c0\right) \cdot 2}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{w}{\left(d \cdot d\right) \cdot c0}\right), -0.5, c0 \cdot \left(\left(d \cdot d\right) \cdot \frac{2}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{{\left(d \cdot c0\right)}^{2}}{\left(D \cdot h\right) \cdot D}}{w \cdot w}\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 2 \cdot 10^{-99}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{w}{\left(d \cdot d\right) \cdot c0} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \left(D \cdot D\right), -0.5, \left(\frac{2}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot \left(d \cdot d\right)\right) \cdot c0\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot 2}\\ t_1 := \left(d \cdot d\right) \cdot c0\\ t_2 := \frac{t\_1}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_3 := \left(\sqrt{t\_2 \cdot t\_2 - M \cdot M} + t\_2\right) \cdot t\_0\\ t_4 := \frac{M}{d} \cdot D\\ t_5 := \left(t\_4 \cdot t\_4\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\frac{t\_1 \cdot 2}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w} \cdot t\_0\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-99}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{w}{t\_1} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \left(D \cdot D\right), -0.5, \left(\frac{2}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot \left(d \cdot d\right)\right) \cdot c0\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w 2.0)))
        (t_1 (* (* d d) c0))
        (t_2 (/ t_1 (* (* D D) (* h w))))
        (t_3 (* (+ (sqrt (- (* t_2 t_2) (* M M))) t_2) t_0))
        (t_4 (* (/ M d) D))
        (t_5 (* (* t_4 t_4) (* 0.25 h))))
   (if (<= t_3 -5e+57)
     (* (/ (* t_1 2.0) (* (* (* D D) h) w)) t_0)
     (if (<= t_3 2e-99)
       t_5
       (if (<= t_3 INFINITY)
         (*
          (fma
           (* (* (/ w t_1) (* (* M M) h)) (* D D))
           -0.5
           (* (* (/ 2.0 (* (* (* D D) w) h)) (* d d)) c0))
          t_0)
         t_5)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * 2.0);
	double t_1 = (d * d) * c0;
	double t_2 = t_1 / ((D * D) * (h * w));
	double t_3 = (sqrt(((t_2 * t_2) - (M * M))) + t_2) * t_0;
	double t_4 = (M / d) * D;
	double t_5 = (t_4 * t_4) * (0.25 * h);
	double tmp;
	if (t_3 <= -5e+57) {
		tmp = ((t_1 * 2.0) / (((D * D) * h) * w)) * t_0;
	} else if (t_3 <= 2e-99) {
		tmp = t_5;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma((((w / t_1) * ((M * M) * h)) * (D * D)), -0.5, (((2.0 / (((D * D) * w) * h)) * (d * d)) * c0)) * t_0;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * 2.0))
	t_1 = Float64(Float64(d * d) * c0)
	t_2 = Float64(t_1 / Float64(Float64(D * D) * Float64(h * w)))
	t_3 = Float64(Float64(sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))) + t_2) * t_0)
	t_4 = Float64(Float64(M / d) * D)
	t_5 = Float64(Float64(t_4 * t_4) * Float64(0.25 * h))
	tmp = 0.0
	if (t_3 <= -5e+57)
		tmp = Float64(Float64(Float64(t_1 * 2.0) / Float64(Float64(Float64(D * D) * h) * w)) * t_0);
	elseif (t_3 <= 2e-99)
		tmp = t_5;
	elseif (t_3 <= Inf)
		tmp = Float64(fma(Float64(Float64(Float64(w / t_1) * Float64(Float64(M * M) * h)) * Float64(D * D)), -0.5, Float64(Float64(Float64(2.0 / Float64(Float64(Float64(D * D) * w) * h)) * Float64(d * d)) * c0)) * t_0);
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+57], N[(N[(N[(t$95$1 * 2.0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$3, 2e-99], t$95$5, If[LessEqual[t$95$3, Infinity], N[(N[(N[(N[(N[(w / t$95$1), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(N[(N[(2.0 / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], t$95$5]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -4.99999999999999972e57

   1. Initial program 89.8%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -4.99999999999999972e57 < (*.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))))) < 2e-99 or +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 6.3%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.7%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 63.8%

