2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 3.9s

Alternatives: 1

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

3. Applied rewrites 98.5%

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

4. Taylor expanded in g around 0

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

6. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-out N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-*.f64 100.0

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

8. Applied rewrites 100.0%

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

Reproduce

herbie shell --seed 2025143 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))