2-ancestry mixing, positive discriminant

Percentage Accurate: 44.2% → 95.7%

Time: 7.3s

Alternatives: 2

Speedup: 6.2×

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: 44.2% 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, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

2. Taylor expanded in g around inf

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

   1. cbrt-unprod N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{-1}{2} \cdot 2} \]

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lower-/.f64 73.1

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

4. Applied rewrites 73.1%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 95.7

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

6. Applied rewrites 95.7%

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

Alternative 2: 73.1% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. cbrt-prod N/A

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

   10. lower-*.f64 N/A

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

3. Applied rewrites 47.3%

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

4. Taylor expanded in g around inf

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

   1. cbrt-unprod N/A

      \[\leadsto \sqrt[3]{\frac{g}{a}} \cdot \sqrt[3]{\frac{-1}{2} \cdot 2} \]

   2. metadata-eval N/A

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

   3. cbrt-unprod N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-/.f64 73.1

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

6. Applied rewrites 73.1%

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

Reproduce

herbie shell --seed 2025101 
(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))))))))