2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 4.6s

Alternatives: 7

Speedup: 1.2×

Specification

Math

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

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

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

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, 0.8× speedup

Math

\[\sin \left(\mathsf{fma}\left(1.0725146985555127, \sqrt[3]{\pi}, \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -0.6666666666666666 \cdot \pi\right)\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin
   (fma
    1.0725146985555127
    (cbrt PI)
    (fma (acos (/ (- g) h)) -0.3333333333333333 (* -0.6666666666666666 PI))))
  2.0))

C

double code(double g, double h) {
	return sin(fma(1.0725146985555127, cbrt(((double) M_PI)), fma(acos((-g / h)), -0.3333333333333333, (-0.6666666666666666 * ((double) M_PI))))) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(1.0725146985555127, cbrt(pi), fma(acos(Float64(Float64(-g) / h)), -0.3333333333333333, Float64(-0.6666666666666666 * pi)))) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(1.0725146985555127 * N[Power[Pi, 1/3], $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] * -0.3333333333333333 + N[(-0.6666666666666666 * Pi), $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-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. lift-PI.f64 N/A

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

   5. add-cube-cbrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\pi \cdot \pi}, \sqrt[3]{\pi} \cdot 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.6666666666666666 \cdot \pi\right)\right)\right)} \]

6. Evaluated real constant 100.0%

   \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{2.1450293971110255}, \sqrt[3]{\pi} \cdot 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.6666666666666666 \cdot \pi\right)\right)\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \sin \left(\mathsf{fma}\left(\frac{4830176796763987}{2251799813685248}, \sqrt[3]{\pi} \cdot \frac{1}{2}, \mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{-2}{3} \cdot \pi\right)\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\frac{4830176796763987}{2251799813685248}, \sqrt[3]{\pi} \cdot \frac{1}{2}, \mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{-2}{3} \cdot \pi\right)\right)\right) \cdot 2} \]

   3. lower-*.f64 100.0%

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(2.1450293971110255, \sqrt[3]{\pi} \cdot 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.6666666666666666 \cdot \pi\right)\right)\right) \cdot 2} \]

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(1.0725146985555127, \sqrt[3]{\pi}, \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, -0.6666666666666666 \cdot \pi\right)\right)\right) \cdot 2} \]
9. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin
   (*
    (*
     (fma (/ (acos (/ (- g) h)) PI) -0.6666666666666666 -0.3333333333333333)
     PI)
    0.5))))

C

double code(double g, double h) {
	return 2.0 * sin(((fma((acos((-g / h)) / ((double) M_PI)), -0.6666666666666666, -0.3333333333333333) * ((double) M_PI)) * 0.5));
}

Julia

function code(g, h)
	return Float64(2.0 * sin(Float64(Float64(fma(Float64(acos(Float64(Float64(-g) / h)) / pi), -0.6666666666666666, -0.3333333333333333) * pi) * 0.5)))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Sin[N[(N[(N[(N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * -0.6666666666666666 + -0.3333333333333333), $MachinePrecision] * Pi), $MachinePrecision] * 0.5), $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. lift-PI.f64 N/A

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

   5. add-cube-cbrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\pi \cdot \pi}, \sqrt[3]{\pi} \cdot 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.6666666666666666 \cdot \pi\right)\right)\right)} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. cbrt-prod N/A

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

   8. rem-3cbrt-lft N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. sum-to-mult-rev N/A

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

7. Applied rewrites 100.0%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin
   (fma
    -0.6666666666666666
    PI
    (fma -0.3333333333333333 (acos (/ (- g) h)) (* 0.5 PI))))
  2.0))

C

double code(double g, double h) {
	return sin(fma(-0.6666666666666666, ((double) M_PI), fma(-0.3333333333333333, acos((-g / h)), (0.5 * ((double) M_PI))))) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(-0.6666666666666666, pi, fma(-0.3333333333333333, acos(Float64(Float64(-g) / h)), Float64(0.5 * pi)))) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(-0.6666666666666666 * Pi + N[(-0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + N[(0.5 * Pi), $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%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval 98.4%

       \[\leadsto \cos \left(\mathsf{fma}\left(\color{blue}{0.3333333333333333}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{2 \cdot \pi}{3}\right)\right) \cdot 2 \]

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 98.4%

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

3. Applied rewrites 98.4%

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

5. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), -0.3333333333333333, 1.5707963267948966\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin
   (fma
    (fma PI 2.0 (acos (/ (- g) h)))
    -0.3333333333333333
    1.5707963267948966))
  2.0))

