2-ancestry mixing, positive discriminant

Percentage Accurate: 43.9% → 95.5%

Time: 13.7s

Alternatives: 5

Speedup: 4.2×

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: 43.9% 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, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Simplified 50.2%

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

5. Taylor expanded in g around inf 75.2%

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

7. Applied egg-rr 95.3%

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

   1. associate-/r/ 95.3%

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

   2. *-commutative 95.3%

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

   3. associate-*r/ 95.4%

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

9. Simplified 95.4%

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

10. Taylor expanded in g around 0 95.4%

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

    1. neg-mul-1 95.4%

       \[\leadsto \frac{\color{blue}{-\sqrt[3]{g}}}{\sqrt[3]{a}} \]

12. Simplified 95.4%

    \[\leadsto \frac{\color{blue}{-\sqrt[3]{g}}}{\sqrt[3]{a}} \]
13. Add Preprocessing

Alternative 2: 73.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Simplified 50.2%

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

5. Taylor expanded in g around inf 75.2%

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

7. Applied egg-rr 95.3%

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

   1. associate-/r/ 95.3%

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

   2. *-commutative 95.3%

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

   3. associate-*r/ 95.4%

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

9. Simplified 95.4%

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

10. Taylor expanded in g around 0 75.8%

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

    1. mul-1-neg 75.8%

       \[\leadsto \color{blue}{-\sqrt[3]{\frac{g}{a}}} \]

12. Simplified 75.8%

    \[\leadsto \color{blue}{-\sqrt[3]{\frac{g}{a}}} \]
13. Add Preprocessing

Alternative 3: 5.8% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Simplified 50.2%

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

5. Taylor expanded in g around inf 75.2%

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

7. Applied egg-rr 6.0%

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

   1. *-commutative 6.0%

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

   2. neg-mul-1 6.0%

      \[\leadsto \sqrt[3]{\color{blue}{-g \cdot a}} \]

   3. distribute-lft-neg-in 6.0%

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

9. Simplified 6.0%

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

10. Final simplification 6.0%

    \[\leadsto \sqrt[3]{g \cdot \left(-a\right)} \]
11. Add Preprocessing

Alternative 4: 1.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Simplified 50.2%

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

5. Taylor expanded in g around inf 75.2%

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

7. Applied egg-rr 95.3%

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

   1. associate-/r/ 95.3%

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

   2. *-commutative 95.3%

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

   3. associate-*r/ 95.4%

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

9. Simplified 95.4%

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

    1. *-un-lft-identity 95.4%

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

    2. add-cbrt-cube 95.2%

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

    3. cbrt-undiv 75.5%

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

    4. pow3 75.5%

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

    5. add-sqr-sqrt 36.2%

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

    6. sqrt-unprod 36.9%

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

    7. *-commutative 36.9%

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

    8. neg-mul-1 36.9%

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

    9. *-commutative 36.9%

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

    10. neg-mul-1 36.9%

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

    11. sqr-neg 36.9%

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

    12. sqrt-unprod 0.7%

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

    13. add-sqr-sqrt 1.4%

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

    14. pow3 1.4%

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

    15. add-cube-cbrt 1.4%

        \[\leadsto 1 \cdot \sqrt[3]{\frac{\color{blue}{g}}{a}} \]

11. Applied egg-rr 1.4%

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

    1. *-lft-identity 1.4%

       \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}}} \]

13. Simplified 1.4%

    \[\leadsto \color{blue}{\sqrt[3]{\frac{g}{a}}} \]

15. Applied egg-rr 0.5%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\log a - \log g}}} \]
16. Step-by-step derivation

    1. exp-diff 0.5%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{e^{\log a}}{e^{\log g}}}} \]

    2. rem-exp-log 0.9%

       \[\leadsto \sqrt[3]{\frac{\color{blue}{a}}{e^{\log g}}} \]

    3. rem-exp-log 1.7%

       \[\leadsto \sqrt[3]{\frac{a}{\color{blue}{g}}} \]

17. Simplified 1.7%

    \[\leadsto \sqrt[3]{\color{blue}{\frac{a}{g}}} \]
18. Add Preprocessing

Alternative 5: 1.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{g \cdot a} \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 (cbrt (* g a)))

C

double code(double g, double h, double a) {
	return cbrt((g * a));
}

Java

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

Julia

function code(g, h, a)
	return cbrt(Float64(g * a))
end

Wolfram

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

Derivation

1. Initial program 50.2%

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

3. Simplified 50.2%

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

5. Taylor expanded in g around inf 75.2%

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

7. Applied egg-rr 1.3%

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

   1. rem-cube-cbrt 1.3%

      \[\leadsto \color{blue}{\sqrt[3]{g \cdot a}} \]

9. Simplified 1.3%

   \[\leadsto \color{blue}{\sqrt[3]{g \cdot a}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(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))))))))