2-ancestry mixing, positive discriminant

Percentage Accurate: 44.0% → 95.6%

Time: 20.8s

Alternatives: 8

Speedup: 2.1×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t_0 \cdot \left(\left(-g\right) + t_1\right)} + \sqrt[3]{t_0 \cdot \left(\left(-g\right) - t_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Initial Program: 44.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t_0 \cdot \left(\left(-g\right) + t_1\right)} + \sqrt[3]{t_0 \cdot \left(\left(-g\right) - t_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* 2.0 a))) (t_1 (sqrt (- (* g g) (* h h)))))
   (+ (cbrt (* t_0 (+ (- g) t_1))) (cbrt (* t_0 (- (- g) t_1))))))

C

double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = sqrt(((g * g) - (h * h)));
	return cbrt((t_0 * (-g + t_1))) + cbrt((t_0 * (-g - t_1)));
}

Java

public static double code(double g, double h, double a) {
	double t_0 = 1.0 / (2.0 * a);
	double t_1 = Math.sqrt(((g * g) - (h * h)));
	return Math.cbrt((t_0 * (-g + t_1))) + Math.cbrt((t_0 * (-g - t_1)));
}

Julia

function code(g, h, a)
	t_0 = Float64(1.0 / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(cbrt(Float64(t_0 * Float64(Float64(-g) + t_1))) + cbrt(Float64(t_0 * Float64(Float64(-g) - t_1))))
end

Wolfram

code[g_, h_, a_] := Block[{t$95$0 = N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[((-g) + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[((-g) - t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 95.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\mathsf{hypot}\left(g, h\right) - g} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (* (cbrt (/ 0.5 a)) (cbrt (- (hypot g h) g)))
  (* (cbrt (/ -0.5 a)) (cbrt (+ g (hypot g h))))))

C

double code(double g, double h, double a) {
	return (cbrt((0.5 / a)) * cbrt((hypot(g, h) - g))) + (cbrt((-0.5 / a)) * cbrt((g + hypot(g, h))));
}

Java

public static double code(double g, double h, double a) {
	return (Math.cbrt((0.5 / a)) * Math.cbrt((Math.hypot(g, h) - g))) + (Math.cbrt((-0.5 / a)) * Math.cbrt((g + Math.hypot(g, h))));
}

Julia

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(hypot(g, h) - g))) + Float64(cbrt(Float64(-0.5 / a)) * cbrt(Float64(g + hypot(g, h)))))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Sqrt[g ^ 2 + h ^ 2], $MachinePrecision] - g), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g + N[Sqrt[g ^ 2 + h ^ 2], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Step-by-step derivation

   1. associate-/r* 49.8%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   2. metadata-eval 49.8%

      \[\leadsto \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   3. +-commutative 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   4. unsub-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   5. fma-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(g, g, -h \cdot h\right)}} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. sub-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-g\right) + \left(-\sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   7. distribute-neg-out 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-\left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   8. neg-mul-1 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-1 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   9. associate-*r* 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\color{blue}{\left(\frac{1}{2 \cdot a} \cdot -1\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}} \]

3. Simplified 49.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 43.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)}\right)\right)} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

   2. expm1-udef 33.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)}\right)} - 1\right)} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

5. Applied egg-rr 27.7%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}\right)} - 1\right)} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]
6. Step-by-step derivation

   1. expm1-def 27.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}\right)\right)} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

   2. expm1-log1p 27.3%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

7. Simplified 27.3%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]
8. Step-by-step derivation

   1. div-inv 27.3%

      \[\leadsto \sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)} + \sqrt[3]{\color{blue}{\left(g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}\right) \cdot \frac{1}{\frac{a}{-0.5}}}} \]

   2. clear-num 27.3%

      \[\leadsto \sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)} + \sqrt[3]{\left(g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}\right) \cdot \color{blue}{\frac{-0.5}{a}}} \]

   3. cbrt-prod 30.0%

      \[\leadsto \sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)} + \color{blue}{\sqrt[3]{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}} \cdot \sqrt[3]{\frac{-0.5}{a}}} \]

