bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.4%

Time: 8.1s

Alternatives: 10

Speedup: 1.0×

Specification

\[0 \leq x \land x \leq 0.5\]

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Initial Program: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- 1.0 x)))

C

double code(double x) {
	return acos((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos((1.0d0 - x))
end function

Java

public static double code(double x) {
	return Math.acos((1.0 - x));
}

Python

def code(x):
	return math.acos((1.0 - x))

Julia

function code(x)
	return acos(Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = acos((1.0 - x));
end

Wolfram

code[x_] := N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_1 := \cos^{-1} \left(1 - x\right)\\ \frac{\mathsf{fma}\left(t\_0, t\_0 \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(t\_1 \cdot \left(t\_1 - \frac{\mathsf{PI}\left(\right)}{-2}\right), \mathsf{PI}\left(\right), -2 \cdot \left({t\_0}^{3} - {t\_1}^{3}\right)\right)\right)}{2 \cdot \left({\left(\frac{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}{2}\right)}^{2} + \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_1\right) \cdot t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (PI) 2.0)) (t_1 (acos (- 1.0 x))))
   (/
    (fma
     t_0
     (* t_0 (PI))
     (fma
      (* t_1 (- t_1 (/ (PI) -2.0)))
      (PI)
      (* -2.0 (- (pow t_0 3.0) (pow t_1 3.0)))))
    (*
     2.0
     (+ (pow (/ (cbrt (pow (PI) 3.0)) 2.0) 2.0) (* (fma 0.5 (PI) t_1) t_1))))))

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

   2. acos-asin N/A

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

   3. asin-acos N/A

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

   4. lift-acos.f64 N/A

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

   5. flip3-- N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 7.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

6. Applied rewrites 10.6%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower-acos.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-acos.f64 N/A

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

   17. lower--.f64 10.6

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{2}, \frac{\mathsf{PI}\left(\right)}{2} \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(\cos^{-1} \left(1 - x\right) \cdot \left(\cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right), \mathsf{PI}\left(\right), -2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\cos^{-1} \left(1 - x\right)}^{3}\right)\right)\right)}{2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} + \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \color{blue}{\left(1 - x\right)}\right)} \]

9. Applied rewrites 10.6%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{2}, \frac{\mathsf{PI}\left(\right)}{2} \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(\cos^{-1} \left(1 - x\right) \cdot \left(\cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right), \mathsf{PI}\left(\right), -2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\cos^{-1} \left(1 - x\right)}^{3}\right)\right)\right)}{2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} + \color{blue}{\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)}\right)} \]
10. Step-by-step derivation

    1. unpow1 N/A

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

    2. metadata-eval N/A

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

    3. pow-pow N/A

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

    4. pow1/3 N/A

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

    5. lower-cbrt.f64 N/A

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

    6. lower-pow.f64 10.6

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{2}, \frac{\mathsf{PI}\left(\right)}{2} \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(\cos^{-1} \left(1 - x\right) \cdot \left(\cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right), \mathsf{PI}\left(\right), -2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\cos^{-1} \left(1 - x\right)}^{3}\right)\right)\right)}{2 \cdot \left({\left(\frac{\sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}}{2}\right)}^{2} + \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)\right)} \]

11. Applied rewrites 10.6%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{2}, \frac{\mathsf{PI}\left(\right)}{2} \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(\cos^{-1} \left(1 - x\right) \cdot \left(\cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right), \mathsf{PI}\left(\right), -2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\cos^{-1} \left(1 - x\right)}^{3}\right)\right)\right)}{2 \cdot \left({\left(\frac{\color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}}{2}\right)}^{2} + \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)\right)} \]
12. Add Preprocessing