C

double code(double g, double h) {
	return sin(fma(fma(((double) M_PI), 2.0, acos((-g / h))), -0.3333333333333333, 1.5707963267948966)) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(fma(pi, 2.0, acos(Float64(Float64(-g) / h))), -0.3333333333333333, 1.5707963267948966)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(N[(Pi * 2.0 + N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + 1.5707963267948966), $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. Evaluated real constant 98.5%

   \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \color{blue}{1.5707963267948966}\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \frac{884279719003555}{562949953421312}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \frac{884279719003555}{562949953421312}\right) \cdot 2} \]

   3. lower-*.f64 98.5%

      \[\leadsto \color{blue}{\sin \left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + 1.5707963267948966\right) \cdot 2} \]

   4. lift-+.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \frac{884279719003555}{562949953421312}\right)} \cdot 2 \]

   5. lift-/.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\frac{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3}} + \frac{884279719003555}{562949953421312}\right) \cdot 2 \]

   6. mult-flip N/A

      \[\leadsto \sin \left(\color{blue}{\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{-3}} + \frac{884279719003555}{562949953421312}\right) \cdot 2 \]

   7. metadata-eval N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \color{blue}{\frac{-1}{3}} + \frac{884279719003555}{562949953421312}\right) \cdot 2 \]

   8. lower-fma.f64 99.9%

      \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), -0.3333333333333333, 1.5707963267948966\right)\right)} \cdot 2 \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\pi, 2, \cos^{-1} \left(\frac{-g}{h}\right)\right), -0.3333333333333333, 1.5707963267948966\right)\right) \cdot 2} \]
7. Add Preprocessing

Alternative 5: 98.5% accurate, 1.1× speedup

Math

\[\cos \left(\mathsf{fma}\left(0.3333333333333333, \pi - \cos^{-1} \left(\frac{g}{h}\right), 2.0943951023931957\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (fma 0.3333333333333333 (- PI (acos (/ g h))) 2.0943951023931957))
  2.0))

C

double code(double g, double h) {
	return cos(fma(0.3333333333333333, (((double) M_PI) - acos((g / h))), 2.0943951023931957)) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(fma(0.3333333333333333, Float64(pi - acos(Float64(g / h))), 2.0943951023931957)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(0.3333333333333333 * N[(Pi - N[ArcCos[N[(g / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 2.0943951023931957), $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%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval 98.4%

       \[\leadsto \cos \left(\mathsf{fma}\left(\color{blue}{0.3333333333333333}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{2 \cdot \pi}{3}\right)\right) \cdot 2 \]

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 98.4%

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

3. Applied rewrites 98.4%

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

4. Evaluated real constant 98.5%

   \[\leadsto \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{2.0943951023931957}\right)\right) \cdot 2 \]
5. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   2. lift-/.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \color{blue}{\left(\frac{-g}{h}\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   3. lift-neg.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{h}\right), \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   4. distribute-frac-neg N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{g}{h}\right)\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   5. acos-neg N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   6. lift-PI.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\pi} - \cos^{-1} \left(\frac{g}{h}\right), \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   7. lower--.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\pi - \cos^{-1} \left(\frac{g}{h}\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   8. lower-acos.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{1}{3}, \pi - \color{blue}{\cos^{-1} \left(\frac{g}{h}\right)}, \frac{2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   9. lower-/.f64 98.5%

      \[\leadsto \cos \left(\mathsf{fma}\left(0.3333333333333333, \pi - \cos^{-1} \color{blue}{\left(\frac{g}{h}\right)}, 2.0943951023931957\right)\right) \cdot 2 \]

6. Applied rewrites 98.5%

   \[\leadsto \cos \left(\mathsf{fma}\left(0.3333333333333333, \color{blue}{\pi - \cos^{-1} \left(\frac{g}{h}\right)}, 2.0943951023931957\right)\right) \cdot 2 \]
7. Add Preprocessing

Alternative 6: 98.5% accurate, 1.1× speedup

Math

\[\cos \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587\right) \cdot 0.3333333333333333\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (* (- (acos (/ (- g) h)) -6.283185307179587) 0.3333333333333333))
  2.0))

C

double code(double g, double h) {
	return cos(((acos((-g / h)) - -6.283185307179587) * 0.3333333333333333)) * 2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(g, h)
use fmin_fmax_functions
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    code = cos(((acos((-g / h)) - (-6.283185307179587d0)) * 0.3333333333333333d0)) * 2.0d0
end function

Java

public static double code(double g, double h) {
	return Math.cos(((Math.acos((-g / h)) - -6.283185307179587) * 0.3333333333333333)) * 2.0;
}

