Ian Simplification

Percentage Accurate: 7.0% → 8.4%

Time: 11.9s

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: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 8.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1 - x}{2}}\\ \left(\frac{\mathsf{PI}\left(\right)}{2} - \left(\sin^{-1} t\_0 - \frac{\mathsf{PI}\left(\right)}{-2}\right)\right) + \cos^{-1} t\_0 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- 1.0 x) 2.0))))
   (+ (- (/ (PI) 2.0) (- (asin t_0) (/ (PI) -2.0))) (acos t_0))))

Derivation

1. Initial program 7.2%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 9.0%

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

Alternative 2: 8.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.2%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 7.1

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

5. Applied rewrites 7.1%

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

7. Applied rewrites 9.0%

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

Alternative 3: 7.0% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.2%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 7.1

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

5. Applied rewrites 7.1%

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

7. Applied rewrites 7.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5 \cdot \left(1 - x\right)}\right), -2, 0.5 \cdot \mathsf{PI}\left(\right)\right)} \]
8. 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 7.2%

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

4. Applied rewrites 3.8%

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

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 4.3%

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

Alternative 5: 4.1% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.2%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
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 7.1

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

5. Applied rewrites 7.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 3.8%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 3.8%

    \[\leadsto \mathsf{fma}\left(-2, \sin^{-1} \left(\sqrt{0.5}\right), \mathsf{PI}\left(\right) \cdot 0.5\right) \]
12. 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

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

Java

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

Python

def code(x):
	return 0.0 / 0.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

4. Applied rewrites 0.0%

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

Developer Target 1: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asin x))

C

double code(double x) {
	return asin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = asin(x)
end function

Java

public static double code(double x) {
	return Math.asin(x);
}

Python

def code(x):
	return math.asin(x)

Julia

function code(x)
	return asin(x)
end

MATLAB

function tmp = code(x)
	tmp = asin(x);
end

Wolfram

code[x_] := N[ArcSin[x], $MachinePrecision]

Reproduce

herbie shell --seed 2024337 
(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))))))