Ian Simplification

Percentage Accurate: 6.7% → 8.1%

Time: 13.5s

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: 8.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_1 := {t\_0}^{2}\\ t_2 := \sqrt{\frac{1 - x}{2}}\\ t_3 := t\_0 + \sin^{-1} t\_2\\ t_4 := \cos^{-1} t\_2\\ t_5 := \mathsf{fma}\left(t\_4, t\_4 + t\_0, t\_1\right)\\ \frac{\left(t\_1 - {\left(t\_0 - t\_4\right)}^{2}\right) \cdot t\_5 - t\_3 \cdot \left({t\_0}^{3} - {t\_4}^{3}\right)}{t\_3 \cdot t\_5} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (PI) 2.0))
        (t_1 (pow t_0 2.0))
        (t_2 (sqrt (/ (- 1.0 x) 2.0)))
        (t_3 (+ t_0 (asin t_2)))
        (t_4 (acos t_2))
        (t_5 (fma t_4 (+ t_4 t_0) t_1)))
   (/
    (-
     (* (- t_1 (pow (- t_0 t_4) 2.0)) t_5)
     (* t_3 (- (pow t_0 3.0) (pow t_4 3.0))))
    (* t_3 t_5))))

Derivation

1. Initial program 9.0%

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

   1. lift-PI.f64 N/A

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

   2. add-sqr-sqrt N/A

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

   3. lower-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-sqrt.f64 8.7

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

4. Applied rewrites 8.7%

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

6. Applied rewrites 8.9%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-acos.f64 N/A

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

   6. lower--.f64 10.5

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

8. Applied rewrites 10.5%

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

Alternative 2: 8.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1 - x}{2}}\\ t_1 := \sin^{-1} t\_0\\ t_2 := \cos^{-1} t\_0\\ t_3 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_4 := {t\_3}^{2}\\ t_5 := t\_4 - {t\_1}^{2}\\ \frac{t\_2 \cdot t\_5 + \left(t\_5 \cdot t\_3 - \left(t\_4 - {t\_2}^{2}\right) \cdot \left(t\_1 + t\_3\right)\right)}{\left(t\_3 + t\_1\right) \cdot \left(t\_2 + t\_3\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- 1.0 x) 2.0)))
        (t_1 (asin t_0))
        (t_2 (acos t_0))
        (t_3 (/ (PI) 2.0))
        (t_4 (pow t_3 2.0))
        (t_5 (- t_4 (pow t_1 2.0))))
   (/
    (+ (* t_2 t_5) (- (* t_5 t_3) (* (- t_4 (pow t_2 2.0)) (+ t_1 t_3))))
    (* (+ t_3 t_1) (+ t_2 t_3)))))

Derivation

1. Initial program 9.0%

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

   1. lift-PI.f64 N/A

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

   2. add-sqr-sqrt N/A

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

   3. lower-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-sqrt.f64 8.7

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

4. Applied rewrites 8.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. associate--r+ N/A

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

   11. flip-- N/A

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

   12. lift-asin.f64 N/A

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

   13. asin-acos-rev N/A

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

   14. lift-acos.f64 N/A

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

6. Applied rewrites 10.5%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-lft-in N/A

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

8. Applied rewrites 10.5%

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

Alternative 3: 8.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_1 := {t\_0}^{2}\\ t_2 := \sqrt{\frac{1 - x}{2}}\\ t_3 := \cos^{-1} t\_2\\ t_4 := t\_3 + t\_0\\ t_5 := \sin^{-1} t\_2\\ t_6 := t\_0 + t\_5\\ \frac{\left(t\_1 - {t\_5}^{2}\right) \cdot t\_4 - t\_6 \cdot \left(t\_1 - {t\_3}^{2}\right)}{t\_6 \cdot t\_4} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (PI) 2.0))
        (t_1 (pow t_0 2.0))
        (t_2 (sqrt (/ (- 1.0 x) 2.0)))
        (t_3 (acos t_2))
        (t_4 (+ t_3 t_0))
        (t_5 (asin t_2))
        (t_6 (+ t_0 t_5)))
   (/
    (- (* (- t_1 (pow t_5 2.0)) t_4) (* t_6 (- t_1 (pow t_3 2.0))))
    (* t_6 t_4))))

Derivation

1. Initial program 9.0%

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

   1. lift-PI.f64 N/A

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

   2. add-sqr-sqrt N/A

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

   3. lower-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-sqrt.f64 8.7

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

4. Applied rewrites 8.7%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. associate--r+ N/A

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

   11. flip-- N/A

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

   12. lift-asin.f64 N/A

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

   13. asin-acos-rev N/A

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

   14. lift-acos.f64 N/A

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

6. Applied rewrites 10.5%

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

Alternative 4: 8.2% 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 9.0%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 10.5

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

4. Applied rewrites 10.5%

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

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 10.4

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

7. Applied rewrites 10.4%

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

9. Applied rewrites 10.5%

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

Alternative 5: 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 9.0%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 8.8

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

5. Applied rewrites 8.8%

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

7. Applied rewrites 9.0%

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

Alternative 6: 4.2% 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 9.0%

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

4. Applied rewrites 3.7%

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

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 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

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 9.0%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 2024358 
(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))))))