Python

def code(g, h):
	return math.cos(((math.acos((-g / h)) - -6.283185307179587) * 0.3333333333333333)) * 2.0

Julia

function code(g, h)
	return Float64(cos(Float64(Float64(acos(Float64(Float64(-g) / h)) - -6.283185307179587) * 0.3333333333333333)) * 2.0)
end

MATLAB

function tmp = code(g, h)
	tmp = cos(((acos((-g / h)) - -6.283185307179587) * 0.3333333333333333)) * 2.0;
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] - -6.283185307179587), $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-*.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%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval 98.4%

       \[\leadsto \cos \left(\mathsf{fma}\left(\color{blue}{0.3333333333333333}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{2 \cdot \pi}{3}\right)\right) \cdot 2 \]

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 98.4%

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

3. Applied rewrites 98.4%

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

4. Evaluated real constant 98.5%

   \[\leadsto \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{2.0943951023931957}\right)\right) \cdot 2 \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \frac{2358079250676147}{1125899906842624}\right)} \cdot 2 \]

   2. metadata-eval N/A

      \[\leadsto \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \color{blue}{\frac{1}{3} \cdot \frac{7074237752028441}{1125899906842624}}\right) \cdot 2 \]

   3. distribute-lft-in N/A

      \[\leadsto \cos \color{blue}{\left(\frac{1}{3} \cdot \left(\cos^{-1} \left(\frac{-g}{h}\right) + \frac{7074237752028441}{1125899906842624}\right)\right)} \cdot 2 \]

   4. +-commutative N/A

      \[\leadsto \cos \left(\frac{1}{3} \cdot \color{blue}{\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right)}\right) \cdot 2 \]

   5. lift-+.f64 N/A

      \[\leadsto \cos \left(\frac{1}{3} \cdot \color{blue}{\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right)}\right) \cdot 2 \]

   6. *-commutative N/A

      \[\leadsto \cos \color{blue}{\left(\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{1}{3}\right)} \cdot 2 \]

   7. lift-*.f64 98.5%

      \[\leadsto \cos \color{blue}{\left(\left(6.283185307179587 + \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot 0.3333333333333333\right)} \cdot 2 \]

   8. lift-+.f64 N/A

      \[\leadsto \cos \left(\color{blue}{\left(\frac{7074237752028441}{1125899906842624} + \cos^{-1} \left(\frac{-g}{h}\right)\right)} \cdot \frac{1}{3}\right) \cdot 2 \]

   9. +-commutative N/A

      \[\leadsto \cos \left(\color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) + \frac{7074237752028441}{1125899906842624}\right)} \cdot \frac{1}{3}\right) \cdot 2 \]

   10. add-flip N/A

       \[\leadsto \cos \left(\color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{7074237752028441}{1125899906842624}\right)\right)\right)} \cdot \frac{1}{3}\right) \cdot 2 \]

   11. lower--.f64 N/A

       \[\leadsto \cos \left(\color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) - \left(\mathsf{neg}\left(\frac{7074237752028441}{1125899906842624}\right)\right)\right)} \cdot \frac{1}{3}\right) \cdot 2 \]

   12. metadata-eval 98.5%

       \[\leadsto \cos \left(\left(\cos^{-1} \left(\frac{-g}{h}\right) - \color{blue}{-6.283185307179587}\right) \cdot 0.3333333333333333\right) \cdot 2 \]

6. Applied rewrites 98.5%

   \[\leadsto \cos \color{blue}{\left(\left(\cos^{-1} \left(\frac{-g}{h}\right) - -6.283185307179587\right) \cdot 0.3333333333333333\right)} \cdot 2 \]
7. Add Preprocessing

Alternative 7: 98.5% accurate, 1.2× speedup

Math

\[\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 2.0943951023931957\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (fma 0.3333333333333333 (acos (/ (- g) h)) 2.0943951023931957)) 2.0))

C

double code(double g, double h) {
	return cos(fma(0.3333333333333333, acos((-g / h)), 2.0943951023931957)) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(fma(0.3333333333333333, acos(Float64(Float64(-g) / h)), 2.0943951023931957)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + 2.0943951023931957), $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%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval 98.4%

       \[\leadsto \cos \left(\mathsf{fma}\left(\color{blue}{0.3333333333333333}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{2 \cdot \pi}{3}\right)\right) \cdot 2 \]

   11. lift-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 98.4%

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

3. Applied rewrites 98.4%

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

4. Evaluated real constant 98.5%

   \[\leadsto \cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{2.0943951023931957}\right)\right) \cdot 2 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))