Ian Simplification

Percentage Accurate: 7.0% → 8.4%

Time: 11.3s

Alternatives: 8

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: 7.0% 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: 8.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{PI}\left(\right)\\ t_1 := \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\\ t_2 := t\_0 - t\_1\\ \frac{\mathsf{fma}\left({t\_2}^{3}, -8, {t\_0}^{3}\right)}{\mathsf{fma}\left({\left(0.5 \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} - t\_1\right)}^{2}, 4, t\_0 \cdot \left(t\_0 - t\_2 \cdot -2\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.5 (PI)))
        (t_1 (acos (sqrt (* (- 1.0 x) 0.5))))
        (t_2 (- t_0 t_1)))
   (/
    (fma (pow t_2 3.0) -8.0 (pow t_0 3.0))
    (fma
     (pow (- (* 0.5 (cbrt (pow (PI) 3.0))) t_1) 2.0)
     4.0
     (* t_0 (- t_0 (* t_2 -2.0)))))))

Derivation

1. Initial program 7.0%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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. lower--.f64 N/A

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

   6. lower-acos.f64 8.5

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

4. Applied rewrites 8.5%

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

5. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-acos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-PI.f64 8.4

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

7. Applied rewrites 8.4%

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

9. Applied rewrites 8.5%

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

11. Applied rewrites 8.5%

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

Alternative 2: 8.5% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.5 (PI))) (t_1 (- t_0 (acos (sqrt (* (- 1.0 x) 0.5))))))
   (/
    (fma (pow t_1 3.0) -8.0 (pow t_0 3.0))
    (fma (pow t_1 2.0) 4.0 (* t_0 (- t_0 (* t_1 -2.0)))))))

Derivation

1. Initial program 7.0%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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. lower--.f64 N/A

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

   6. lower-acos.f64 8.5

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

4. Applied rewrites 8.5%

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

5. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-acos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-PI.f64 8.4

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

7. Applied rewrites 8.4%

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

9. Applied rewrites 8.5%

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

Alternative 3: 8.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.5\\ t_1 := \cos^{-1} \left(\sqrt{\left(1 - x\right) \cdot 0.5}\right)\\ t_2 := 0.5 \cdot \mathsf{PI}\left(\right)\\ t_3 := t\_0 - t\_1\\ \frac{\mathsf{fma}\left({\left(t\_2 - t\_1\right)}^{3}, -8, {t\_2}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, 2 \cdot t\_3\right), t\_0, 4 \cdot {t\_3}^{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (PI) 0.5))
        (t_1 (acos (sqrt (* (- 1.0 x) 0.5))))
        (t_2 (* 0.5 (PI)))
        (t_3 (- t_0 t_1)))
   (/
    (fma (pow (- t_2 t_1) 3.0) -8.0 (pow t_2 3.0))
    (fma (fma (PI) 0.5 (* 2.0 t_3)) t_0 (* 4.0 (pow t_3 2.0))))))

Derivation

1. Initial program 7.0%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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. lower--.f64 N/A

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

   6. lower-acos.f64 8.5

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

4. Applied rewrites 8.5%

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

5. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-acos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-PI.f64 8.4

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

7. Applied rewrites 8.4%

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

9. Applied rewrites 8.5%

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

11. Applied rewrites 8.5%

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

Alternative 4: 8.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.5 (PI))))
   (fma (- t_0 (acos (sqrt (* (- 1.0 x) 0.5)))) -2.0 t_0)))

Derivation

1. Initial program 7.0%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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. lower--.f64 N/A

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

   6. lower-acos.f64 8.5

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

4. Applied rewrites 8.5%

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

5. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-acos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-PI.f64 8.4

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

7. Applied rewrites 8.4%

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

9. Applied rewrites 8.5%

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

Alternative 5: 5.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(\sqrt{0.5}\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -2, \frac{\mathsf{PI}\left(\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 2, \frac{\mathsf{PI}\left(\right)}{-2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (sqrt 0.5))))
   (if (<= x -5e-310)
     (fma t_0 -2.0 (/ (PI) 2.0))
     (fma t_0 2.0 (/ (PI) -2.0)))))

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 7.0%

      \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
   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. lower--.f64 N/A

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

      6. lower-acos.f64 7.3

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

   4. Applied rewrites 7.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 5.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 5.6

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. asin-acos-rev N/A

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

      13. lift-asin.f64 5.5

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

   9. Applied rewrites 5.5%

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

   if -4.999999999999985e-310 < x

   1. Initial program 7.1%

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

   4. Applied rewrites 5.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{2}}}\right), 2, \frac{\mathsf{PI}\left(\right)}{-2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 5.9%

      \[\leadsto \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\color{blue}{0.5}}\right), 2, \frac{\mathsf{PI}\left(\right)}{-2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 7.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fma (asin (sqrt (fma -0.5 x 0.5))) -2.0 (/ (PI) 2.0)))

Derivation

1. Initial program 7.0%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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. lower--.f64 N/A

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

   6. lower-acos.f64 8.5

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

4. Applied rewrites 8.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 5.4%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 5.4

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-acos.f64 N/A

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

   12. asin-acos-rev N/A

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

   13. lift-asin.f64 4.0

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

9. Applied rewrites 4.0%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 7.0

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

12. Applied rewrites 7.0%

    \[\leadsto \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, x, 0.5\right)}}\right), -2, \frac{\mathsf{PI}\left(\right)}{2}\right) \]
13. Add Preprocessing

Alternative 7: 4.1% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fma (asin (sqrt 0.5)) -2.0 (/ (PI) 2.0)))

Derivation

1. Initial program 7.0%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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. lower--.f64 N/A

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

   6. lower-acos.f64 8.5

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

4. Applied rewrites 8.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 5.4%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 5.4

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-acos.f64 N/A

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

   12. asin-acos-rev N/A

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

   13. lift-asin.f64 4.0

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

9. Applied rewrites 4.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5}\right), -2, \frac{\mathsf{PI}\left(\right)}{2}\right)} \]
10. Add Preprocessing

Alternative 8: 0.0% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \frac{0}{0} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.0 0.0))

C

double code(double x) {
	return 0.0 / 0.0;
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.0d0 / 0.0d0
end function

Java

public static double code(double x) {
	return 0.0 / 0.0;
}

Python

def code(x):
	return 0.0 / 0.0

Julia

function code(x)
	return Float64(0.0 / 0.0)
end

MATLAB

function tmp = code(x)
	tmp = 0.0 / 0.0;
end

Wolfram

code[x_] := N[(0.0 / 0.0), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

4. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\frac{0}{0}} \]
5. 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

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(x)
use fmin_fmax_functions
    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 2024346 
(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))))))