Ian Simplification

Percentage Accurate: 6.8% → 8.2%

Time: 2.2min

Alternatives: 5

Speedup: 1.1×

Specification

Math

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

FPCore

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 8.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot -0.5\\ \frac{\mathsf{fma}\left({\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}\right)}^{1.5}, 8, {t\_1}^{3}\right)}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.25, 4 \cdot {t\_0}^{2} - \left(t\_1 \cdot 2\right) \cdot t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma -0.5 x 0.5)))) (t_1 (* (PI) -0.5)))
   (/
    (fma (pow (pow (acos (sqrt (fma x -0.5 0.5))) 2.0) 1.5) 8.0 (pow t_1 3.0))
    (fma (* (PI) (PI)) 0.25 (- (* 4.0 (pow t_0 2.0)) (* (* t_1 2.0) t_0))))))

Derivation

1. Initial program 7.2%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-asin.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \]

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-acos.f64 8.8

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

   12. lift-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. div-sub N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. distribute-rgt-neg-in N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-fma.f64 N/A

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

4. Applied rewrites 8.8%

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

5. Taylor expanded in x around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. associate-+r+ N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

7. Applied rewrites 8.8%

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

9. Applied rewrites 8.9%

   \[\leadsto \frac{\mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{3}, 8, {\left(\mathsf{PI}\left(\right) \cdot -0.5\right)}^{3}\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.25, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot -0.5\right)\right)\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 8.9%

    \[\leadsto \frac{\mathsf{fma}\left({\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}\right)}^{1.5}, 8, {\left(\mathsf{PI}\left(\right) \cdot -0.5\right)}^{3}\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 0.25, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot -0.5\right)\right)\right)} \]

12. Final simplification 8.9%

    \[\leadsto \frac{\mathsf{fma}\left({\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)}^{2}\right)}^{1.5}, 8, {\left(\mathsf{PI}\left(\right) \cdot -0.5\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.25, 4 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} - \left(\left(\mathsf{PI}\left(\right) \cdot -0.5\right) \cdot 2\right) \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\right)} \]
13. Add Preprocessing

Alternative 2: 8.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\\ \frac{\mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{3}, 8, {\left(\mathsf{PI}\left(\right) \cdot -0.5\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.25, \mathsf{fma}\left(t\_0, 4, \mathsf{PI}\left(\right)\right) \cdot t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma x -0.5 0.5)))))
   (/
    (fma (pow (acos (sqrt (fma -0.5 x 0.5))) 3.0) 8.0 (pow (* (PI) -0.5) 3.0))
    (fma (* (PI) (PI)) 0.25 (* (fma t_0 4.0 (PI)) t_0)))))

Derivation

1. Initial program 7.2%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-asin.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \]

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-acos.f64 8.8

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

   12. lift-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. div-sub N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. distribute-rgt-neg-in N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-fma.f64 N/A

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

4. Applied rewrites 8.8%

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

5. Taylor expanded in x around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. associate-+r+ N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

7. Applied rewrites 8.8%

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

9. Applied rewrites 8.9%

   \[\leadsto \frac{\mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{3}, 8, {\left(\mathsf{PI}\left(\right) \cdot -0.5\right)}^{3}\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.25, {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4 - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right) \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot -0.5\right)\right)\right)}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{\mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{2}\right)}\right)}^{3}, 8, {\left(\mathsf{PI}\left(\right) \cdot \frac{-1}{2}\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{1}{4}, 4 \cdot {\cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)}^{2} - -1 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos^{-1} \left(\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot x}\right)\right)\right)} \]
11. Step-by-step derivation

12. Applied rewrites 8.9%

    \[\leadsto \frac{\mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{3}, 8, {\left(\mathsf{PI}\left(\right) \cdot -0.5\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.25, \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right) \cdot \mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \mathsf{PI}\left(\right)\right)\right)} \]

13. Final simplification 8.9%

    \[\leadsto \frac{\mathsf{fma}\left({\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{3}, 8, {\left(\mathsf{PI}\left(\right) \cdot -0.5\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.25, \mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right), 4, \mathsf{PI}\left(\right)\right) \cdot \cos^{-1} \left(\sqrt{\mathsf{fma}\left(x, -0.5, 0.5\right)}\right)\right)} \]
14. Add Preprocessing

Alternative 3: 8.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_1 := \mathsf{fma}\left(t\_0, 2, \mathsf{PI}\left(\right) \cdot 0.5\right)\\ \frac{4 \cdot {t\_0}^{2}}{t\_1} - \frac{0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma -0.5 x 0.5)))) (t_1 (fma t_0 2.0 (* (PI) 0.5))))
   (- (/ (* 4.0 (pow t_0 2.0)) t_1) (/ (* 0.25 (* (PI) (PI))) t_1))))

Derivation

1. Initial program 7.2%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-asin.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \]

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-acos.f64 8.8

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

   12. lift-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. div-sub N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. distribute-rgt-neg-in N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-fma.f64 N/A

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

4. Applied rewrites 8.8%

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

5. Taylor expanded in x around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. associate-+r+ N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

7. Applied rewrites 8.8%

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

9. Applied rewrites 8.9%

   \[\leadsto \frac{{\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2} \cdot 4}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, \mathsf{PI}\left(\right) \cdot 0.5\right)} - \color{blue}{\frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, \mathsf{PI}\left(\right) \cdot 0.5\right)}} \]

10. Final simplification 8.9%

    \[\leadsto \frac{4 \cdot {\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}^{2}}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, \mathsf{PI}\left(\right) \cdot 0.5\right)} - \frac{0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), 2, \mathsf{PI}\left(\right) \cdot 0.5\right)} \]
11. Add Preprocessing

Alternative 4: 8.2% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.2%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-asin.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \]

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-acos.f64 8.8

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

   12. lift-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. div-sub N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. distribute-rgt-neg-in N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-fma.f64 N/A

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

4. Applied rewrites 8.8%

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

5. Taylor expanded in x around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. associate-+r+ N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

7. Applied rewrites 8.8%

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

8. Final simplification 8.8%

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

Alternative 5: 5.3% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.2%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-asin.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)} \]

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-acos.f64 8.8

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

   12. lift-/.f64 N/A

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

   13. lift--.f64 N/A

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

   14. div-sub N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. distribute-rgt-neg-in N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-fma.f64 N/A

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

4. Applied rewrites 8.8%

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

5. Taylor expanded in x around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. associate-+r+ N/A

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

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. neg-mul-1 N/A

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

7. Applied rewrites 8.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.4%

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

11. Final simplification 5.4%

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

Developer Target 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asin x))

C

double code(double x) {
	return asin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = asin(x)
end function

Java

public static double code(double x) {
	return Math.asin(x);
}

Python

def code(x):
	return math.asin(x)

Julia

function code(x)
	return asin(x)
end

MATLAB

function tmp = code(x)
	tmp = asin(x);
end

Wolfram

code[x_] := N[ArcSin[x], $MachinePrecision]

Reproduce

herbie shell --seed 2024249 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (! :herbie-platform default (asin x))

  (- (/ (PI) 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))