2-ancestry mixing, positive discriminant

Percentage Accurate: 44.2% → 95.5%

Time: 7.0s

Alternatives: 3

Speedup: 3.9×

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: 95.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)} \]
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 74.3

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

4. Applied rewrites 74.3%

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

   1. lift-/.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. cbrt-div N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-cbrt.f64 95.5

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

6. Applied rewrites 95.5%

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

Alternative 2: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{2 \cdot a} \leq 10^{+230}:\\ \;\;\;\;-\sqrt[3]{\frac{g}{a}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(g \cdot \sqrt[3]{\frac{1}{a \cdot \left(g \cdot g\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (g h a)
 :precision binary64
 (if (<= (/ 1.0 (* 2.0 a)) 1e+230)
   (- (cbrt (/ g a)))
   (* -1.0 (* g (cbrt (/ 1.0 (* a (* g g))))))))

C

double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (2.0 * a)) <= 1e+230) {
		tmp = -cbrt((g / a));
	} else {
		tmp = -1.0 * (g * cbrt((1.0 / (a * (g * g)))));
	}
	return tmp;
}

Java

public static double code(double g, double h, double a) {
	double tmp;
	if ((1.0 / (2.0 * a)) <= 1e+230) {
		tmp = -Math.cbrt((g / a));
	} else {
		tmp = -1.0 * (g * Math.cbrt((1.0 / (a * (g * g)))));
	}
	return tmp;
}

Julia

function code(g, h, a)
	tmp = 0.0
	if (Float64(1.0 / Float64(2.0 * a)) <= 1e+230)
		tmp = Float64(-cbrt(Float64(g / a)));
	else
		tmp = Float64(-1.0 * Float64(g * cbrt(Float64(1.0 / Float64(a * Float64(g * g))))));
	end
	return tmp
end

Wolfram

code[g_, h_, a_] := If[LessEqual[N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 1e+230], (-N[Power[N[(g / a), $MachinePrecision], 1/3], $MachinePrecision]), N[(-1.0 * N[(g * N[Power[N[(1.0 / N[(a * N[(g * g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a)) < 1.0000000000000001e230

   1. Initial program 45.2%

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

   2. Taylor expanded in g around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)} \]
   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 77.0

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

   4. Applied rewrites 77.0%

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

   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 77.0

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

   7. Applied rewrites 77.0%

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

   if 1.0000000000000001e230 < (/.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 28.1%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\sqrt[3]{\frac{g + \sqrt{{g}^{2} - {h}^{2}}}{a}} \cdot \sqrt[3]{\frac{-1}{2}} + \sqrt[3]{\frac{\sqrt{{g}^{2} - {h}^{2}} - g}{a}} \cdot \sqrt[3]{\frac{1}{2}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in g around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

   7. Applied rewrites 33.8%

      \[\leadsto -1 \cdot \color{blue}{\left(g \cdot \left(\sqrt[3]{\frac{0}{a \cdot \left(g \cdot g\right)} \cdot 0.5} + \sqrt[3]{\frac{-2}{a \cdot \left(g \cdot g\right)} \cdot -0.5}\right)\right)} \]

   8. Taylor expanded in g around 0

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

      1. lower-cbrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 34.6

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

   10. Applied rewrites 34.6%

       \[\leadsto -1 \cdot \left(g \cdot \sqrt[3]{\frac{1}{a \cdot \left(g \cdot g\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.3% accurate, 3.9× 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 44.2%

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

2. Taylor expanded in g around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt[3]{\frac{g}{a}} \cdot \left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{2}\right)\right)} \]
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 74.3

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

4. Applied rewrites 74.3%

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

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 74.3

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

7. Applied rewrites 74.3%

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

Reproduce

herbie shell --seed 2025117 
(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))))))))