2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 2.9s

Alternatives: 4

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, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin \left(\mathsf{fma}\left(\pi, -0.16666666666666666, \cos^{-1} \left(\frac{-g}{h}\right) \cdot -0.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin
   (fma PI -0.16666666666666666 (* (acos (/ (- g) h)) -0.3333333333333333)))))

C

double code(double g, double h) {
	return 2.0 * sin(fma(((double) M_PI), -0.16666666666666666, (acos((-g / h)) * -0.3333333333333333)));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(fma(pi, -0.16666666666666666, Float64(acos(Float64(Float64(-g) / h)) * -0.3333333333333333))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(Pi * -0.16666666666666666 + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * -0.3333333333333333), $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%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \pi \cdot 0.5\right)} \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 98.5

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi, 0.5, \frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3}\right)\right)} \]

   5. lift-/.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. div-add N/A

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. associate-/l* N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 98.4

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot \color{blue}{-0.6666666666666666}\right)\right)\right) \]

5. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi, 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot -0.6666666666666666\right)\right)\right)} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. *-commutative N/A

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

   6. unpow1 N/A

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

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. lift-pow.f64 N/A

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

   10. pow1/3 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi, -0.16666666666666666, \cos^{-1} \left(\frac{-g}{h}\right) \cdot -0.3333333333333333\right)\right)} \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.16666666666666666 \cdot \pi\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin
   (fma -0.3333333333333333 (acos (/ (- g) h)) (* -0.16666666666666666 PI)))
  2.0))

C

double code(double g, double h) {
	return sin(fma(-0.3333333333333333, acos((-g / h)), (-0.16666666666666666 * ((double) M_PI)))) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(-0.3333333333333333, acos(Float64(Float64(-g) / h)), Float64(-0.16666666666666666 * pi))) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(-0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + N[(-0.16666666666666666 * Pi), $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-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%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \pi \cdot 0.5\right)} \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 98.5

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi, 0.5, \frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3}\right)\right)} \]

   5. lift-/.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. div-add N/A

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. associate-/l* N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 98.4

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot \color{blue}{-0.6666666666666666}\right)\right)\right) \]

5. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi, 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot -0.6666666666666666\right)\right)\right)} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. *-commutative N/A

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

   6. unpow1 N/A

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

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. lift-pow.f64 N/A

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

   10. pow1/3 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi, -0.16666666666666666, \cos^{-1} \left(\frac{-g}{h}\right) \cdot -0.3333333333333333\right)\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 100.0

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\pi, -0.16666666666666666, \cos^{-1} \left(\frac{-g}{h}\right) \cdot -0.3333333333333333\right)\right) \cdot 2} \]

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 100.0

       \[\leadsto \sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{-0.16666666666666666 \cdot \pi}\right)\right) \cdot 2 \]

9. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.16666666666666666 \cdot \pi\right)\right) \cdot 2} \]
10. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos \left(\mathsf{fma}\left(\sin^{-1} \left(\frac{g}{h}\right), 0.3333333333333333, 0.8333333333333334 \cdot \pi\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (fma (asin (/ g h)) 0.3333333333333333 (* 0.8333333333333334 PI)))
  2.0))

C

double code(double g, double h) {
	return cos(fma(asin((g / h)), 0.3333333333333333, (0.8333333333333334 * ((double) M_PI)))) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(fma(asin(Float64(g / h)), 0.3333333333333333, Float64(0.8333333333333334 * pi))) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(N[ArcSin[N[(g / h), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333 + N[(0.8333333333333334 * Pi), $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-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%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \pi \cdot 0.5\right)} \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 98.5

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi, 0.5, \frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3}\right)\right)} \]

   5. lift-/.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. div-add N/A

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. associate-/l* N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 98.4

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot \color{blue}{-0.6666666666666666}\right)\right)\right) \]

5. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi, 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot -0.6666666666666666\right)\right)\right)} \]

7. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(\pi, 2.5, \sin^{-1} \left(\frac{g}{h}\right)\right) \cdot 0.3333333333333333\right) \cdot 2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\mathsf{fma}\left(\pi, \frac{5}{2}, \sin^{-1} \left(\frac{g}{h}\right)\right) \cdot \frac{1}{3}\right)} \cdot 2 \]

   2. *-commutative N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{3} \cdot \mathsf{fma}\left(\pi, \frac{5}{2}, \sin^{-1} \left(\frac{g}{h}\right)\right)\right)} \cdot 2 \]

