2-ancestry mixing, positive discriminant

Percentage Accurate: 43.5% → 95.6%

Time: 18.8s

Alternatives: 5

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: 43.5% 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

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

FPCore

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

C

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

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((g + Math.hypot(g, h))) / Math.cbrt((a * -2.0)));
}

Julia

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(hypot(g, h) - g))) + Float64(cbrt(Float64(g + hypot(g, h))) / cbrt(Float64(a * -2.0))))
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[(g + N[Sqrt[g ^ 2 + h ^ 2], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.9%

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

   1. associate-/r* 41.9%

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

   2. metadata-eval 41.9%

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

   3. +-commutative 41.9%

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

   4. unsub-neg 41.9%

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

   5. fma-neg 41.9%

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

   6. sub-neg 41.9%

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

   7. distribute-neg-out 41.9%

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

   8. neg-mul-1 41.9%

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

   9. associate-*r* 41.9%

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

3. Simplified 41.9%

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

   1. cbrt-div 46.4%

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

   2. fma-udef 46.4%

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

   3. add-sqr-sqrt 19.7%

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

   4. hypot-def 21.5%

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

   5. add-sqr-sqrt 21.5%

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

   6. sqrt-unprod 47.0%

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

   7. sqr-neg 47.0%

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

   8. sqrt-unprod 48.5%

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

   9. add-sqr-sqrt 48.5%

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

   10. sqrt-prod 23.5%

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

   11. add-sqr-sqrt 48.5%

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

   12. div-inv 48.5%

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

   13. metadata-eval 48.5%

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

5. Applied egg-rr 48.5%

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

   1. add-exp-log 33.4%

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

   2. fma-udef 33.4%

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

   3. add-sqr-sqrt 14.7%

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

   4. hypot-def 23.0%

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

   5. add-sqr-sqrt 23.0%

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

   6. sqrt-unprod 48.9%

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

   7. sqr-neg 48.9%

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

   8. sqrt-unprod 51.4%

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

   9. add-sqr-sqrt 51.4%

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

   10. sqrt-prod 27.3%

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

   11. add-sqr-sqrt 54.1%

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

7. Applied egg-rr 54.1%

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

   1. cbrt-prod 64.6%

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

   2. add-exp-log 96.4%

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

9. Applied egg-rr 96.4%

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

10. Final simplification 96.4%

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

Alternative 2: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.9%

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

3. Simplified 41.9%

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

4. Taylor expanded in g around inf 25.3%

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

   1. distribute-rgt1-in 25.3%

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

   2. metadata-eval 25.3%

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

   3. mul0-lft 25.3%

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

   4. metadata-eval 25.3%

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

6. Simplified 25.3%

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

7. Taylor expanded in g around -inf 0.0%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{0.5 \cdot \frac{\left({\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot g}{a}}} \]
8. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{0.5 \cdot \frac{\color{blue}{g \cdot \left({\left(\sqrt{-1}\right)}^{2} - 1\right)}}{a}} \]

   2. unpow2 0.0%

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

   3. rem-square-sqrt 72.3%

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

   4. metadata-eval 72.3%

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

9. Simplified 72.3%

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

10. Taylor expanded in g around 0 72.3%

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

    1. associate-*r/ 72.3%

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

    2. neg-mul-1 72.3%

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

12. Simplified 72.3%

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

    1. frac-2neg 72.3%

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

    2. cbrt-div 96.3%

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

    3. remove-double-neg 96.3%

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

14. Applied egg-rr 96.3%

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

15. Final simplification 96.3%

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

Alternative 3: 73.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{0.5 \cdot \frac{g \cdot -2}{a}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+ (cbrt (* 0.0 (/ -0.5 a))) (cbrt (* 0.5 (/ (* g -2.0) a)))))

C

double code(double g, double h, double a) {
	return cbrt((0.0 * (-0.5 / a))) + cbrt((0.5 * ((g * -2.0) / a)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt((0.0 * (-0.5 / a))) + Math.cbrt((0.5 * ((g * -2.0) / a)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(0.0 * Float64(-0.5 / a))) + cbrt(Float64(0.5 * Float64(Float64(g * -2.0) / a))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(0.0 * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(0.5 * N[(N[(g * -2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.9%