9. Applied egg-rr 51.8%

   \[\leadsto \sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)} + \color{blue}{\sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \cdot \sqrt[3]{\frac{-0.5}{a}}} \]
10. Step-by-step derivation

    1. *-commutative 51.8%

       \[\leadsto \sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)} + \color{blue}{\sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)}} \]

11. Simplified 51.8%

    \[\leadsto \sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)} + \color{blue}{\sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)}} \]
12. Step-by-step derivation

    1. add-sqr-sqrt 51.5%

       \[\leadsto \color{blue}{\sqrt{\sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}} \cdot \sqrt{\sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

    2. sqrt-unprod 69.7%

       \[\leadsto \color{blue}{\sqrt{\sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)} \cdot \sqrt[3]{\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

    3. cbrt-unprod 61.6%

       \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)\right) \cdot \left(\frac{-0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)\right)}}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

    4. swap-sqr 39.6%

       \[\leadsto \sqrt{\sqrt[3]{\color{blue}{\left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right) \cdot \left(\left(\mathsf{hypot}\left(g, h\right) - g\right) \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)\right)}}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

    5. frac-times 39.6%

       \[\leadsto \sqrt{\sqrt[3]{\color{blue}{\frac{-0.5 \cdot -0.5}{a \cdot a}} \cdot \left(\left(\mathsf{hypot}\left(g, h\right) - g\right) \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)\right)}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

    6. metadata-eval 39.6%

       \[\leadsto \sqrt{\sqrt[3]{\frac{\color{blue}{0.25}}{a \cdot a} \cdot \left(\left(\mathsf{hypot}\left(g, h\right) - g\right) \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)\right)}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

    7. metadata-eval 39.6%

       \[\leadsto \sqrt{\sqrt[3]{\frac{\color{blue}{0.5 \cdot 0.5}}{a \cdot a} \cdot \left(\left(\mathsf{hypot}\left(g, h\right) - g\right) \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)\right)}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

    8. frac-times 39.6%

       \[\leadsto \sqrt{\sqrt[3]{\color{blue}{\left(\frac{0.5}{a} \cdot \frac{0.5}{a}\right)} \cdot \left(\left(\mathsf{hypot}\left(g, h\right) - g\right) \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)\right)}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

    9. swap-sqr 61.6%

       \[\leadsto \sqrt{\sqrt[3]{\color{blue}{\left(\frac{0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)\right) \cdot \left(\frac{0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)\right)}}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

    10. expm1-log1p-u 61.2%

        \[\leadsto \sqrt{\sqrt[3]{\left(\frac{0.5}{a} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(g, h\right) - g\right)\right)}\right) \cdot \left(\frac{0.5}{a} \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)\right)}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

    11. expm1-log1p-u 61.1%

        \[\leadsto \sqrt{\sqrt[3]{\left(\frac{0.5}{a} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(g, h\right) - g\right)\right)\right) \cdot \left(\frac{0.5}{a} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(g, h\right) - g\right)\right)}\right)}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

    12. cbrt-unprod 69.0%

        \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(g, h\right) - g\right)\right)} \cdot \sqrt[3]{\frac{0.5}{a} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(g, h\right) - g\right)\right)}}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

13. Applied egg-rr 96.4%

    \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\mathsf{hypot}\left(g, h\right) - g}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

14. Final simplification 96.4%

    \[\leadsto \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\mathsf{hypot}\left(g, h\right) - g} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]

Alternative 2: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0.5} \cdot \frac{\sqrt[3]{g \cdot -2}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g - g}{\frac{a}{-0.5}}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (* (cbrt 0.5) (/ (cbrt (* g -2.0)) (cbrt a)))
  (cbrt (/ (- g g) (/ a -0.5)))))

C

double code(double g, double h, double a) {
	return (cbrt(0.5) * (cbrt((g * -2.0)) / cbrt(a))) + cbrt(((g - g) / (a / -0.5)));
}

