2-ancestry mixing, positive discriminant

Percentage Accurate: 44.1% → 95.5%

Time: 16.7s

Alternatives: 2

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: 44.1% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - g}, \sqrt[3]{\frac{0.5}{a} \cdot 0}\right) \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (fma (cbrt (/ 0.5 a)) (cbrt (- (- g) g)) (cbrt (* (/ 0.5 a) 0.0))))

C

double code(double g, double h, double a) {
	return fma(cbrt((0.5 / a)), cbrt((-g - g)), cbrt(((0.5 / a) * 0.0)));
}

Julia

function code(g, h, a)
	return fma(cbrt(Float64(0.5 / a)), cbrt(Float64(Float64(-g) - g)), cbrt(Float64(Float64(0.5 / a) * 0.0)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[((-g) - g), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(0.5 / a), $MachinePrecision] * 0.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.1%

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

3. Applied rewrites 47.1%

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

4. Taylor expanded in g around inf

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

6. Applied rewrites 32.3%

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

7. Taylor expanded in h around 0

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

   1. lower-+.f64 N/A

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

   2. lift-*.f64 95.5

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

9. Applied rewrites 95.5%

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

10. Taylor expanded in g around 0

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{2}}{a}}, \sqrt[3]{\left(-g\right) - g}, \sqrt[3]{\frac{\frac{1}{2}}{a} \cdot 0}\right) \]
11. Step-by-step derivation

12. Applied rewrites 95.5%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{a}}, \sqrt[3]{\left(-g\right) - g}, \sqrt[3]{\frac{0.5}{a} \cdot 0}\right) \]
13. Add Preprocessing

Alternative 2: 73.4% 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 Float64(-cbrt(Float64(g / a)))
end

Wolfram

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

Derivation

1. Initial program 44.1%

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

2. Taylor expanded in g around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. cbrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cbrt.f64 N/A

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

   8. lower-/.f64 73.4

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

4. Applied rewrites 73.4%

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

5. Taylor expanded in g around 0

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

   1. lift-cbrt.f64 N/A

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

   2. lift-/.f64 73.4

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

7. Applied rewrites 73.4%

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

Reproduce

herbie shell --seed 2025092 
(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))))))))