Ian Simplification

Percentage Accurate: 6.8% → 8.3%

Time: 11.4s

Alternatives: 6

Speedup: 1.1×

Specification

Math

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

FPCore

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 8.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.7%

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

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

   4. lift-asin.f64 N/A

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

   5. asin-acos N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-+r- N/A

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

   9. associate--r- N/A

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

   10. lower-+.f64 N/A

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

4. Applied rewrites 7.8%

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

5. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sqrt.f64 7.1

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

7. Applied rewrites 7.1%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--r- N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-asin.f64 N/A

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

   9. acos-asin-rev N/A

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

   10. lower-acos.f64 7.2

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

9. Applied rewrites 7.2%

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

10. Taylor expanded in x around 0

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

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

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

    2. *-commutative N/A

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

    3. *-lft-identity N/A

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

    4. metadata-eval N/A

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

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

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

    6. *-commutative N/A

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

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

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

    8. +-commutative N/A

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

12. Applied rewrites 7.8%

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

Alternative 2: 8.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.7%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-acos.f64 7.8

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

4. Applied rewrites 7.8%

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

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 7.8

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

7. Applied rewrites 7.8%

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

9. Applied rewrites 7.8%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 7.8%

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

Alternative 3: 5.4% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.7%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-acos.f64 7.8

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

4. Applied rewrites 7.8%

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

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 7.8

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

7. Applied rewrites 7.8%

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

9. Applied rewrites 7.8%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 5.8%

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

Alternative 4: 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 5.7%

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

4. Applied rewrites 3.9%

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

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 4.0%

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

Alternative 5: 4.0% 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 5.7%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-acos.f64 7.8

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

4. Applied rewrites 7.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 5.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 5.8

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-acos.f64 N/A

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

   12. asin-acos-rev N/A

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

   13. lift-asin.f64 3.9

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

9. Applied rewrites 3.9%

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

Alternative 6: 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 5.7%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 2024363 
(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))))))