2-ancestry mixing, positive discriminant

Percentage Accurate: 43.8% → 95.6%

Time: 25.0s

Alternatives: 6

Speedup: 2.0×

• could not determine a ground truth

  1. g = -2.2694980605276193e-172
  2. h = 2.286276913952702e-195
  3. a = 7.532458446417837e-88

Specification

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 43.8% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g}}} \cdot \left(\sqrt[3]{-0.5} \cdot {2}^{0.3333333333333333}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (/ 0.5 a) (- g g)))
  (*
   (/ 1.0 (/ (cbrt a) (cbrt g)))
   (* (cbrt -0.5) (pow 2.0 0.3333333333333333)))))

C

double code(double g, double h, double a) {
	return cbrt(((0.5 / a) * (g - g))) + ((1.0 / (cbrt(a) / cbrt(g))) * (cbrt(-0.5) * pow(2.0, 0.3333333333333333)));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((0.5 / a) * (g - g))) + ((1.0 / (Math.cbrt(a) / Math.cbrt(g))) * (Math.cbrt(-0.5) * Math.pow(2.0, 0.3333333333333333)));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + Float64(Float64(1.0 / Float64(cbrt(a) / cbrt(g))) * Float64(cbrt(-0.5) * (2.0 ^ 0.3333333333333333))))
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[(1.0 / N[(N[Power[a, 1/3], $MachinePrecision] / N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.2%

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

3. Simplified 47.2%

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

5. Taylor expanded in h around 0 17.9%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \color{blue}{{\left(\frac{1 \cdot g}{a}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)} \]
6. Step-by-step derivation

   1. unpow1/3 31.1%

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

   2. *-lft-identity 31.1%

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

7. Simplified 31.1%

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

   1. cbrt-div 35.2%

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

   2. clear-num 35.3%

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

9. Applied egg-rr 35.3%

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

10. Taylor expanded in g around inf 96.5%

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

    1. pow1/3 97.1%

       \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g}}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{{2}^{0.3333333333333333}}\right) \]

12. Applied egg-rr 97.1%

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g}}} \cdot \left(\sqrt[3]{-0.5} \cdot \color{blue}{{2}^{0.3333333333333333}}\right) \]

13. Final simplification 97.1%

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{1}{\frac{\sqrt[3]{a}}{\sqrt[3]{g}}} \cdot \left(\sqrt[3]{-0.5} \cdot {2}^{0.3333333333333333}\right) \]
14. Add Preprocessing

Alternative 2: 95.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + Float64(Float64(1.0 / Float64(cbrt(a) / cbrt(g))) * Float64(cbrt(-0.5) * cbrt(2.0))))
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[(1.0 / N[(N[Power[a, 1/3], $MachinePrecision] / N[Power[g, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.2%

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

3. Simplified 47.2%

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

5. Taylor expanded in h around 0 17.9%

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \color{blue}{{\left(\frac{1 \cdot g}{a}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)} \]
6. Step-by-step derivation

   1. unpow1/3 31.1%

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

   2. *-lft-identity 31.1%

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

7. Simplified 31.1%

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

   1. cbrt-div 35.2%

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

   2. clear-num 35.3%

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

9. Applied egg-rr 35.3%

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

10. Taylor expanded in g around inf 96.5%

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

11. Final simplification 96.5%

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

Alternative 3: 91.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.2%

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

3. Simplified 47.2%

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

   1. associate-*l/ 47.2%

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

   2. cbrt-div 49.0%

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

   3. pow2 49.0%

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

   4. pow2 49.0%

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

6. Applied egg-rr 49.0%

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

7. Taylor expanded in g around -inf 25.0%

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

   1. associate-*r/ 25.0%

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

9. Simplified 25.0%

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

10. Taylor expanded in g around -inf 93.5%

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

    1. neg-mul-1 93.5%

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

12. Simplified 93.5%

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

13. Final simplification 93.5%

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

Alternative 4: 73.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Derivation

1. Initial program 47.2%

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

3. Simplified 47.2%

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

5. Taylor expanded in g around inf 28.3%

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

6. Taylor expanded in g around inf 79.0%

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

7. Final simplification 79.0%

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

Alternative 5: 73.0% 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(g / 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[Power[N[(g / (-a)), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.2%

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

3. Simplified 47.2%

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

5. Taylor expanded in g around inf 28.3%

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

6. Taylor expanded in g around inf 79.0%

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

7. Taylor expanded in g around 0 79.0%

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

8. Final simplification 79.0%

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

Alternative 6: 4.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.2%

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

3. Simplified 47.2%

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

5. Taylor expanded in g around inf 28.3%

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

6. Taylor expanded in g around inf 79.0%

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

   1. cbrt-prod 97.1%

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

8. Applied egg-rr 0.0%

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

10. Simplified 5.2%

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

11. Final simplification 5.2%

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

Reproduce

herbie shell --seed 2024047 
(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))))))))