2-ancestry mixing, positive discriminant

Percentage Accurate: 44.0% → 95.6%

Time: 20.3s

Alternatives: 7

Speedup: 2.0×

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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[g_, h_, a_] := N[(N[(N[Power[N[(0.5 * N[(N[Sqrt[g ^ 2 + h ^ 2], $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Power[N[Sqrt[N[(g + N[Sqrt[g ^ 2 + h ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(a * -2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.4%

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

   1. associate-/r* 49.4%

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

   2. metadata-eval 49.4%

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

   3. +-commutative 49.4%

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

   4. unsub-neg 49.4%

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

   5. fma-neg 49.4%

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

   6. sub-neg 49.4%

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

   7. distribute-neg-out 49.4%

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

   8. neg-mul-1 49.4%

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

   9. associate-*r* 49.4%

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

3. Simplified 49.4%

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

   1. associate-*l/ 49.4%

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

   2. cbrt-div 52.8%

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

5. Applied egg-rr 53.7%

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

   1. cbrt-div 54.4%

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

   2. fma-udef 54.4%

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

   3. add-sqr-sqrt 30.3%

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

   4. hypot-def 44.6%

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

   5. add-sqr-sqrt 44.6%

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

   6. sqrt-unprod 83.2%

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

   7. sqr-neg 83.2%

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

   8. sqrt-unprod 89.7%

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

   9. add-sqr-sqrt 89.7%

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

   10. sqrt-prod 44.6%

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

   11. add-sqr-sqrt 94.8%

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

   12. div-inv 94.8%

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

   13. metadata-eval 94.8%

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

7. Applied egg-rr 94.8%

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

   1. add-sqr-sqrt 94.8%

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

   2. pow2 94.8%

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

9. Applied egg-rr 94.8%

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

10. Final simplification 94.8%

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

Alternative 2: 95.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

   1. associate-/r* 49.4%

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

   2. metadata-eval 49.4%

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

   3. +-commutative 49.4%

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

   4. unsub-neg 49.4%

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

   5. fma-neg 49.4%

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

   6. sub-neg 49.4%

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

   7. distribute-neg-out 49.4%

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

   8. neg-mul-1 49.4%

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

   9. associate-*r* 49.4%

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

3. Simplified 49.4%

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

   1. associate-*l/ 49.4%

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

   2. cbrt-div 52.8%

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

5. Applied egg-rr 53.7%

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

   1. div-inv 53.6%

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

   2. clear-num 53.6%

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

   3. cbrt-prod 54.4%

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

7. Applied egg-rr 94.8%

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

   1. *-commutative 94.8%

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

9. Simplified 94.8%

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

10. Final simplification 94.8%

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

Alternative 3: 95.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double g, double h, double a) {
	return (cbrt((0.5 * (hypot(g, h) - g))) / cbrt(a)) + (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 * (Math.hypot(g, h) - g))) / Math.cbrt(a)) + (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 * Float64(hypot(g, h) - g))) / cbrt(a)) + 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 * N[(N[Sqrt[g ^ 2 + h ^ 2], $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 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 49.4%

