2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 4.2s

Alternatives: 5

Speedup: 1.1×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(N[(N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(-3.0 * Pi), $MachinePrecision]), $MachinePrecision] / -6.0), $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-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

3. Applied rewrites 98.5%

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

   1. lift-+.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-add N/A

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

   6. lower-/.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. metadata-eval 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), -0.3333333333333333, 0.5 \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin (fma (fma PI 2.0 (acos (/ (- g) h))) -0.3333333333333333 (* 0.5 PI)))))

C

double code(double g, double h) {
	return 2.0 * sin(fma(fma(((double) M_PI), 2.0, acos((-g / h))), -0.3333333333333333, (0.5 * ((double) M_PI))));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(fma(fma(pi, 2.0, acos(Float64(Float64(-g) / h))), -0.3333333333333333, Float64(0.5 * pi))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + N[(0.5 * Pi), $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-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in g around 0

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{-1}{3} \cdot \left(\cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. mul-1-neg N/A

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

   7. distribute-frac-neg N/A

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

   8. lift-/.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. lift-acos.f64 N/A

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

   11. lift-fma.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-*.f64 98.5

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -0.3333333333333333 \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(0.5, \pi, -0.3333333333333333 \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 99.9

      \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), -0.3333333333333333, 0.5 \cdot \pi\right)\right) \]

8. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), \color{blue}{-0.3333333333333333}, 0.5 \cdot \pi\right)\right) \]
9. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -0.3333333333333333 \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin (fma 0.5 PI (* -0.3333333333333333 (fma PI 2.0 (acos (/ (- g) h))))))))

C

double code(double g, double h) {
	return 2.0 * sin(fma(0.5, ((double) M_PI), (-0.3333333333333333 * fma(((double) M_PI), 2.0, acos((-g / h))))));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(fma(0.5, pi, Float64(-0.3333333333333333 * fma(pi, 2.0, acos(Float64(Float64(-g) / h)))))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(0.5 * Pi + N[(-0.3333333333333333 * N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $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-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in g around 0

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{-1}{3} \cdot \left(\cos^{-1} \left(-1 \cdot \frac{g}{h}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. mul-1-neg N/A

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

   7. distribute-frac-neg N/A

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

   8. lift-/.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. lift-acos.f64 N/A

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

   11. lift-fma.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-*.f64 98.5

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -0.3333333333333333 \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(0.5, \pi, -0.3333333333333333 \cdot \mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)\right)} \]
7. Add Preprocessing

Alternative 4: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(N[Cos[N[(0.3333333333333333 * N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in g around 0

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-acos.f64 N/A

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

   4. div-add N/A

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

   5. *-commutative N/A

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

   6. lift-acos.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. mul-1-neg N/A

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

   12. distribute-frac-neg N/A

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

   13. lift-/.f64 N/A

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

6. Applied rewrites 98.4%

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

Alternative 5: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + N[(0.6666666666666666 * Pi), $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. Taylor expanded in g around 0

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 98.4

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \pi\right)\right) \]

4. Applied rewrites 98.4%

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

Reproduce

herbie shell --seed 2025097 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))