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

   if 2e-99 < (*.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 85.5%

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

   3. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 81.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{w}{\left(d \cdot d\right) \cdot c0}\right), -0.5, \frac{\left(\left(d \cdot d\right) \cdot c0\right) \cdot 2}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{w}{\left(d \cdot d\right) \cdot c0}\right), -0.5, c0 \cdot \left(\left(d \cdot d\right) \cdot \frac{2}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left(\left(d \cdot d\right) \cdot c0\right) \cdot 2}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w} \cdot \frac{c0}{w \cdot 2}\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 2 \cdot 10^{-99}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{w}{\left(d \cdot d\right) \cdot c0} \cdot \left(\left(M \cdot M\right) \cdot h\right)\right) \cdot \left(D \cdot D\right), -0.5, \left(\frac{2}{\left(\left(D \cdot D\right) \cdot w\right) \cdot h} \cdot \left(d \cdot d\right)\right) \cdot c0\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot 2}\\ t_1 := \frac{M}{d} \cdot D\\ t_2 := \left(t\_1 \cdot t\_1\right) \cdot \left(0.25 \cdot h\right)\\ t_3 := \left(d \cdot d\right) \cdot c0\\ t_4 := \frac{t\_3}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_5 := \left(\sqrt{t\_4 \cdot t\_4 - M \cdot M} + t\_4\right) \cdot t\_0\\ t_6 := \frac{t\_3 \cdot 2}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w} \cdot t\_0\\ \mathbf{if}\;t\_5 \leq -5 \cdot 10^{+57}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{-99}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* w 2.0)))
        (t_1 (* (/ M d) D))
        (t_2 (* (* t_1 t_1) (* 0.25 h)))
        (t_3 (* (* d d) c0))
        (t_4 (/ t_3 (* (* D D) (* h w))))
        (t_5 (* (+ (sqrt (- (* t_4 t_4) (* M M))) t_4) t_0))
        (t_6 (* (/ (* t_3 2.0) (* (* (* D D) h) w)) t_0)))
   (if (<= t_5 -5e+57)
     t_6
     (if (<= t_5 2e-99) t_2 (if (<= t_5 INFINITY) t_6 t_2)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * 2.0);
	double t_1 = (M / d) * D;
	double t_2 = (t_1 * t_1) * (0.25 * h);
	double t_3 = (d * d) * c0;
	double t_4 = t_3 / ((D * D) * (h * w));
	double t_5 = (sqrt(((t_4 * t_4) - (M * M))) + t_4) * t_0;