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

3. Simplified 41.9%

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

4. Taylor expanded in g around inf 25.3%

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

   1. distribute-rgt1-in 25.3%

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

   2. metadata-eval 25.3%

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

   3. mul0-lft 25.3%

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

   4. metadata-eval 25.3%

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

6. Simplified 25.3%

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

7. Taylor expanded in g around -inf 0.0%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{0.5 \cdot \frac{\left({\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot g}{a}}} \]
8. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{0.5 \cdot \frac{\color{blue}{g \cdot \left({\left(\sqrt{-1}\right)}^{2} - 1\right)}}{a}} \]

   2. unpow2 0.0%

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

   3. rem-square-sqrt 72.3%

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

   4. metadata-eval 72.3%

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

9. Simplified 72.3%

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

10. Final simplification 72.3%

    \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{0.5 \cdot \frac{g \cdot -2}{a}} \]

Alternative 4: 73.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.9%

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

3. Simplified 41.9%

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

4. Taylor expanded in g around inf 25.3%

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

   1. distribute-rgt1-in 25.3%

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

   2. metadata-eval 25.3%

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

   3. mul0-lft 25.3%

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

   4. metadata-eval 25.3%

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

6. Simplified 25.3%

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

7. Taylor expanded in g around -inf 0.0%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{0.5 \cdot \frac{\left({\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot g}{a}}} \]
8. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{0.5 \cdot \frac{\color{blue}{g \cdot \left({\left(\sqrt{-1}\right)}^{2} - 1\right)}}{a}} \]

   2. unpow2 0.0%

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

   3. rem-square-sqrt 72.3%

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

   4. metadata-eval 72.3%

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

9. Simplified 72.3%

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

10. Taylor expanded in g around 0 72.3%

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

    1. associate-*r/ 72.3%

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

    2. neg-mul-1 72.3%

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

12. Simplified 72.3%

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

13. Final simplification 72.3%

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

Alternative 5: 1.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.9%

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

3. Simplified 41.9%

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

4. Taylor expanded in g around inf 25.3%

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

   1. distribute-rgt1-in 25.3%

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

   2. metadata-eval 25.3%

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

   3. mul0-lft 25.3%

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

   4. metadata-eval 25.3%

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

6. Simplified 25.3%

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

7. Taylor expanded in g around -inf 0.0%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{0.5 \cdot \frac{\left({\left(\sqrt{-1}\right)}^{2} - 1\right) \cdot g}{a}}} \]
8. Step-by-step derivation

   1. *-commutative 0.0%

      \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{0.5 \cdot \frac{\color{blue}{g \cdot \left({\left(\sqrt{-1}\right)}^{2} - 1\right)}}{a}} \]

   2. unpow2 0.0%

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

   3. rem-square-sqrt 72.3%

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

   4. metadata-eval 72.3%

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

9. Simplified 72.3%

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

10. Taylor expanded in g around 0 72.3%

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

    1. associate-*r/ 72.3%

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

    2. neg-mul-1 72.3%

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

12. Simplified 72.3%

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

    1. expm1-log1p-u 47.1%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{-g}{a}}\right)\right)} \]

    2. expm1-udef 26.6%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{-g}{a}}\right)} - 1\right)} \]

    3. frac-2neg 26.6%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\color{blue}{\frac{-\left(-g\right)}{-a}}}\right)} - 1\right) \]

    4. frac-2neg 26.6%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\color{blue}{\frac{-g}{a}}}\right)} - 1\right) \]

    5. add-sqr-sqrt 15.0%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{\sqrt{-g} \cdot \sqrt{-g}}}{a}}\right)} - 1\right) \]

    6. sqrt-unprod 8.8%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{\sqrt{\left(-g\right) \cdot \left(-g\right)}}}{a}}\right)} - 1\right) \]

    7. sqr-neg 8.8%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\sqrt{\color{blue}{g \cdot g}}}{a}}\right)} - 1\right) \]

    8. sqrt-unprod 0.9%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{\sqrt{g} \cdot \sqrt{g}}}{a}}\right)} - 1\right) \]

    9. add-sqr-sqrt 1.3%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\color{blue}{g}}{a}}\right)} - 1\right) \]

14. Applied egg-rr 1.3%

    \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{g}{a}}\right)} - 1\right)} \]
15. Step-by-step derivation

    1. expm1-def 1.0%

       \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\frac{g}{a}}\right)\right)} \]

    2. expm1-log1p 1.3%

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

16. Simplified 1.3%

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

17. Final simplification 1.3%

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

Reproduce

herbie shell --seed 2023214 
(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))))))))