Java

public static double code(double g, double h, double a) {
	return (Math.cbrt(0.5) * (Math.cbrt((g * -2.0)) / Math.cbrt(a))) + Math.cbrt(((g - g) / (a / -0.5)));
}

Julia

function code(g, h, a)
	return Float64(Float64(cbrt(0.5) * Float64(cbrt(Float64(g * -2.0)) / cbrt(a))) + cbrt(Float64(Float64(g - g) / Float64(a / -0.5))))
end

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[0.5, 1/3], $MachinePrecision] * N[(N[Power[N[(g * -2.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(g - g), $MachinePrecision] / N[(a / -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Step-by-step derivation

   1. associate-/r* 49.8%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   2. metadata-eval 49.8%

      \[\leadsto \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   3. +-commutative 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   4. unsub-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   5. fma-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(g, g, -h \cdot h\right)}} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. sub-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-g\right) + \left(-\sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   7. distribute-neg-out 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-\left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   8. neg-mul-1 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-1 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   9. associate-*r* 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\color{blue}{\left(\frac{1}{2 \cdot a} \cdot -1\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}} \]

3. Simplified 49.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}}} \]
4. Step-by-step derivation

   1. associate-*l/ 49.8%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)}{a}}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

   2. cbrt-div 54.4%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)}}{\sqrt[3]{a}}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

5. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}}{\sqrt[3]{a}}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

6. Taylor expanded in g around -inf 37.5%

   \[\leadsto \frac{\sqrt[3]{0.5 \cdot \color{blue}{\left(-2 \cdot g\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]
7. Step-by-step derivation

   1. *-commutative 37.5%

      \[\leadsto \frac{\sqrt[3]{0.5 \cdot \color{blue}{\left(g \cdot -2\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

8. Simplified 37.5%

   \[\leadsto \frac{\sqrt[3]{0.5 \cdot \color{blue}{\left(g \cdot -2\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

9. Taylor expanded in g around -inf 96.4%

   \[\leadsto \frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \color{blue}{-1 \cdot g}}{\frac{a}{-0.5}}} \]
10. Step-by-step derivation

    1. neg-mul-1 96.4%

       \[\leadsto \frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \color{blue}{\left(-g\right)}}{\frac{a}{-0.5}}} \]

11. Simplified 96.4%

    \[\leadsto \frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \color{blue}{\left(-g\right)}}{\frac{a}{-0.5}}} \]
12. Step-by-step derivation

    1. cbrt-prod 96.4%

       \[\leadsto \frac{\color{blue}{\sqrt[3]{0.5} \cdot \sqrt[3]{g \cdot -2}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    2. *-un-lft-identity 96.4%

       \[\leadsto \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{g \cdot -2}}{\color{blue}{1 \cdot \sqrt[3]{a}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    3. times-frac 96.4%

       \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5}}{1} \cdot \frac{\sqrt[3]{g \cdot -2}}{\sqrt[3]{a}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

13. Applied egg-rr 96.4%

    \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5}}{1} \cdot \frac{\sqrt[3]{g \cdot -2}}{\sqrt[3]{a}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]
14. Step-by-step derivation

    1. /-rgt-identity 96.4%

       \[\leadsto \color{blue}{\sqrt[3]{0.5}} \cdot \frac{\sqrt[3]{g \cdot -2}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

15. Simplified 96.4%

    \[\leadsto \color{blue}{\sqrt[3]{0.5} \cdot \frac{\sqrt[3]{g \cdot -2}}{\sqrt[3]{a}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

16. Final simplification 96.4%

    \[\leadsto \sqrt[3]{0.5} \cdot \frac{\sqrt[3]{g \cdot -2}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g - g}{\frac{a}{-0.5}}} \]

Alternative 3: 95.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g - g}{\frac{a}{-0.5}}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (/ (- g g) (/ a -0.5))) (/ (cbrt (- g)) (cbrt a))))