	double t_6 = ((t_3 * 2.0) / (((D * D) * h) * w)) * t_0;
	double tmp;
	if (t_5 <= -5e+57) {
		tmp = t_6;
	} else if (t_5 <= 2e-99) {
		tmp = t_2;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_6;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w * 2.0);
	double t_1 = (M / d) * D;
	double t_2 = (t_1 * t_1) * (0.25 * h);
	double t_3 = (d * d) * c0;
	double t_4 = t_3 / ((D * D) * (h * w));
	double t_5 = (Math.sqrt(((t_4 * t_4) - (M * M))) + t_4) * t_0;
	double t_6 = ((t_3 * 2.0) / (((D * D) * h) * w)) * t_0;
	double tmp;
	if (t_5 <= -5e+57) {
		tmp = t_6;
	} else if (t_5 <= 2e-99) {
		tmp = t_2;
	} else if (t_5 <= Double.POSITIVE_INFINITY) {
		tmp = t_6;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w * 2.0)
	t_1 = (M / d) * D
	t_2 = (t_1 * t_1) * (0.25 * h)
	t_3 = (d * d) * c0
	t_4 = t_3 / ((D * D) * (h * w))
	t_5 = (math.sqrt(((t_4 * t_4) - (M * M))) + t_4) * t_0
	t_6 = ((t_3 * 2.0) / (((D * D) * h) * w)) * t_0
	tmp = 0
	if t_5 <= -5e+57:
		tmp = t_6
	elif t_5 <= 2e-99:
		tmp = t_2
	elif t_5 <= math.inf:
		tmp = t_6
	else:
		tmp = t_2
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w * 2.0))
	t_1 = Float64(Float64(M / d) * D)
	t_2 = Float64(Float64(t_1 * t_1) * Float64(0.25 * h))
	t_3 = Float64(Float64(d * d) * c0)
	t_4 = Float64(t_3 / Float64(Float64(D * D) * Float64(h * w)))
	t_5 = Float64(Float64(sqrt(Float64(Float64(t_4 * t_4) - Float64(M * M))) + t_4) * t_0)
	t_6 = Float64(Float64(Float64(t_3 * 2.0) / Float64(Float64(Float64(D * D) * h) * w)) * t_0)
	tmp = 0.0
	if (t_5 <= -5e+57)
		tmp = t_6;
	elseif (t_5 <= 2e-99)
		tmp = t_2;
	elseif (t_5 <= Inf)
		tmp = t_6;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w * 2.0);
	t_1 = (M / d) * D;
	t_2 = (t_1 * t_1) * (0.25 * h);
	t_3 = (d * d) * c0;
	t_4 = t_3 / ((D * D) * (h * w));
	t_5 = (sqrt(((t_4 * t_4) - (M * M))) + t_4) * t_0;
	t_6 = ((t_3 * 2.0) / (((D * D) * h) * w)) * t_0;
	tmp = 0.0;
	if (t_5 <= -5e+57)
		tmp = t_6;
	elseif (t_5 <= 2e-99)
		tmp = t_2;
	elseif (t_5 <= Inf)
		tmp = t_6;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sqrt[N[(N[(t$95$4 * t$95$4), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$4), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(t$95$3 * 2.0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$5, -5e+57], t$95$6, If[LessEqual[t$95$5, 2e-99], t$95$2, If[LessEqual[t$95$5, Infinity], t$95$6, t$95$2]]]]]]]]]]