Alternative 2: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_1 := \cos^{-1} \left(1 - x\right)\\ \frac{\mathsf{fma}\left(t\_0, t\_0 \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(t\_1 \cdot \left(t\_1 - \frac{\mathsf{PI}\left(\right)}{-2}\right), \mathsf{PI}\left(\right), -2 \cdot \left({t\_0}^{3} - {t\_1}^{3}\right)\right)\right)}{2 \cdot \left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} + \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_1\right) \cdot t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (PI) 2.0)) (t_1 (acos (- 1.0 x))))
   (/
    (fma
     t_0
     (* t_0 (PI))
     (fma
      (* t_1 (- t_1 (/ (PI) -2.0)))
      (PI)
      (* -2.0 (- (pow t_0 3.0) (pow t_1 3.0)))))
    (* 2.0 (+ (/ (* (PI) (PI)) 4.0) (* (fma 0.5 (PI) t_1) t_1))))))

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

   2. acos-asin N/A

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

   3. asin-acos N/A

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

   4. lift-acos.f64 N/A

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

   5. flip3-- N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 7.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

6. Applied rewrites 10.6%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

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

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower-acos.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-acos.f64 N/A

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

   17. lower--.f64 10.6

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{2}, \frac{\mathsf{PI}\left(\right)}{2} \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(\cos^{-1} \left(1 - x\right) \cdot \left(\cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right), \mathsf{PI}\left(\right), -2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\cos^{-1} \left(1 - x\right)}^{3}\right)\right)\right)}{2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} + \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \color{blue}{\left(1 - x\right)}\right)} \]

9. Applied rewrites 10.6%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{2}, \frac{\mathsf{PI}\left(\right)}{2} \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(\cos^{-1} \left(1 - x\right) \cdot \left(\cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right), \mathsf{PI}\left(\right), -2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\cos^{-1} \left(1 - x\right)}^{3}\right)\right)\right)}{2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} + \color{blue}{\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)}\right)} \]
10. Step-by-step derivation

    1. lift-pow.f64 N/A

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

    2. unpow2 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. frac-times N/A

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

    6. lift-*.f64 N/A

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

    7. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{2}, \frac{\mathsf{PI}\left(\right)}{2} \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(\cos^{-1} \left(1 - x\right) \cdot \left(\cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right), \mathsf{PI}\left(\right), -2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\cos^{-1} \left(1 - x\right)}^{3}\right)\right)\right)}{2 \cdot \left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\color{blue}{4}} + \mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)\right)} \]

    8. lower-/.f64 10.6

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{2}, \frac{\mathsf{PI}\left(\right)}{2} \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(\cos^{-1} \left(1 - x\right) \cdot \left(\cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right), \mathsf{PI}\left(\right), -2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\cos^{-1} \left(1 - x\right)}^{3}\right)\right)\right)}{2 \cdot \left(\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4}} + \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)\right)} \]

11. Applied rewrites 10.6%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{2}, \frac{\mathsf{PI}\left(\right)}{2} \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(\cos^{-1} \left(1 - x\right) \cdot \left(\cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right), \mathsf{PI}\left(\right), -2 \cdot \left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\cos^{-1} \left(1 - x\right)}^{3}\right)\right)\right)}{2 \cdot \left(\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4}} + \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right) \cdot \cos^{-1} \left(1 - x\right)\right)} \]
12. Add Preprocessing

Alternative 3: 10.4% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))
        (t_1 (/ (PI) -2.0))
        (t_2 (fma (- t_0 t_1) t_0 (pow t_1 2.0))))
   (fma t_2 (/ (PI) (* t_2 2.0)) (- (asin (- 1.0 x))))))

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

   2. acos-asin N/A

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

   3. asin-acos N/A

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

   4. lift-acos.f64 N/A

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

   5. flip3-- N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 7.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

6. Applied rewrites 10.6%

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

8. Applied rewrites 10.6%

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

9. Final simplification 10.6%

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

Alternative 4: 10.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ \frac{\left(\left(t\_0 \cdot t\_0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25 - {\left(\sqrt{t\_1}\right)}^{4}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))) (t_1 (asin (- 1.0 x))))
   (/
    (- (* (* (* t_0 t_0) (PI)) 0.25) (pow (sqrt t_1) 4.0))
    (fma (PI) 0.5 t_1))))