C

double code(double g, double h, double a) {
	return cbrt(((g - g) / (a / -0.5))) + (cbrt(-g) / cbrt(a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((g - g) / (a / -0.5))) + (Math.cbrt(-g) / Math.cbrt(a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(g - g) / Float64(a / -0.5))) + Float64(cbrt(Float64(-g)) / cbrt(a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(g - g), $MachinePrecision] / N[(a / -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[(-g), 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Step-by-step derivation

   1. associate-/r* 49.8%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   2. metadata-eval 49.8%

      \[\leadsto \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   3. +-commutative 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   4. unsub-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   5. fma-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(g, g, -h \cdot h\right)}} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. sub-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-g\right) + \left(-\sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   7. distribute-neg-out 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-\left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   8. neg-mul-1 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-1 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   9. associate-*r* 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\color{blue}{\left(\frac{1}{2 \cdot a} \cdot -1\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}} \]

3. Simplified 49.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}}} \]
4. Step-by-step derivation

   1. associate-*l/ 49.8%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)}{a}}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

   2. cbrt-div 54.4%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)}}{\sqrt[3]{a}}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

5. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}}{\sqrt[3]{a}}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

6. Taylor expanded in g around -inf 37.5%

   \[\leadsto \frac{\sqrt[3]{0.5 \cdot \color{blue}{\left(-2 \cdot g\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]
7. Step-by-step derivation

   1. *-commutative 37.5%

      \[\leadsto \frac{\sqrt[3]{0.5 \cdot \color{blue}{\left(g \cdot -2\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

8. Simplified 37.5%

   \[\leadsto \frac{\sqrt[3]{0.5 \cdot \color{blue}{\left(g \cdot -2\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

9. Taylor expanded in g around -inf 96.4%

   \[\leadsto \frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \color{blue}{-1 \cdot g}}{\frac{a}{-0.5}}} \]
10. Step-by-step derivation

    1. neg-mul-1 96.4%

       \[\leadsto \frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \color{blue}{\left(-g\right)}}{\frac{a}{-0.5}}} \]

11. Simplified 96.4%

    \[\leadsto \frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \color{blue}{\left(-g\right)}}{\frac{a}{-0.5}}} \]
12. Step-by-step derivation

    1. expm1-log1p-u 60.4%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}\right)\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    2. expm1-udef 38.8%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}\right)} - 1}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

13. Applied egg-rr 38.8%

    \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}\right)} - 1}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]
14. Step-by-step derivation

    1. expm1-def 60.4%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}\right)\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    2. expm1-log1p 96.4%

       \[\leadsto \frac{\color{blue}{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    3. *-commutative 96.4%

       \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(g \cdot -2\right) \cdot 0.5}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    4. associate-*l* 96.4%

       \[\leadsto \frac{\sqrt[3]{\color{blue}{g \cdot \left(-2 \cdot 0.5\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    5. metadata-eval 96.4%

       \[\leadsto \frac{\sqrt[3]{g \cdot \color{blue}{-1}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    6. *-commutative 96.4%

       \[\leadsto \frac{\sqrt[3]{\color{blue}{-1 \cdot g}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    7. neg-mul-1 96.4%

       \[\leadsto \frac{\sqrt[3]{\color{blue}{-g}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

15. Simplified 96.4%

    \[\leadsto \frac{\color{blue}{\sqrt[3]{-g}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

16. Final simplification 96.4%

    \[\leadsto \sqrt[3]{\frac{g - g}{\frac{a}{-0.5}}} + \frac{\sqrt[3]{-g}}{\sqrt[3]{a}} \]

Alternative 4: 74.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\frac{0.5 \cdot \frac{h}{\frac{g}{h}}}{\frac{a}{-0.5}}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (/ 0.5 a) (* g -2.0)))
  (cbrt (/ (* 0.5 (/ h (/ g h))) (/ a -0.5)))))

C

double code(double g, double h, double a) {
	return cbrt(((0.5 / a) * (g * -2.0))) + cbrt(((0.5 * (h / (g / h))) / (a / -0.5)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((0.5 / a) * (g * -2.0))) + Math.cbrt(((0.5 * (h / (g / h))) / (a / -0.5)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g * -2.0))) + cbrt(Float64(Float64(0.5 * Float64(h / Float64(g / h))) / Float64(a / -0.5))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 * N[(h / N[(g / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a / -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Step-by-step derivation