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))))) < -4.99999999999999972e57 or 2e-99 < (*.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 87.4%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if -4.99999999999999972e57 < (*.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))))) < 2e-99 or +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 6.3%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.7%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 63.8%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left(\left(d \cdot d\right) \cdot c0\right) \cdot 2}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w} \cdot \frac{c0}{w \cdot 2}\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 2 \cdot 10^{-99}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\frac{\left(\left(d \cdot d\right) \cdot c0\right) \cdot 2}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w} \cdot \frac{c0}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{D} \cdot d\\ t_1 := \frac{M}{d} \cdot D\\ t_2 := \left(t\_1 \cdot t\_1\right) \cdot \left(0.25 \cdot h\right)\\ t_3 := \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_4 := \left(\sqrt{t\_3 \cdot t\_3 - M \cdot M} + t\_3\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{\frac{t\_0}{\left(D \cdot h\right) \cdot w}}{w}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-99}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{t\_0 \cdot c0}{\left(\left(w \cdot w\right) \cdot D\right) \cdot h} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ d D) d))
        (t_1 (* (/ M d) D))
        (t_2 (* (* t_1 t_1) (* 0.25 h)))
        (t_3 (/ (* (* d d) c0) (* (* D D) (* h w))))
        (t_4 (* (+ (sqrt (- (* t_3 t_3) (* M M))) t_3) (/ c0 (* w 2.0)))))
   (if (<= t_4 -5e+57)
     (* (* c0 c0) (/ (/ t_0 (* (* D h) w)) w))
     (if (<= t_4 2e-99)
       t_2
       (if (<= t_4 INFINITY) (* (/ (* t_0 c0) (* (* (* w w) D) h)) c0) t_2)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * d;
	double t_1 = (M / d) * D;
	double t_2 = (t_1 * t_1) * (0.25 * h);
	double t_3 = ((d * d) * c0) / ((D * D) * (h * w));
	double t_4 = (sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0));
	double tmp;
	if (t_4 <= -5e+57) {
		tmp = (c0 * c0) * ((t_0 / ((D * h) * w)) / w);
	} else if (t_4 <= 2e-99) {
		tmp = t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = ((t_0 * c0) / (((w * w) * D) * h)) * c0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (d / D) * d;
	double t_1 = (M / d) * D;
	double t_2 = (t_1 * t_1) * (0.25 * h);
	double t_3 = ((d * d) * c0) / ((D * D) * (h * w));
	double t_4 = (Math.sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0));
	double tmp;
	if (t_4 <= -5e+57) {
		tmp = (c0 * c0) * ((t_0 / ((D * h) * w)) / w);
	} else if (t_4 <= 2e-99) {
		tmp = t_2;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = ((t_0 * c0) / (((w * w) * D) * h)) * c0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (d / D) * d
	t_1 = (M / d) * D
	t_2 = (t_1 * t_1) * (0.25 * h)
	t_3 = ((d * d) * c0) / ((D * D) * (h * w))
	t_4 = (math.sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0))
	tmp = 0
	if t_4 <= -5e+57:
		tmp = (c0 * c0) * ((t_0 / ((D * h) * w)) / w)
	elif t_4 <= 2e-99:
		tmp = t_2
	elif t_4 <= math.inf:
		tmp = ((t_0 * c0) / (((w * w) * D) * h)) * c0
	else:
		tmp = t_2
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(d / D) * d)
	t_1 = Float64(Float64(M / d) * D)
	t_2 = Float64(Float64(t_1 * t_1) * Float64(0.25 * h))
	t_3 = Float64(Float64(Float64(d * d) * c0) / Float64(Float64(D * D) * Float64(h * w)))
	t_4 = Float64(Float64(sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))) + t_3) * Float64(c0 / Float64(w * 2.0)))
	tmp = 0.0
	if (t_4 <= -5e+57)
		tmp = Float64(Float64(c0 * c0) * Float64(Float64(t_0 / Float64(Float64(D * h) * w)) / w));
	elseif (t_4 <= 2e-99)
		tmp = t_2;
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(Float64(t_0 * c0) / Float64(Float64(Float64(w * w) * D) * h)) * c0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (d / D) * d;
	t_1 = (M / d) * D;
	t_2 = (t_1 * t_1) * (0.25 * h);
	t_3 = ((d * d) * c0) / ((D * D) * (h * w));
	t_4 = (sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0));
	tmp = 0.0;
	if (t_4 <= -5e+57)
		tmp = (c0 * c0) * ((t_0 / ((D * h) * w)) / w);
	elseif (t_4 <= 2e-99)
		tmp = t_2;
	elseif (t_4 <= Inf)
		tmp = ((t_0 * c0) / (((w * w) * D) * h)) * c0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d / D), $MachinePrecision] * d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+57], N[(N[(c0 * c0), $MachinePrecision] * N[(N[(t$95$0 / N[(N[(D * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e-99], t$95$2, If[LessEqual[t$95$4, Infinity], N[(N[(N[(t$95$0 * c0), $MachinePrecision] / N[(N[(N[(w * w), $MachinePrecision] * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -4.99999999999999972e57

