2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 3.1s

Alternatives: 5

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos((((2.0 * Math.PI) / 3.0) + (Math.acos((-g / h)) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(2.0 * pi) / 3.0) + Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(2.0 * Pi), $MachinePrecision] / 3.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos((((2.0 * Math.PI) / 3.0) + (Math.acos((-g / h)) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(2.0 * pi) / 3.0) + Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(2.0 * Pi), $MachinePrecision] / 3.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{-g}{h}\right)\\ \cos \left(\mathsf{fma}\left(0.3333333333333333, t\_0, \pi \cdot 0.6666666666666666\right)\right) - \mathsf{fma}\left(\cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \pi\right)\right), \cos \left(-0.3333333333333333 \cdot t\_0\right), \sin \left(0.3333333333333333 \cdot t\_0\right) \cdot \sin \left(\pi \cdot 0.6666666666666666\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h))))
   (-
    (cos (fma 0.3333333333333333 t_0 (* PI 0.6666666666666666)))
    (fma
     (cos (fma PI 0.6666666666666666 PI))
     (cos (* -0.3333333333333333 t_0))
     (* (sin (* 0.3333333333333333 t_0)) (sin (* PI 0.6666666666666666)))))))

C

double code(double g, double h) {
	double t_0 = acos((-g / h));
	return cos(fma(0.3333333333333333, t_0, (((double) M_PI) * 0.6666666666666666))) - fma(cos(fma(((double) M_PI), 0.6666666666666666, ((double) M_PI))), cos((-0.3333333333333333 * t_0)), (sin((0.3333333333333333 * t_0)) * sin((((double) M_PI) * 0.6666666666666666))));
}

Julia

function code(g, h)
	t_0 = acos(Float64(Float64(-g) / h))
	return Float64(cos(fma(0.3333333333333333, t_0, Float64(pi * 0.6666666666666666))) - fma(cos(fma(pi, 0.6666666666666666, pi)), cos(Float64(-0.3333333333333333 * t_0)), Float64(sin(Float64(0.3333333333333333 * t_0)) * sin(Float64(pi * 0.6666666666666666)))))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]}, N[(N[Cos[N[(0.3333333333333333 * t$95$0 + N[(Pi * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[Cos[N[(Pi * 0.6666666666666666 + Pi), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.3333333333333333 * t$95$0), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(0.3333333333333333 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(Pi * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-acos.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-add N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, \pi \cdot 6\right)}{9}\right)} \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 0.6666666666666666\right)\right) - \mathsf{fma}\left(\cos \left(\mathsf{fma}\left(\pi, 0.6666666666666666, \pi\right)\right), \cos \left(-0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right), \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \sin \left(\pi \cdot 0.6666666666666666\right)\right)} \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin \left(\mathsf{fma}\left(\sqrt{\pi}, \frac{\sqrt{\pi}}{2}, \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -0.6666666666666666 \cdot \pi\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin
   (fma
    (sqrt PI)
    (/ (sqrt PI) 2.0)
    (fma (acos (/ (- g) h)) -0.3333333333333333 (* -0.6666666666666666 PI))))))

C

double code(double g, double h) {
	return 2.0 * sin(fma(sqrt(((double) M_PI)), (sqrt(((double) M_PI)) / 2.0), fma(acos((-g / h)), -0.3333333333333333, (-0.6666666666666666 * ((double) M_PI)))));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(fma(sqrt(pi), Float64(sqrt(pi) / 2.0), fma(acos(Float64(Float64(-g) / h)), -0.3333333333333333, Float64(-0.6666666666666666 * pi)))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * -0.3333333333333333 + N[(-0.6666666666666666 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-acos.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-add N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, \pi \cdot 6\right)}{9}\right)} \]

5. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-0.3333333333333333, \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), \frac{\pi}{2}\right)\right)} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) + \frac{\pi}{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{\pi}{2} + \frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)} \]

   3. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2} + \frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} + \frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \]

   5. add-sqr-sqrt N/A

      \[\leadsto 2 \cdot \sin \left(\frac{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}}{2} + \frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \]

   6. associate-/l* N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}} + \frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}, \frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}, \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}, \frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right) \]

   9. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\sqrt{\color{blue}{\pi}}, \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}, \frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\sqrt{\pi}, \color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}}, \frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right) \]

   11. lower-sqrt.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\sqrt{\pi}, \frac{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}{2}, \frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\sqrt{\pi}, \frac{\sqrt{\color{blue}{\pi}}}{2}, \frac{-1}{3} \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right) \]

   13. lift-PI.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\sqrt{\pi}, \frac{\sqrt{\pi}}{2}, \frac{-1}{3} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)}, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right) \]

   14. lift-fma.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\sqrt{\pi}, \frac{\sqrt{\pi}}{2}, \frac{-1}{3} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2 + \cos^{-1} \left(\frac{-g}{h}\right)\right)}\right)\right) \]

   15. distribute-lft-in N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\sqrt{\pi}, \frac{\sqrt{\pi}}{2}, \color{blue}{\frac{-1}{3} \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right) + \frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\sqrt{\pi}, \frac{\sqrt{\pi}}{2}, \frac{-1}{3} \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} + \frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \]

   17. lift-*.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\sqrt{\pi}, \frac{\sqrt{\pi}}{2}, \frac{-1}{3} \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)}\right)\right) \]

7. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\sqrt{\pi}, \frac{\sqrt{\pi}}{2}, \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -0.6666666666666666 \cdot \pi\right)\right)\right)} \]
8. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin \left(\mathsf{fma}\left(-0.3333333333333333, \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), \frac{\pi}{2}\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin (fma -0.3333333333333333 (fma PI 2.0 (acos (/ (- g) h))) (/ PI 2.0)))))

C

double code(double g, double h) {
	return 2.0 * sin(fma(-0.3333333333333333, fma(((double) M_PI), 2.0, acos((-g / h))), (((double) M_PI) / 2.0)));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(fma(-0.3333333333333333, fma(pi, 2.0, acos(Float64(Float64(-g) / h))), Float64(pi / 2.0))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(-0.3333333333333333 * N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-acos.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-add N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, \pi \cdot 6\right)}{9}\right)} \]

5. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-0.3333333333333333, \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), \frac{\pi}{2}\right)\right)} \]
6. Add Preprocessing

Alternative 4: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, \pi \cdot 6\right)}{9}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (/ (fma (acos (/ (- g) h)) 3.0 (* PI 6.0)) 9.0))))

C

double code(double g, double h) {
	return 2.0 * cos((fma(acos((-g / h)), 3.0, (((double) M_PI) * 6.0)) / 9.0));
}

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(fma(acos(Float64(Float64(-g) / h)), 3.0, Float64(pi * 6.0)) / 9.0)))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * 3.0 + N[(Pi * 6.0), $MachinePrecision]), $MachinePrecision] / 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-acos.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-add N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, \pi \cdot 6\right)}{9}\right)} \]
4. Add Preprocessing

Alternative 5: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{3}\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (/ (fma PI 2.0 (acos (/ (- g) h))) 3.0)) 2.0))

C

double code(double g, double h) {
	return cos((fma(((double) M_PI), 2.0, acos((-g / h))) / 3.0)) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(Float64(fma(pi, 2.0, acos(Float64(Float64(-g) / h))) / 3.0)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.5

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

3. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\cos \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{3}\right) \cdot 2} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))