Ian Simplification

Percentage Accurate: 6.7% → 8.2%

Time: 12.6s

Alternatives: 7

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: 6.7% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)\\ t_1 := {\left(\frac{\mathsf{PI}\left(\right)}{-2}\right)}^{2} - {t\_0}^{2}\\ t_2 := \frac{\mathsf{PI}\left(\right)}{2} + t\_0\\ t_3 := t\_2 \cdot 2\\ t_4 := \mathsf{fma}\left(t\_1, -2, t\_2 \cdot \mathsf{PI}\left(\right)\right) \cdot t\_2\\ \frac{{t\_4}^{2} - {\left(t\_3 \cdot t\_1\right)}^{2}}{\mathsf{fma}\left(t\_3, t\_1, t\_4\right) \cdot \left({t\_2}^{2} \cdot 2\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (fma -0.5 x 0.5))))
        (t_1 (- (pow (/ (PI) -2.0) 2.0) (pow t_0 2.0)))
        (t_2 (+ (/ (PI) 2.0) t_0))
        (t_3 (* t_2 2.0))
        (t_4 (* (fma t_1 -2.0 (* t_2 (PI))) t_2)))
   (/
    (- (pow t_4 2.0) (pow (* t_3 t_1) 2.0))
    (* (fma t_3 t_1 t_4) (* (pow t_2 2.0) 2.0)))))

Derivation

1. Initial program 6.9%

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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 8.5

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

7. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2-rev N/A

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

   4. associate--r+ N/A

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

9. Applied rewrites 8.6%

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

    1. lift-/.f64 N/A

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

11. Applied rewrites 8.6%

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

Alternative 2: 8.2% accurate, 0.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.9%

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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 8.5

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

7. Applied rewrites 8.5%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. associate-*r/ N/A

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

9. Applied rewrites 6.9%

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

    1. lift-fma.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. distribute-rgt-in N/A

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

    4. associate-+l+ N/A

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

    5. *-commutative N/A

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

11. Applied rewrites 8.6%

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

Alternative 3: 8.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-asin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-PI.f64 6.8

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

5. Applied rewrites 6.8%

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

7. Applied rewrites 8.5%

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

Alternative 4: 6.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fma (asin (sqrt (* (- 1.0 x) 0.5))) -2.0 (* 0.5 (PI))))

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-asin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-PI.f64 6.8

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

5. Applied rewrites 6.8%

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

7. Applied rewrites 6.9%

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

Alternative 5: 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 6.9%

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

4. Applied rewrites 4.0%

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

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 4.2%

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

Alternative 6: 4.1% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-asin.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-PI.f64 6.8

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

5. Applied rewrites 6.8%

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

7. Applied rewrites 6.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 3.9%

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

Alternative 7: 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

real(8) function code(x)
    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 6.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

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

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 2024326 
(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))))))