   1. Initial program 89.8%

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

   4. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   9. Applied rewrites 76.6%

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

   11. Applied rewrites 84.4%

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

   if -4.99999999999999972e57 < (*.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))))) < 2e-99 or +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 6.3%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.7%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 63.8%

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

   if 2e-99 < (*.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 85.5%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   9. Applied rewrites 72.9%

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

   11. Applied rewrites 82.2%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq -5 \cdot 10^{+57}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{\frac{\frac{d}{D} \cdot d}{\left(D \cdot h\right) \cdot w}}{w}\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 2 \cdot 10^{-99}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\frac{\left(\frac{d}{D} \cdot d\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot D\right) \cdot h} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ t_1 := \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_2 := \left(\sqrt{t\_1 \cdot t\_1 - M \cdot M} + t\_1\right) \cdot \frac{c0}{w \cdot 2}\\ t_3 := \left(t\_0 \cdot t\_0\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(\frac{d}{h} \cdot \frac{d}{D \cdot D}\right) \cdot c0}{w \cdot w} \cdot c0\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-99}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\left(\frac{d}{D} \cdot d\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot D\right) \cdot h} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ M d) D))
        (t_1 (/ (* (* d d) c0) (* (* D D) (* h w))))
        (t_2 (* (+ (sqrt (- (* t_1 t_1) (* M M))) t_1) (/ c0 (* w 2.0))))
        (t_3 (* (* t_0 t_0) (* 0.25 h))))
   (if (<= t_2 -2e+125)
     (* (/ (* (* (/ d h) (/ d (* D D))) c0) (* w w)) c0)
     (if (<= t_2 2e-99)
       t_3
       (if (<= t_2 INFINITY)
         (* (/ (* (* (/ d D) d) c0) (* (* (* w w) D) h)) c0)
         t_3)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (M / d) * D;
	double t_1 = ((d * d) * c0) / ((D * D) * (h * w));
	double t_2 = (sqrt(((t_1 * t_1) - (M * M))) + t_1) * (c0 / (w * 2.0));
	double t_3 = (t_0 * t_0) * (0.25 * h);
	double tmp;
	if (t_2 <= -2e+125) {
		tmp = ((((d / h) * (d / (D * D))) * c0) / (w * w)) * c0;
	} else if (t_2 <= 2e-99) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = ((((d / D) * d) * c0) / (((w * w) * D) * h)) * c0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (M / d) * D;
	double t_1 = ((d * d) * c0) / ((D * D) * (h * w));
	double t_2 = (Math.sqrt(((t_1 * t_1) - (M * M))) + t_1) * (c0 / (w * 2.0));
	double t_3 = (t_0 * t_0) * (0.25 * h);
	double tmp;
	if (t_2 <= -2e+125) {
		tmp = ((((d / h) * (d / (D * D))) * c0) / (w * w)) * c0;
	} else if (t_2 <= 2e-99) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = ((((d / D) * d) * c0) / (((w * w) * D) * h)) * c0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (M / d) * D
	t_1 = ((d * d) * c0) / ((D * D) * (h * w))
	t_2 = (math.sqrt(((t_1 * t_1) - (M * M))) + t_1) * (c0 / (w * 2.0))
	t_3 = (t_0 * t_0) * (0.25 * h)
	tmp = 0
	if t_2 <= -2e+125:
		tmp = ((((d / h) * (d / (D * D))) * c0) / (w * w)) * c0
	elif t_2 <= 2e-99:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = ((((d / D) * d) * c0) / (((w * w) * D) * h)) * c0
	else:
		tmp = t_3
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(M / d) * D)
	t_1 = Float64(Float64(Float64(d * d) * c0) / Float64(Float64(D * D) * Float64(h * w)))
	t_2 = Float64(Float64(sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))) + t_1) * Float64(c0 / Float64(w * 2.0)))
	t_3 = Float64(Float64(t_0 * t_0) * Float64(0.25 * h))
	tmp = 0.0
	if (t_2 <= -2e+125)
		tmp = Float64(Float64(Float64(Float64(Float64(d / h) * Float64(d / Float64(D * D))) * c0) / Float64(w * w)) * c0);
	elseif (t_2 <= 2e-99)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(Float64(Float64(d / D) * d) * c0) / Float64(Float64(Float64(w * w) * D) * h)) * c0);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (M / d) * D;
	t_1 = ((d * d) * c0) / ((D * D) * (h * w));
	t_2 = (sqrt(((t_1 * t_1) - (M * M))) + t_1) * (c0 / (w * 2.0));
	t_3 = (t_0 * t_0) * (0.25 * h);
	tmp = 0.0;
	if (t_2 <= -2e+125)
		tmp = ((((d / h) * (d / (D * D))) * c0) / (w * w)) * c0;
	elseif (t_2 <= 2e-99)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = ((((d / D) * d) * c0) / (((w * w) * D) * h)) * c0;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+125], N[(N[(N[(N[(N[(d / h), $MachinePrecision] * N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$2, 2e-99], t$95$3, If[LessEqual[t$95$2, Infinity], N[(N[(N[(N[(N[(d / D), $MachinePrecision] * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(N[(w * w), $MachinePrecision] * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -1.9999999999999998e125

