2-ancestry mixing, positive discriminant

Percentage Accurate: 45.4% → 95.7%

Time: 14.2s

Alternatives: 4

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: 45.4% 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: 95.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double g, double h, double a) {
	return (cbrt((0.5 * pow(sqrt((hypot(g, h) - g)), 2.0))) / 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.pow(Math.sqrt((Math.hypot(g, h) - g)), 2.0))) / 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 * (sqrt(Float64(hypot(g, h) - g)) ^ 2.0))) / 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[Power[N[Sqrt[N[(N[Sqrt[g ^ 2 + h ^ 2], $MachinePrecision] - g), $MachinePrecision]], $MachinePrecision], 2.0], $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 45.6%

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

   1. associate-/r* 45.6%

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

   2. metadata-eval 45.6%

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

   3. +-commutative 45.6%

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

   4. unsub-neg 45.6%

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

   5. fma-neg 45.6%

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

   6. sub-neg 45.6%

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

   7. distribute-neg-out 45.6%

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

   8. neg-mul-1 45.6%

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

   9. associate-*r* 45.6%

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

3. Simplified 45.6%

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

   1. cbrt-div 48.9%

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

   2. fma-udef 48.9%

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

   3. add-sqr-sqrt 23.5%

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

   4. hypot-def 24.4%

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

   5. add-sqr-sqrt 24.4%

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

   6. sqrt-unprod 48.4%

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

   7. sqr-neg 48.4%

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

   8. sqrt-unprod 50.0%

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

   9. add-sqr-sqrt 50.0%

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

   10. sqrt-prod 23.2%

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

   11. add-sqr-sqrt 50.0%

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

   12. div-inv 50.0%

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

   13. metadata-eval 50.0%

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

5. Applied egg-rr 50.0%

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

   1. associate-*l/ 50.0%

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

   2. cbrt-div 51.3%

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

7. Applied egg-rr 96.5%

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

   1. add-sqr-sqrt 96.5%

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

   2. pow2 96.5%

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

9. Applied egg-rr 96.5%

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

10. Final simplification 96.5%

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

Alternative 2: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.6%

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

3. Simplified 45.6%

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

4. Taylor expanded in g around inf 24.7%

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

5. Taylor expanded in g around -inf 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 73.7%

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

   4. metadata-eval 73.7%

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

7. Simplified 73.7%

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

   1. associate-*r/ 73.7%

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

   2. cbrt-div 96.5%

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

9. Applied egg-rr 96.5%

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

10. Final simplification 96.5%

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

Alternative 3: 74.7% 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 45.6%

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

3. Simplified 45.6%

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

   1. *-commutative 45.6%

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

   2. sqrt-prod 22.4%

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

5. Applied egg-rr 22.4%

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

6. Taylor expanded in g around inf 36.3%

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

   1. *-rgt-identity 36.3%

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

   2. *-commutative 36.3%

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

   3. distribute-lft-out 36.3%

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

   4. metadata-eval 36.3%

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

   5. mul0-lft 36.3%

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

   6. metadata-eval 36.3%

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

   7. distribute-lft1-in 36.3%

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

   8. distribute-lft1-in 36.3%

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

   9. distribute-rgt1-in 36.3%

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

   10. *-commutative 36.3%

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

   11. distribute-rgt1-in 36.3%

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

   12. metadata-eval 36.3%

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

   13. mul0-lft 36.3%

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

   14. mul0-lft 36.3%

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

   15. metadata-eval 36.3%

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

8. Simplified 36.3%

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

9. Taylor expanded in g around inf 73.7%

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

10. Final simplification 73.7%

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

Alternative 4: 74.8% 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 45.6%

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

3. Simplified 45.6%

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

   1. *-commutative 45.6%

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

   2. sqrt-prod 22.4%

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

5. Applied egg-rr 22.4%

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

6. Taylor expanded in g around inf 36.3%

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

   1. *-rgt-identity 36.3%

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

   2. *-commutative 36.3%

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

   3. distribute-lft-out 36.3%

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

   4. metadata-eval 36.3%

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

   5. mul0-lft 36.3%

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

   6. metadata-eval 36.3%

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

   7. distribute-lft1-in 36.3%

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

   8. distribute-lft1-in 36.3%

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

   9. distribute-rgt1-in 36.3%

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

   10. *-commutative 36.3%

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

   11. distribute-rgt1-in 36.3%

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

   12. metadata-eval 36.3%

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

   13. mul0-lft 36.3%

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

   14. mul0-lft 36.3%

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

   15. metadata-eval 36.3%

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

8. Simplified 36.3%

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

9. Taylor expanded in g around inf 73.7%

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

    1. associate-*r/ 73.7%

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

    2. neg-mul-1 73.7%

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

11. Simplified 73.7%

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

12. Final simplification 73.7%

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

Reproduce

herbie shell --seed 2023229 
(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))))))))