2-ancestry mixing, positive discriminant

Percentage Accurate: 44.5% → 95.7%

Time: 18.7s

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: 44.5% 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, 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(cbrt(g) / cbrt(Float64(-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 46.9%

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

3. Simplified 46.9%

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

5. Taylor expanded in g around inf 78.5%

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

7. Applied egg-rr 96.3%

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

Alternative 2: 73.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt[3]{\frac{a}{-g}}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.9%

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

3. Simplified 46.9%

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

5. Taylor expanded in g around inf 78.5%

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

7. Applied egg-rr 96.3%

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

   1. pow1/3 45.4%

      \[\leadsto \frac{\color{blue}{{g}^{0.3333333333333333}}}{\sqrt[3]{-a}} \]

   2. add-cube-cbrt 45.4%

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

   3. unpow-prod-down 45.4%

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

   4. pow2 45.4%

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

   5. pow1/3 90.9%

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

9. Applied egg-rr 90.9%

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

    1. *-commutative 90.9%

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

    2. unpow1/3 95.9%

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

11. Simplified 95.9%

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

    1. clear-num 95.9%

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

    2. inv-pow 95.9%

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

    3. *-commutative 95.9%

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

    4. cbrt-prod 96.2%

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

    5. unpow2 96.2%

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

    6. add-cube-cbrt 96.4%

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

    7. cbrt-undiv 80.0%

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

13. Applied egg-rr 80.0%

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

    1. unpow-1 80.0%

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

15. Simplified 80.0%

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

16. Final simplification 80.0%

    \[\leadsto \frac{1}{\sqrt[3]{\frac{a}{-g}}} \]
17. Add Preprocessing

Alternative 3: 73.2% 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 46.9%

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

3. Simplified 46.9%

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

5. Taylor expanded in g around inf 78.5%

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

7. Applied egg-rr 96.3%

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

8. Taylor expanded in g around -inf 79.2%

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

   1. mul-1-neg 79.2%

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

10. Simplified 79.2%

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

Alternative 4: 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 46.9%

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

3. Simplified 46.9%

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

5. Taylor expanded in g around inf 78.5%

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

7. Applied egg-rr 5.6%

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

   1. *-commutative 5.6%

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

   2. neg-mul-1 5.6%

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

   3. distribute-lft-neg-in 5.6%

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

9. Simplified 5.6%

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

10. Final simplification 5.6%

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

Alternative 5: 2.9% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (g h a) :precision binary64 0.0)

C

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

Fortran

real(8) function code(g, h, a)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

public static double code(double g, double h, double a) {
	return 0.0;
}

Python

def code(g, h, a):
	return 0.0

Julia

function code(g, h, a)
	return 0.0
end

MATLAB

function tmp = code(g, h, a)
	tmp = 0.0;
end

Wolfram

code[g_, h_, a_] := 0.0

Derivation

1. Initial program 46.9%

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

3. Simplified 46.9%

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

   1. add-cube-cbrt 46.9%

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

   2. pow3 46.9%

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

   3. pow2 46.9%

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

   4. pow2 46.9%

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

6. Applied egg-rr 46.9%

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

8. Applied egg-rr 0.0%

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

   1. +-inverses 3.0%

      \[\leadsto \color{blue}{0} \]

10. Simplified 3.0%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(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))))))))