Ian Simplification

Percentage Accurate: 7.0% → 8.5%

Time: 6.8s

Alternatives: 4

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: 8.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} - \left(\sin^{-1} \left(\frac{\sqrt{1 - x}}{\sqrt{2}}\right) \cdot 2\right) \cdot \left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \cdot 2\right)}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right), 2, 0.5 \cdot \mathsf{PI}\left(\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (/ (* (PI) (PI)) 4.0)
   (*
    (* (asin (/ (sqrt (- 1.0 x)) (sqrt 2.0))) 2.0)
    (* (asin (sqrt (/ (- 1.0 x) 2.0))) 2.0)))
  (fma (asin (sqrt (* 0.5 (- 1.0 x)))) 2.0 (* 0.5 (PI)))))

Derivation

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
3. Step-by-step derivation

   1. flip-- N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 7.2%

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

   1. sqrt-div N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-sqrt.f64 8.6

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

6. Applied rewrites 8.6%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lower-asin.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 8.6

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

9. Applied rewrites 8.6%

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

    1. frac-times N/A

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

    2. unpow2 N/A

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

    3. metadata-eval N/A

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

    4. lower-/.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 N/A

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

    7. lower-PI.f64 N/A

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

    8. lower-PI.f64 8.6

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

11. Applied rewrites 8.6%

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

Alternative 2: 8.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

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
3. Step-by-step derivation

   1. asin-acos N/A

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

   2. lower--.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-acos.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 8.5

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

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. asin-acos-rev N/A

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

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

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

   3. lower-fma.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. sqrt-prod N/A

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

   11. lower-acos.f64 N/A

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

7. Applied rewrites 8.5%

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

Alternative 3: 7.0% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

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

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

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

   2. lower-fma.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-asin.f64 N/A

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

   7. sqrt-unprod N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 7.1

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

5. Applied rewrites 7.1%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 7.1

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

8. Applied rewrites 7.1%

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

Alternative 4: 4.1% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

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

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

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

   2. lower-fma.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-asin.f64 N/A

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

   7. sqrt-unprod N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 7.1

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

5. Applied rewrites 7.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 4.2%

   \[\leadsto \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), -2 \cdot \sin^{-1} \left(\sqrt{0.5}\right)\right) \]
9. 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 2025044 
(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))))))