   1. Initial program 91.6%

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

   4. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.5

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

   7. Applied rewrites 82.5%

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

   9. Applied rewrites 87.3%

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

   10. Taylor expanded in h around 0

       \[\leadsto \frac{\frac{{d}^{2}}{{D}^{2} \cdot h} \cdot c0}{w \cdot w} \cdot c0 \]
   11. Step-by-step derivation

   12. Applied rewrites 87.4%

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

   if -1.9999999999999998e125 < (*.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))))) < 2e-99 or +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 7.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 63.6%

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

   if 2e-99 < (*.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 85.5%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   9. Applied rewrites 72.9%

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

   11. Applied rewrites 82.2%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(\frac{d}{h} \cdot \frac{d}{D \cdot D}\right) \cdot c0}{w \cdot w} \cdot c0\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 2 \cdot 10^{-99}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\frac{\left(\frac{d}{D} \cdot d\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot D\right) \cdot h} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot \left(0.25 \cdot h\right)\\ t_2 := \left(d \cdot d\right) \cdot c0\\ t_3 := \frac{t\_2}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_4 := \left(\sqrt{t\_3 \cdot t\_3 - M \cdot M} + t\_3\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{t\_2}{\left(\left(\left(w \cdot w\right) \cdot h\right) \cdot D\right) \cdot D} \cdot c0\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\left(\frac{d}{D} \cdot d\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot D\right) \cdot h} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ M d) D))
        (t_1 (* (* t_0 t_0) (* 0.25 h)))
        (t_2 (* (* d d) c0))
        (t_3 (/ t_2 (* (* D D) (* h w))))
        (t_4 (* (+ (sqrt (- (* t_3 t_3) (* M M))) t_3) (/ c0 (* w 2.0)))))
   (if (<= t_4 -2e+125)
     (* (/ t_2 (* (* (* (* w w) h) D) D)) c0)
     (if (<= t_4 2e-99)
       t_1
       (if (<= t_4 INFINITY)
         (* (/ (* (* (/ d D) d) c0) (* (* (* w w) D) h)) c0)
         t_1)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (M / d) * D;
	double t_1 = (t_0 * t_0) * (0.25 * h);
	double t_2 = (d * d) * c0;
	double t_3 = t_2 / ((D * D) * (h * w));
	double t_4 = (sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0));
	double tmp;
	if (t_4 <= -2e+125) {
		tmp = (t_2 / ((((w * w) * h) * D) * D)) * c0;
	} else if (t_4 <= 2e-99) {
		tmp = t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = ((((d / D) * d) * c0) / (((w * w) * D) * h)) * c0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (M / d) * D;
	double t_1 = (t_0 * t_0) * (0.25 * h);
	double t_2 = (d * d) * c0;
	double t_3 = t_2 / ((D * D) * (h * w));
	double t_4 = (Math.sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0));
	double tmp;
	if (t_4 <= -2e+125) {
		tmp = (t_2 / ((((w * w) * h) * D) * D)) * c0;
	} else if (t_4 <= 2e-99) {
		tmp = t_1;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = ((((d / D) * d) * c0) / (((w * w) * D) * h)) * c0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (M / d) * D
	t_1 = (t_0 * t_0) * (0.25 * h)
	t_2 = (d * d) * c0
	t_3 = t_2 / ((D * D) * (h * w))
	t_4 = (math.sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0))
	tmp = 0
	if t_4 <= -2e+125:
		tmp = (t_2 / ((((w * w) * h) * D) * D)) * c0
	elif t_4 <= 2e-99:
		tmp = t_1
	elif t_4 <= math.inf:
		tmp = ((((d / D) * d) * c0) / (((w * w) * D) * h)) * c0
	else:
		tmp = t_1
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(M / d) * D)
	t_1 = Float64(Float64(t_0 * t_0) * Float64(0.25 * h))
	t_2 = Float64(Float64(d * d) * c0)
	t_3 = Float64(t_2 / Float64(Float64(D * D) * Float64(h * w)))
	t_4 = Float64(Float64(sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))) + t_3) * Float64(c0 / Float64(w * 2.0)))
	tmp = 0.0
	if (t_4 <= -2e+125)
		tmp = Float64(Float64(t_2 / Float64(Float64(Float64(Float64(w * w) * h) * D) * D)) * c0);
	elseif (t_4 <= 2e-99)
		tmp = t_1;
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(Float64(Float64(Float64(d / D) * d) * c0) / Float64(Float64(Float64(w * w) * D) * h)) * c0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (M / d) * D;
	t_1 = (t_0 * t_0) * (0.25 * h);
	t_2 = (d * d) * c0;
	t_3 = t_2 / ((D * D) * (h * w));
	t_4 = (sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0));
	tmp = 0.0;
	if (t_4 <= -2e+125)
		tmp = (t_2 / ((((w * w) * h) * D) * D)) * c0;
	elseif (t_4 <= 2e-99)
		tmp = t_1;
	elseif (t_4 <= Inf)
		tmp = ((((d / D) * d) * c0) / (((w * w) * D) * h)) * c0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+125], N[(N[(t$95$2 / N[(N[(N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$4, 2e-99], t$95$1, If[LessEqual[t$95$4, Infinity], N[(N[(N[(N[(N[(d / D), $MachinePrecision] * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(N[(w * w), $MachinePrecision] * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -1.9999999999999998e125

