Ian Simplification

Percentage Accurate: 7.0% → 8.5%

Time: 7.0s

Alternatives: 2

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 7.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))

C

double code(double x) {
	return (((double) M_PI) / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
}

Java

public static double code(double x) {
	return (Math.PI / 2.0) - (2.0 * Math.asin(Math.sqrt(((1.0 - x) / 2.0))));
}

Python

def code(x):
	return (math.pi / 2.0) - (2.0 * math.asin(math.sqrt(((1.0 - x) / 2.0))))

Julia

function code(x)
	return Float64(Float64(pi / 2.0) - Float64(2.0 * asin(sqrt(Float64(Float64(1.0 - x) / 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi / 2.0) - (2.0 * asin(sqrt(((1.0 - x) / 2.0))));
end

Wolfram

code[x_] := N[(N[(Pi / 2.0), $MachinePrecision] - N[(2.0 * N[ArcSin[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 8.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, -0.5 \cdot \pi\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (acos (sqrt (fma -0.5 x 0.5))) 2.0 (* -0.5 PI)))

C

double code(double x) {
	return fma(acos(sqrt(fma(-0.5, x, 0.5))), 2.0, (-0.5 * ((double) M_PI)));
}

Julia

function code(x)
	return fma(acos(sqrt(fma(-0.5, x, 0.5))), 2.0, Float64(-0.5 * pi))
end

Wolfram

code[x_] := N[(N[ArcCos[N[Sqrt[N[(-0.5 * x + 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0 + N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

3. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. sub-flip N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-lft-neg-out N/A

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

   9. remove-double-neg N/A

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

   10. sub-flip N/A

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

   11. lift-/.f64 N/A

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

   12. mult-flip N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. lift-PI.f64 N/A

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

   16. add-log-exp N/A

       \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2 + \left(\frac{1}{2} \cdot \color{blue}{\log \left(e^{\mathsf{PI}\left(\right)}\right)} + \left(\mathsf{neg}\left(\pi\right)\right)\right) \]

   17. log-pow-rev N/A

       \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2 + \left(\color{blue}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)} + \left(\mathsf{neg}\left(\pi\right)\right)\right) \]

   18. lift-PI.f64 N/A

       \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2 + \left(\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   19. add-log-exp N/A

       \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2 + \left(\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\log \left(e^{\mathsf{PI}\left(\right)}\right)}\right)\right)\right) \]

5. Applied rewrites 8.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, -0.5 \cdot \pi\right)} \]
6. Add Preprocessing

Alternative 2: 5.3% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fma (acos (sqrt 0.5)) 2.0 (* -0.5 PI)))

C

double code(double x) {
	return fma(acos(sqrt(0.5)), 2.0, (-0.5 * ((double) M_PI)));
}

Julia

function code(x)
	return fma(acos(sqrt(0.5)), 2.0, Float64(-0.5 * pi))
end

Wolfram

code[x_] := N[(N[ArcCos[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * 2.0 + N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

   \[\frac{\pi}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

3. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. sub-flip N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-lft-neg-out N/A

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

   9. remove-double-neg N/A

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

   10. sub-flip N/A

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

   11. lift-/.f64 N/A

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

   12. mult-flip N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. lift-PI.f64 N/A

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

   16. add-log-exp N/A

       \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2 + \left(\frac{1}{2} \cdot \color{blue}{\log \left(e^{\mathsf{PI}\left(\right)}\right)} + \left(\mathsf{neg}\left(\pi\right)\right)\right) \]

   17. log-pow-rev N/A

       \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2 + \left(\color{blue}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)} + \left(\mathsf{neg}\left(\pi\right)\right)\right) \]

   18. lift-PI.f64 N/A

       \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2 + \left(\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   19. add-log-exp N/A

       \[\leadsto \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, \frac{-1}{2}, \frac{1}{2}\right)}\right) \cdot 2 + \left(\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\log \left(e^{\mathsf{PI}\left(\right)}\right)}\right)\right)\right) \]

5. Applied rewrites 8.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, -0.5 \cdot \pi\right)} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 5.3%

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

Reproduce

herbie shell --seed 2025156 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64
  (- (/ PI 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))