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

   1. associate-/r* 49.4%

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

   2. metadata-eval 49.4%

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

   3. +-commutative 49.4%

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

   4. unsub-neg 49.4%

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

   5. fma-neg 49.4%

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

   6. sub-neg 49.4%

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

   7. distribute-neg-out 49.4%

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

   8. neg-mul-1 49.4%

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

   9. associate-*r* 49.4%

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

3. Simplified 49.4%

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

   1. associate-*l/ 49.4%

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

   2. cbrt-div 52.8%

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

5. Applied egg-rr 53.7%

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

   1. cbrt-div 54.4%

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

   2. fma-udef 54.4%

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

   3. add-sqr-sqrt 30.3%

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

   4. hypot-def 44.6%

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

   5. add-sqr-sqrt 44.6%

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

   6. sqrt-unprod 83.2%

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

   7. sqr-neg 83.2%

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

   8. sqrt-unprod 89.7%

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

   9. add-sqr-sqrt 89.7%

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

   10. sqrt-prod 44.6%

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

   11. add-sqr-sqrt 94.8%

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

   12. div-inv 94.8%

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

   13. metadata-eval 94.8%

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

7. Applied egg-rr 94.8%

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

8. Final simplification 94.8%

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

Alternative 4: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

3. Simplified 49.4%

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

4. Taylor expanded in g around inf 21.5%

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

5. Taylor expanded in g around inf 76.9%

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

   1. associate-*r/ 76.9%

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

   2. neg-mul-1 76.9%

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

7. Simplified 76.9%

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

   1. frac-2neg 76.9%

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

   2. cbrt-div 94.7%

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

   3. add-sqr-sqrt 53.0%

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

   4. sqrt-unprod 33.8%

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

   5. sqr-neg 33.8%

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

   6. sqrt-unprod 0.6%

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

   7. add-sqr-sqrt 1.5%

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

   8. add-sqr-sqrt 0.8%

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

   9. sqrt-unprod 21.6%

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

   10. sqr-neg 21.6%

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

   11. sqrt-unprod 41.8%

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

   12. add-sqr-sqrt 94.7%

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

9. Applied egg-rr 94.7%

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

10. Final simplification 94.7%

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

Alternative 5: 75.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{h}{\frac{g}{h}}\right)} + \frac{1}{\sqrt[3]{\frac{a \cdot -2}{g \cdot 2}}} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (/ 0.5 a) (* -0.5 (/ h (/ g h)))))
  (/ 1.0 (cbrt (/ (* a -2.0) (* g 2.0))))))

C

double code(double g, double h, double a) {
	return cbrt(((0.5 / a) * (-0.5 * (h / (g / h))))) + (1.0 / cbrt(((a * -2.0) / (g * 2.0))));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((0.5 / a) * (-0.5 * (h / (g / h))))) + (1.0 / Math.cbrt(((a * -2.0) / (g * 2.0))));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(-0.5 * Float64(h / Float64(g / h))))) + Float64(1.0 / cbrt(Float64(Float64(a * -2.0) / Float64(g * 2.0)))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(-0.5 * N[(h / N[(g / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(1.0 / N[Power[N[(N[(a * -2.0), $MachinePrecision] / N[(g * 2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.4%

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

   1. associate-/r* 49.4%

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

   2. metadata-eval 49.4%

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

   3. +-commutative 49.4%

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

   4. unsub-neg 49.4%

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

   5. fma-neg 49.4%

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

   6. sub-neg 49.4%

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

   7. distribute-neg-out 49.4%

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

   8. neg-mul-1 49.4%

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

   9. associate-*r* 49.4%

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

3. Simplified 49.4%

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

   1. clear-num 49.1%

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

   2. cbrt-div 49.1%

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

   3. metadata-eval 49.1%

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

   4. div-inv 49.1%

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

   5. metadata-eval 49.1%

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

   6. fma-udef 49.1%

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

   7. add-sqr-sqrt 28.0%

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

   8. hypot-def 28.2%

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

   9. add-sqr-sqrt 28.2%

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

   10. sqrt-unprod 48.0%

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

   11. sqr-neg 48.0%

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

   12. sqrt-unprod 49.4%

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

   13. add-sqr-sqrt 49.4%

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

   14. sqrt-prod 22.8%

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

   15. add-sqr-sqrt 49.4%

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

5. Applied egg-rr 49.4%

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

6. Taylor expanded in g around inf 25.8%

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

   1. *-commutative 25.8%

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

8. Simplified 25.8%

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

9. Taylor expanded in g around inf 74.6%

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

    1. unpow2 74.6%

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

    2. associate-/l* 79.3%

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

11. Simplified 79.3%

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

12. Final simplification 79.3%

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

Alternative 6: 73.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

3. Simplified 49.4%

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

4. Taylor expanded in g around inf 21.5%

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

5. Taylor expanded in g around inf 76.9%

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

   1. associate-*r/ 76.9%

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

   2. neg-mul-1 76.9%

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

7. Simplified 76.9%

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

8. Final simplification 76.9%

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

Alternative 7: 1.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

3. Simplified 49.4%

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

4. Taylor expanded in g around inf 21.5%

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

5. Taylor expanded in g around inf 76.9%

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

   1. associate-*r/ 76.9%

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

   2. neg-mul-1 76.9%

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

7. Simplified 76.9%

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

   1. expm1-log1p-u 51.8%

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

   2. expm1-udef 26.6%

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

   3. add-sqr-sqrt 13.4%

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

   4. sqrt-unprod 9.4%

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

   5. sqr-neg 9.4%

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

   6. sqrt-unprod 0.6%

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

   7. add-sqr-sqrt 1.6%

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

9. Applied egg-rr 1.6%

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

    1. expm1-def 1.2%

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

    2. expm1-log1p 1.5%

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

11. Simplified 1.5%

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

12. Final simplification 1.5%

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

Reproduce

herbie shell --seed 2023199 
(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))))))))