   1. Initial program 91.6%

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

   4. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.5

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

   7. Applied rewrites 82.5%

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

   9. Applied rewrites 87.3%

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

   10. Taylor expanded in c0 around 0

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

   12. Applied rewrites 87.3%

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

   if -1.9999999999999998e125 < (*.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))))) < 2e-99 or +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 7.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 63.6%

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

   if 2e-99 < (*.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 85.5%

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

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   9. Applied rewrites 72.9%

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

   11. Applied rewrites 82.2%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(\left(w \cdot w\right) \cdot h\right) \cdot D\right) \cdot D} \cdot c0\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 2 \cdot 10^{-99}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\frac{\left(\frac{d}{D} \cdot d\right) \cdot c0}{\left(\left(w \cdot w\right) \cdot D\right) \cdot h} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot \left(0.25 \cdot h\right)\\ t_2 := \left(d \cdot d\right) \cdot c0\\ t_3 := \frac{t\_2}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_4 := \left(\sqrt{t\_3 \cdot t\_3 - M \cdot M} + t\_3\right) \cdot \frac{c0}{w \cdot 2}\\ t_5 := \frac{t\_2}{\left(\left(\left(w \cdot w\right) \cdot h\right) \cdot D\right) \cdot D} \cdot c0\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ M d) D))
        (t_1 (* (* t_0 t_0) (* 0.25 h)))
        (t_2 (* (* d d) c0))
        (t_3 (/ t_2 (* (* D D) (* h w))))
        (t_4 (* (+ (sqrt (- (* t_3 t_3) (* M M))) t_3) (/ c0 (* w 2.0))))
        (t_5 (* (/ t_2 (* (* (* (* w w) h) D) D)) c0)))
   (if (<= t_4 -2e+125)
     t_5
     (if (<= t_4 2e-99) t_1 (if (<= t_4 INFINITY) t_5 t_1)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (M / d) * D;
	double t_1 = (t_0 * t_0) * (0.25 * h);
	double t_2 = (d * d) * c0;
	double t_3 = t_2 / ((D * D) * (h * w));
	double t_4 = (sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0));
	double t_5 = (t_2 / ((((w * w) * h) * D) * D)) * c0;
	double tmp;
	if (t_4 <= -2e+125) {
		tmp = t_5;
	} else if (t_4 <= 2e-99) {
		tmp = t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (M / d) * D;
	double t_1 = (t_0 * t_0) * (0.25 * h);
	double t_2 = (d * d) * c0;
	double t_3 = t_2 / ((D * D) * (h * w));
	double t_4 = (Math.sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0));
	double t_5 = (t_2 / ((((w * w) * h) * D) * D)) * c0;
	double tmp;
	if (t_4 <= -2e+125) {
		tmp = t_5;
	} else if (t_4 <= 2e-99) {
		tmp = t_1;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (M / d) * D
	t_1 = (t_0 * t_0) * (0.25 * h)
	t_2 = (d * d) * c0
	t_3 = t_2 / ((D * D) * (h * w))
	t_4 = (math.sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0))
	t_5 = (t_2 / ((((w * w) * h) * D) * D)) * c0
	tmp = 0
	if t_4 <= -2e+125:
		tmp = t_5
	elif t_4 <= 2e-99:
		tmp = t_1
	elif t_4 <= math.inf:
		tmp = t_5
	else:
		tmp = t_1
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(M / d) * D)
	t_1 = Float64(Float64(t_0 * t_0) * Float64(0.25 * h))
	t_2 = Float64(Float64(d * d) * c0)
	t_3 = Float64(t_2 / Float64(Float64(D * D) * Float64(h * w)))
	t_4 = Float64(Float64(sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))) + t_3) * Float64(c0 / Float64(w * 2.0)))
	t_5 = Float64(Float64(t_2 / Float64(Float64(Float64(Float64(w * w) * h) * D) * D)) * c0)
	tmp = 0.0
	if (t_4 <= -2e+125)
		tmp = t_5;
	elseif (t_4 <= 2e-99)
		tmp = t_1;
	elseif (t_4 <= Inf)
		tmp = t_5;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (M / d) * D;
	t_1 = (t_0 * t_0) * (0.25 * h);
	t_2 = (d * d) * c0;
	t_3 = t_2 / ((D * D) * (h * w));
	t_4 = (sqrt(((t_3 * t_3) - (M * M))) + t_3) * (c0 / (w * 2.0));
	t_5 = (t_2 / ((((w * w) * h) * D) * D)) * c0;
	tmp = 0.0;
	if (t_4 <= -2e+125)
		tmp = t_5;
	elseif (t_4 <= 2e-99)
		tmp = t_1;
	elseif (t_4 <= Inf)
		tmp = t_5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(0.25 * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 / N[(N[(N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+125], t$95$5, If[LessEqual[t$95$4, 2e-99], t$95$1, If[LessEqual[t$95$4, Infinity], t$95$5, t$95$1]]]]]]]]]

