2-ancestry mixing, positive discriminant

Percentage Accurate: 44.9% → 95.5%

Time: 30.7s

Alternatives: 6

Speedup: 2.1×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t\_0 \cdot \left(\left(-g\right) + t\_1\right)} + \sqrt[3]{t\_0 \cdot \left(\left(-g\right) - t\_1\right)} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 44.9% 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: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.8%

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

3. Simplified 42.8%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around -inf 68.7%

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

   1. neg-mul-1 68.7%

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

10. Simplified 68.7%

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

    1. cbrt-prod 95.8%

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

12. Applied egg-rr 95.8%

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

13. Taylor expanded in a around 0 95.8%

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

14. Final simplification 95.8%

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

Alternative 2: 95.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Derivation

1. Initial program 42.8%

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

3. Simplified 42.8%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around -inf 68.7%

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

   1. neg-mul-1 68.7%

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

10. Simplified 68.7%

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

    1. cbrt-prod 95.8%

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

12. Applied egg-rr 95.8%

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

13. Final simplification 95.8%

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

Alternative 3: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Derivation

1. Initial program 42.8%

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

3. Simplified 42.8%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around -inf 68.7%

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

   1. neg-mul-1 68.7%

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

10. Simplified 68.7%

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

    1. associate-*l/ 68.7%

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

    2. cbrt-div 95.8%

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

    3. *-commutative 95.8%

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

    4. associate-*r* 95.8%

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

    5. metadata-eval 95.8%

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

    6. neg-mul-1 95.8%

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

12. Applied egg-rr 95.8%

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

13. Final simplification 95.8%

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

Alternative 4: 75.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.8%

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

3. Simplified 42.8%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around -inf 67.5%

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

   1. unpow2 67.5%

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

   2. associate-/l* 69.5%

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

10. Applied egg-rr 69.5%

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

11. Final simplification 69.5%

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

Alternative 5: 73.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.8%

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

3. Simplified 42.8%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around inf 14.5%

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

   1. add-log-exp 30.6%

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

   2. exp-prod 41.5%

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

   3. metadata-eval 41.5%

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

   4. distribute-rgt-out-- 41.5%

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

   5. neg-mul-1 41.5%

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

   6. add-sqr-sqrt 40.3%

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

   7. sqrt-unprod 52.3%

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

   8. sqr-neg 52.3%

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

   9. unpow2 52.3%

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

   10. sqrt-pow1 68.7%

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

   11. metadata-eval 68.7%

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

   12. pow1 68.7%

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

   13. *-un-lft-identity 68.7%

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

   14. +-inverses 68.7%

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

   15. metadata-eval 68.7%

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

   16. metadata-eval 68.7%

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

10. Applied egg-rr 68.7%

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

11. Final simplification 68.7%

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

Alternative 6: 73.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.8%

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

3. Simplified 42.8%

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

5. Taylor expanded in g around -inf 24.8%

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

   1. *-commutative 24.8%

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

7. Simplified 24.8%

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

8. Taylor expanded in g around inf 14.5%

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

   1. add-log-exp 30.6%

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

   2. exp-prod 41.5%

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

   3. metadata-eval 41.5%

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

   4. distribute-rgt-out-- 41.5%

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

   5. neg-mul-1 41.5%

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

   6. add-sqr-sqrt 40.3%

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

   7. sqrt-unprod 52.3%

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

   8. sqr-neg 52.3%

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

   9. unpow2 52.3%

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

   10. sqrt-pow1 68.7%

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

   11. metadata-eval 68.7%

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

   12. pow1 68.7%

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

   13. *-un-lft-identity 68.7%

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

   14. +-inverses 68.7%

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

   15. metadata-eval 68.7%

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

   16. metadata-eval 68.7%

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

10. Applied egg-rr 68.7%

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

11. Taylor expanded in g around -inf 68.7%

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

    1. mul-1-neg 68.7%

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

13. Simplified 68.7%

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

14. Final simplification 68.7%

    \[\leadsto \sqrt[3]{0} - \sqrt[3]{\frac{g}{a}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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))))))))