2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 2.5s

Alternatives: 2

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

3. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.5

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

5. Applied rewrites 99.9%

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

   1. lift-PI.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. metadata-eval 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 1.1666666666666667\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin (fma 0.3333333333333333 (acos (/ (- g) h)) (* PI 1.1666666666666667)))
  2.0))

C

double code(double g, double h) {
	return sin(fma(0.3333333333333333, acos((-g / h)), (((double) M_PI) * 1.1666666666666667))) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(0.3333333333333333, acos(Float64(Float64(-g) / h)), Float64(pi * 1.1666666666666667))) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + N[(Pi * 1.1666666666666667), $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. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-/.f64 N/A

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

   8. division-flip N/A

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

   9. associate-/r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

3. Applied rewrites 97.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 97.5

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

   4. lift-PI.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. metadata-eval 97.6

       \[\leadsto \sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot \color{blue}{1.1666666666666667}\right)\right) \cdot 2 \]

5. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \pi \cdot 1.1666666666666667\right)\right) \cdot 2} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025121 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))