   3. lift-fma.f64 N/A

      \[\leadsto \cos \left(\frac{1}{3} \cdot \color{blue}{\left(\pi \cdot \frac{5}{2} + \sin^{-1} \left(\frac{g}{h}\right)\right)}\right) \cdot 2 \]

   4. distribute-lft-in N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{3} \cdot \left(\pi \cdot \frac{5}{2}\right) + \frac{1}{3} \cdot \sin^{-1} \left(\frac{g}{h}\right)\right)} \cdot 2 \]

   5. +-commutative N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{3} \cdot \sin^{-1} \left(\frac{g}{h}\right) + \frac{1}{3} \cdot \left(\pi \cdot \frac{5}{2}\right)\right)} \cdot 2 \]

   6. *-commutative N/A

      \[\leadsto \cos \left(\color{blue}{\sin^{-1} \left(\frac{g}{h}\right) \cdot \frac{1}{3}} + \frac{1}{3} \cdot \left(\pi \cdot \frac{5}{2}\right)\right) \cdot 2 \]

   7. lower-fma.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\mathsf{fma}\left(\sin^{-1} \left(\frac{g}{h}\right), \frac{1}{3}, \frac{1}{3} \cdot \left(\pi \cdot \frac{5}{2}\right)\right)\right)} \cdot 2 \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\sin^{-1} \left(\frac{g}{h}\right), \frac{1}{3}, \frac{1}{3} \cdot \color{blue}{\left(\frac{5}{2} \cdot \pi\right)}\right)\right) \cdot 2 \]

   9. associate-*r* N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\sin^{-1} \left(\frac{g}{h}\right), \frac{1}{3}, \color{blue}{\left(\frac{1}{3} \cdot \frac{5}{2}\right) \cdot \pi}\right)\right) \cdot 2 \]

   10. metadata-eval N/A

       \[\leadsto \cos \left(\mathsf{fma}\left(\sin^{-1} \left(\frac{g}{h}\right), \frac{1}{3}, \color{blue}{\frac{5}{6}} \cdot \pi\right)\right) \cdot 2 \]

   11. metadata-eval N/A

       \[\leadsto \cos \left(\mathsf{fma}\left(\sin^{-1} \left(\frac{g}{h}\right), \frac{1}{3}, \color{blue}{\left(\frac{5}{2} \cdot \frac{1}{3}\right)} \cdot \pi\right)\right) \cdot 2 \]

   12. lower-*.f64 N/A

       \[\leadsto \cos \left(\mathsf{fma}\left(\sin^{-1} \left(\frac{g}{h}\right), \frac{1}{3}, \color{blue}{\left(\frac{5}{2} \cdot \frac{1}{3}\right) \cdot \pi}\right)\right) \cdot 2 \]

   13. metadata-eval 98.5

       \[\leadsto \cos \left(\mathsf{fma}\left(\sin^{-1} \left(\frac{g}{h}\right), 0.3333333333333333, \color{blue}{0.8333333333333334} \cdot \pi\right)\right) \cdot 2 \]

9. Applied rewrites 98.5%

   \[\leadsto \cos \color{blue}{\left(\mathsf{fma}\left(\sin^{-1} \left(\frac{g}{h}\right), 0.3333333333333333, 0.8333333333333334 \cdot \pi\right)\right)} \cdot 2 \]
10. Add Preprocessing

Alternative 4: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos \left(\mathsf{fma}\left(\pi, 2.5, \sin^{-1} \left(\frac{g}{h}\right)\right) \cdot 0.3333333333333333\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (* (fma PI 2.5 (asin (/ g h))) 0.3333333333333333)) 2.0))

C

double code(double g, double h) {
	return cos((fma(((double) M_PI), 2.5, asin((g / h))) * 0.3333333333333333)) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(Float64(fma(pi, 2.5, asin(Float64(g / h))) * 0.3333333333333333)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(N[(Pi * 2.5 + N[ArcSin[N[(g / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $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-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%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \pi \cdot 0.5\right)} \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 98.5

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi, 0.5, \frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3}\right)\right)} \]

   5. lift-/.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. div-add N/A

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. associate-/l* N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 98.4

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot \color{blue}{-0.6666666666666666}\right)\right)\right) \]

5. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi, 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot -0.6666666666666666\right)\right)\right)} \]

7. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(\pi, 2.5, \sin^{-1} \left(\frac{g}{h}\right)\right) \cdot 0.3333333333333333\right) \cdot 2} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))