   1. associate-/r* 49.8%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   2. metadata-eval 49.8%

      \[\leadsto \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   3. +-commutative 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   4. unsub-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   5. fma-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(g, g, -h \cdot h\right)}} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. sub-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-g\right) + \left(-\sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   7. distribute-neg-out 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-\left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   8. neg-mul-1 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-1 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   9. associate-*r* 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\color{blue}{\left(\frac{1}{2 \cdot a} \cdot -1\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}} \]

3. Simplified 49.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}}} \]

4. Taylor expanded in g around -inf 30.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot g\right)}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]
5. Step-by-step derivation

   1. *-commutative 30.1%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

6. Simplified 30.1%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

7. Taylor expanded in g around -inf 72.9%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\frac{\color{blue}{0.5 \cdot \frac{{h}^{2}}{g}}}{\frac{a}{-0.5}}} \]
8. Step-by-step derivation

   1. unpow2 72.9%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\frac{0.5 \cdot \frac{\color{blue}{h \cdot h}}{g}}{\frac{a}{-0.5}}} \]

   2. associate-/l* 77.0%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\frac{0.5 \cdot \color{blue}{\frac{h}{\frac{g}{h}}}}{\frac{a}{-0.5}}} \]

9. Simplified 77.0%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\frac{\color{blue}{0.5 \cdot \frac{h}{\frac{g}{h}}}}{\frac{a}{-0.5}}} \]

10. Final simplification 77.0%

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g \cdot -2\right)} + \sqrt[3]{\frac{0.5 \cdot \frac{h}{\frac{g}{h}}}{\frac{a}{-0.5}}} \]

Alternative 5: 74.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{g - g}{\frac{a}{-0.5}}} + \frac{1}{\sqrt[3]{\frac{a}{-g}}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (/ (- g g) (/ a -0.5))) (/ 1.0 (cbrt (/ a (- g))))))

C

double code(double g, double h, double a) {
	return cbrt(((g - g) / (a / -0.5))) + (1.0 / cbrt((a / -g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((g - g) / (a / -0.5))) + (1.0 / Math.cbrt((a / -g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(g - g) / Float64(a / -0.5))) + Float64(1.0 / cbrt(Float64(a / Float64(-g)))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(g - g), $MachinePrecision] / N[(a / -0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(1.0 / N[Power[N[(a / (-g)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Step-by-step derivation

   1. associate-/r* 49.8%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   2. metadata-eval 49.8%

      \[\leadsto \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   3. +-commutative 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   4. unsub-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   5. fma-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(g, g, -h \cdot h\right)}} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

   6. sub-neg 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-g\right) + \left(-\sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   7. distribute-neg-out 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-\left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   8. neg-mul-1 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-1 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]

   9. associate-*r* 49.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\color{blue}{\left(\frac{1}{2 \cdot a} \cdot -1\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}} \]

3. Simplified 49.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}}} \]
4. Step-by-step derivation

   1. associate-*l/ 49.8%

      \[\leadsto \sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)}{a}}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

   2. cbrt-div 54.4%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)}}{\sqrt[3]{a}}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

5. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}}{\sqrt[3]{a}}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

6. Taylor expanded in g around -inf 37.5%

   \[\leadsto \frac{\sqrt[3]{0.5 \cdot \color{blue}{\left(-2 \cdot g\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]
7. Step-by-step derivation

   1. *-commutative 37.5%

      \[\leadsto \frac{\sqrt[3]{0.5 \cdot \color{blue}{\left(g \cdot -2\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

8. Simplified 37.5%

   \[\leadsto \frac{\sqrt[3]{0.5 \cdot \color{blue}{\left(g \cdot -2\right)}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)}}{\frac{a}{-0.5}}} \]

9. Taylor expanded in g around -inf 96.4%

   \[\leadsto \frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \color{blue}{-1 \cdot g}}{\frac{a}{-0.5}}} \]
10. Step-by-step derivation

