2-ancestry mixing, positive discriminant

Percentage Accurate: 44.3% → 95.8%

Time: 9.5s

Alternatives: 3

Speedup: 2.6×

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: 44.3% 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: 95.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(0.5 / a)) * Float64(cbrt(g) * Float64(cbrt(-1.0) * (2.0 ^ 0.3333333333333333))))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] * N[(N[Power[-1.0, 1/3], $MachinePrecision] * N[Power[2.0, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.4%

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

4. Applied rewrites 13.9%

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

5. Taylor expanded in g around inf

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

7. Applied rewrites 95.2%

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

9. Applied rewrites 95.6%

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

Alternative 2: 95.8% accurate, 1.4× 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(Float64(-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 46.4%

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

3. Taylor expanded in g around inf

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

5. Applied rewrites 70.0%

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

7. Applied rewrites 95.5%

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

Alternative 3: 73.7% accurate, 2.6× 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 cbrt(Float64(Float64(-g) / a))
end

Wolfram

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

Derivation

1. Initial program 46.4%

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

3. Taylor expanded in g around inf

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

5. Applied rewrites 70.0%

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

7. Applied rewrites 70.7%

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

8. Taylor expanded in g around 0

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

10. Applied rewrites 70.7%

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

11. Final simplification 70.7%

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

Reproduce

herbie shell --seed 2025018 
(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))))))))