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))))) < -1.9999999999999998e125 or 2e-99 < (*.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 88.0%

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

   4. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 73.2

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

   7. Applied rewrites 73.2%

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

   9. Applied rewrites 82.6%

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

   10. Taylor expanded in c0 around 0

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

   12. Applied rewrites 82.6%

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

   if -1.9999999999999998e125 < (*.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))))) < 2e-99 or +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 7.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 63.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(\left(w \cdot w\right) \cdot h\right) \cdot D\right) \cdot D} \cdot c0\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq 2 \cdot 10^{-99}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \mathbf{elif}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\frac{\left(d \cdot d\right) \cdot c0}{\left(\left(\left(w \cdot w\right) \cdot h\right) \cdot D\right) \cdot D} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(0.25 \cdot h\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 15.7%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 60.2%

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

4. Final simplification 67.0%

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

Alternative 11: 51.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.2%

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

   4. Applied rewrites 65.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(d \cdot d\right) \cdot c0\right) \cdot \left(c0 \cdot 0.5\right), \frac{1}{\left(\left(\left(D \cdot D\right) \cdot w\right) \cdot h\right) \cdot w}, \frac{c0}{2 \cdot w} \cdot \sqrt{\mathsf{fma}\left(-M, M, {\left(\frac{\frac{h \cdot w}{c0}}{{\left(\frac{d}{D}\right)}^{2}}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 61.3

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

   7. Applied rewrites 61.3%

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

   9. Applied rewrites 69.6%

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

   10. Taylor expanded in c0 around 0

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

   12. Applied rewrites 70.9%

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

   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. 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 38.8

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

   5. Applied rewrites 38.8%

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

4. Final simplification 48.3%

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

Alternative 12: 47.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.2%

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

   3. Taylor expanded in c0 around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. mul0-rgt N/A

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

      7. metadata-eval N/A

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

      8. div0 38.8

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

   5. Applied rewrites 38.8%

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

4. Final simplification 45.5%

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

Alternative 13: 33.5% 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 24.7%

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

3. Taylor expanded in c0 around -inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
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 33.2

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

5. Applied rewrites 33.2%

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

Reproduce

herbie shell --seed 2024276 
(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))))))