    1. neg-mul-1 96.4%

       \[\leadsto \frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \color{blue}{\left(-g\right)}}{\frac{a}{-0.5}}} \]

11. Simplified 96.4%

    \[\leadsto \frac{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{g + \color{blue}{\left(-g\right)}}{\frac{a}{-0.5}}} \]
12. Step-by-step derivation

    1. clear-num 96.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    2. inv-pow 96.3%

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{a}}{\sqrt[3]{0.5 \cdot \left(g \cdot -2\right)}}\right)}^{-1}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    3. cbrt-undiv 76.5%

       \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{a}{0.5 \cdot \left(g \cdot -2\right)}}\right)}}^{-1} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

13. Applied egg-rr 76.5%

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{a}{0.5 \cdot \left(g \cdot -2\right)}}\right)}^{-1}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]
14. Step-by-step derivation

    1. unpow-1 76.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{a}{0.5 \cdot \left(g \cdot -2\right)}}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    2. *-commutative 76.5%

       \[\leadsto \frac{1}{\sqrt[3]{\frac{a}{\color{blue}{\left(g \cdot -2\right) \cdot 0.5}}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    3. associate-*l* 76.5%

       \[\leadsto \frac{1}{\sqrt[3]{\frac{a}{\color{blue}{g \cdot \left(-2 \cdot 0.5\right)}}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    4. metadata-eval 76.5%

       \[\leadsto \frac{1}{\sqrt[3]{\frac{a}{g \cdot \color{blue}{-1}}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    5. *-commutative 76.5%

       \[\leadsto \frac{1}{\sqrt[3]{\frac{a}{\color{blue}{-1 \cdot g}}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

    6. neg-mul-1 76.5%

       \[\leadsto \frac{1}{\sqrt[3]{\frac{a}{\color{blue}{-g}}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

15. Simplified 76.5%

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\frac{a}{-g}}}} + \sqrt[3]{\frac{g + \left(-g\right)}{\frac{a}{-0.5}}} \]

16. Final simplification 76.5%

    \[\leadsto \sqrt[3]{\frac{g - g}{\frac{a}{-0.5}}} + \frac{1}{\sqrt[3]{\frac{a}{-g}}} \]

Alternative 6: 73.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{-0.5}{a} \cdot 0} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* (/ -0.5 a) 0.0)) (cbrt (* (/ -0.5 a) (+ g g)))))

C

double code(double g, double h, double a) {
	return cbrt(((-0.5 / a) * 0.0)) + cbrt(((-0.5 / a) * (g + g)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((-0.5 / a) * 0.0)) + Math.cbrt(((-0.5 / a) * (g + g)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(-0.5 / a) * 0.0)) + cbrt(Float64(Float64(-0.5 / a) * Float64(g + g))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(-0.5 / a), $MachinePrecision] * 0.0), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(-0.5 / a), $MachinePrecision] * N[(g + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Step-by-step derivation

3. Simplified 49.8%

   \[\leadsto \color{blue}{\sqrt[3]{\left(g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}}} \]

4. Taylor expanded in g around inf 28.4%

   \[\leadsto \sqrt[3]{\color{blue}{\left(-0.5 \cdot \left(h + -1 \cdot h\right)\right)} \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]
5. Step-by-step derivation

   1. distribute-rgt1-in 28.4%

      \[\leadsto \sqrt[3]{\left(-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot h\right)}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

   2. metadata-eval 28.4%

      \[\leadsto \sqrt[3]{\left(-0.5 \cdot \left(\color{blue}{0} \cdot h\right)\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

   3. mul0-lft 28.4%

      \[\leadsto \sqrt[3]{\left(-0.5 \cdot \color{blue}{0}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. metadata-eval 28.4%

      \[\leadsto \sqrt[3]{\color{blue}{0} \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

6. Simplified 28.4%

   \[\leadsto \sqrt[3]{\color{blue}{0} \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

7. Taylor expanded in g around inf 75.3%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]

8. Final simplification 75.3%

   \[\leadsto \sqrt[3]{\frac{-0.5}{a} \cdot 0} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)} \]

Alternative 7: 73.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{-0.5}{a} \cdot 0} + \sqrt[3]{\frac{-g}{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* (/ -0.5 a) 0.0)) (cbrt (/ (- g) a))))

C

double code(double g, double h, double a) {
	return cbrt(((-0.5 / a) * 0.0)) + cbrt((-g / a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((-0.5 / a) * 0.0)) + Math.cbrt((-g / a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(-0.5 / a) * 0.0)) + cbrt(Float64(Float64(-g) / a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(-0.5 / a), $MachinePrecision] * 0.0), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[((-g) / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Step-by-step derivation

3. Simplified 49.8%

   \[\leadsto \color{blue}{\sqrt[3]{\left(g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}}} \]

4. Taylor expanded in g around inf 28.4%

   \[\leadsto \sqrt[3]{\color{blue}{\left(-0.5 \cdot \left(h + -1 \cdot h\right)\right)} \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]
5. Step-by-step derivation

   1. distribute-rgt1-in 28.4%

      \[\leadsto \sqrt[3]{\left(-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot h\right)}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

   2. metadata-eval 28.4%

      \[\leadsto \sqrt[3]{\left(-0.5 \cdot \left(\color{blue}{0} \cdot h\right)\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

   3. mul0-lft 28.4%

      \[\leadsto \sqrt[3]{\left(-0.5 \cdot \color{blue}{0}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. metadata-eval 28.4%

      \[\leadsto \sqrt[3]{\color{blue}{0} \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

6. Simplified 28.4%

   \[\leadsto \sqrt[3]{\color{blue}{0} \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

7. Taylor expanded in g around inf 75.3%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]
8. Step-by-step derivation

   1. associate-*r/ 75.3%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]

   2. neg-mul-1 75.3%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

9. Simplified 75.3%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]

10. Final simplification 75.3%

    \[\leadsto \sqrt[3]{\frac{-0.5}{a} \cdot 0} + \sqrt[3]{\frac{-g}{a}} \]

Alternative 8: 3.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{-0.5}{a} \cdot 0} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (cbrt (* (/ -0.5 a) 0.0)))

C

double code(double g, double h, double a) {
	return cbrt(((-0.5 / a) * 0.0));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((-0.5 / a) * 0.0));
}

Julia

function code(g, h, a)
	return cbrt(Float64(Float64(-0.5 / a) * 0.0))
end

Wolfram

code[g_, h_, a_] := N[Power[N[(N[(-0.5 / a), $MachinePrecision] * 0.0), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 49.8%

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
2. Step-by-step derivation

3. Simplified 49.8%

   \[\leadsto \color{blue}{\sqrt[3]{\left(g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}}} \]

4. Taylor expanded in g around inf 28.4%

   \[\leadsto \sqrt[3]{\color{blue}{\left(-0.5 \cdot \left(h + -1 \cdot h\right)\right)} \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]
5. Step-by-step derivation

   1. distribute-rgt1-in 28.4%

      \[\leadsto \sqrt[3]{\left(-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot h\right)}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

   2. metadata-eval 28.4%

      \[\leadsto \sqrt[3]{\left(-0.5 \cdot \left(\color{blue}{0} \cdot h\right)\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

   3. mul0-lft 28.4%

      \[\leadsto \sqrt[3]{\left(-0.5 \cdot \color{blue}{0}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

   4. metadata-eval 28.4%

      \[\leadsto \sqrt[3]{\color{blue}{0} \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

6. Simplified 28.4%

   \[\leadsto \sqrt[3]{\color{blue}{0} \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

7. Taylor expanded in g around inf 75.3%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \color{blue}{g}\right) \cdot \frac{-0.5}{a}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 34.8%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{\sqrt{\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}} \cdot \sqrt{\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}}} \]

   2. sqrt-unprod 35.5%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{\sqrt{\sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}} \cdot \sqrt[3]{\left(g + g\right) \cdot \frac{-0.5}{a}}}} \]

   3. cbrt-unprod 17.9%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\color{blue}{\sqrt[3]{\left(\left(g + g\right) \cdot \frac{-0.5}{a}\right) \cdot \left(\left(g + g\right) \cdot \frac{-0.5}{a}\right)}}} \]

   4. swap-sqr 9.2%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\color{blue}{\left(\left(g + g\right) \cdot \left(g + g\right)\right) \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}}} \]

   5. count-2 9.2%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\left(\color{blue}{\left(2 \cdot g\right)} \cdot \left(g + g\right)\right) \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}} \]

   6. count-2 9.2%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\left(\left(2 \cdot g\right) \cdot \color{blue}{\left(2 \cdot g\right)}\right) \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}} \]

   7. swap-sqr 9.2%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\color{blue}{\left(\left(2 \cdot 2\right) \cdot \left(g \cdot g\right)\right)} \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}} \]

   8. metadata-eval 9.2%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\left(\color{blue}{4} \cdot \left(g \cdot g\right)\right) \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}} \]

   9. metadata-eval 9.2%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\left(\color{blue}{\left(-2 \cdot -2\right)} \cdot \left(g \cdot g\right)\right) \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}} \]

   10. swap-sqr 9.2%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\color{blue}{\left(\left(-2 \cdot g\right) \cdot \left(-2 \cdot g\right)\right)} \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}} \]

   11. *-commutative 9.2%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\left(\color{blue}{\left(g \cdot -2\right)} \cdot \left(-2 \cdot g\right)\right) \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}} \]

   12. *-commutative 9.2%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\left(\left(g \cdot -2\right) \cdot \color{blue}{\left(g \cdot -2\right)}\right) \cdot \left(\frac{-0.5}{a} \cdot \frac{-0.5}{a}\right)}} \]

   13. frac-times 9.2%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right) \cdot \color{blue}{\frac{-0.5 \cdot -0.5}{a \cdot a}}}} \]

   14. metadata-eval 9.2%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right) \cdot \frac{\color{blue}{0.25}}{a \cdot a}}} \]

   15. metadata-eval 9.2%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right) \cdot \frac{\color{blue}{0.5 \cdot 0.5}}{a \cdot a}}} \]

   16. frac-times 9.2%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\left(\left(g \cdot -2\right) \cdot \left(g \cdot -2\right)\right) \cdot \color{blue}{\left(\frac{0.5}{a} \cdot \frac{0.5}{a}\right)}}} \]

   17. swap-sqr 17.9%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\color{blue}{\left(\left(g \cdot -2\right) \cdot \frac{0.5}{a}\right) \cdot \left(\left(g \cdot -2\right) \cdot \frac{0.5}{a}\right)}}} \]

   18. *-commutative 17.9%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\color{blue}{\left(\frac{0.5}{a} \cdot \left(g \cdot -2\right)\right)} \cdot \left(\left(g \cdot -2\right) \cdot \frac{0.5}{a}\right)}} \]

   19. associate-*l/ 17.9%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}} \cdot \left(\left(g \cdot -2\right) \cdot \frac{0.5}{a}\right)}} \]

   20. *-commutative 17.9%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\frac{0.5 \cdot \left(g \cdot -2\right)}{a} \cdot \color{blue}{\left(\frac{0.5}{a} \cdot \left(g \cdot -2\right)\right)}}} \]

   21. associate-*l/ 17.9%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\frac{0.5 \cdot \left(g \cdot -2\right)}{a} \cdot \color{blue}{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}}}} \]

   22. frac-times 9.2%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt{\sqrt[3]{\color{blue}{\frac{\left(0.5 \cdot \left(g \cdot -2\right)\right) \cdot \left(0.5 \cdot \left(g \cdot -2\right)\right)}{a \cdot a}}}} \]

9. Applied egg-rr 3.0%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{0} \]

10. Final simplification 3.0%

    \[\leadsto \sqrt[3]{\frac{-0.5}{a} \cdot 0} \]

Reproduce

herbie shell --seed 2023178 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))