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

   2. acos-asin N/A

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

   3. flip-- N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-asin.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-PI.f64 7.2

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

4. Applied rewrites 7.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 7.2%

   \[\leadsto \color{blue}{\frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25 - {\sin^{-1} \left(1 - x\right)}^{2}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 10.5%

   \[\leadsto \frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25 - {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{4}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 10.5%

    \[\leadsto \frac{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25 - {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{4}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
12. Add Preprocessing

Alternative 5: 10.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25 - {\left(\sqrt{t\_0}\right)}^{4}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/ (- (* (* (PI) (PI)) 0.25) (pow (sqrt t_0) 4.0)) (fma (PI) 0.5 t_0))))

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

   2. acos-asin N/A

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

   3. flip-- N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-asin.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-PI.f64 7.2

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

4. Applied rewrites 7.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 7.2%

   \[\leadsto \color{blue}{\frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25 - {\sin^{-1} \left(1 - x\right)}^{2}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 10.5%

   \[\leadsto \frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25 - {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{4}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \sin^{-1} \left(1 - x\right)\right)} \]
10. Add Preprocessing

Alternative 6: 10.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} - \sin^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.5 (cbrt (pow (PI) 3.0))) (asin (- 1.0 x))))

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

   2. acos-asin N/A

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

   3. flip-- N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-asin.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-PI.f64 7.2

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

4. Applied rewrites 7.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 7.2%

   \[\leadsto \color{blue}{\frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25 - {\sin^{-1} \left(1 - x\right)}^{2}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 7.2%

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

11. Applied rewrites 10.5%

    \[\leadsto 0.5 \cdot \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} - \sin^{-1} \left(1 - \color{blue}{x}\right) \]
12. Add Preprocessing

Alternative 7: 9.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot t\_0\right) \cdot t\_0 - \sin^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (if (<= (acos (- 1.0 x)) 0.0)
     (acos (- x))
     (- (* (* 0.5 t_0) t_0) (asin (- 1.0 x))))))

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \cos^{-1} \color{blue}{\left(-1 \cdot x\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. lower-neg.f64 6.5

         \[\leadsto \cos^{-1} \color{blue}{\left(-x\right)} \]

   5. Applied rewrites 6.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(-x\right)} \]

   if 0.0 < (acos.f64 (-.f64 #s(literal 1 binary64) x))

   1. Initial program 74.1%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-acos.f64 N/A

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      2. acos-asin N/A

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

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-asin.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-asin.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-PI.f64 74.0

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25 - {\sin^{-1} \left(1 - x\right)}^{2}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \sin^{-1} \left(1 - x\right)\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 74.1%

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

   11. Applied rewrites 74.3%

       \[\leadsto \left(0.5 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)} - \sin^{-1} \color{blue}{\left(1 - x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 9.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))) (if (<= t_0 0.0) (acos (- x)) t_0)))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = acos(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos((1.0d0 - x))
    if (t_0 <= 0.0d0) then
        tmp = acos(-x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.acos(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.acos(-x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = acos(Float64(-x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = acos(-x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[ArcCos[(-x)], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (-.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 3.9%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \cos^{-1} \color{blue}{\left(-1 \cdot x\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. lower-neg.f64 6.5

         \[\leadsto \cos^{-1} \color{blue}{\left(-x\right)} \]

   5. Applied rewrites 6.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(-x\right)} \]

   if 0.0 < (acos.f64 (-.f64 #s(literal 1 binary64) x))

   1. Initial program 74.1%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 6.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(-x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acos (- x)))

C

double code(double x) {
	return acos(-x);
}

Fortran

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

Java

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

Python

def code(x):
	return math.acos(-x)

Julia

function code(x)
	return acos(Float64(-x))
end

MATLAB

function tmp = code(x)
	tmp = acos(-x);
end

Wolfram

code[x_] := N[ArcCos[(-x)], $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \cos^{-1} \color{blue}{\left(-1 \cdot x\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   2. lower-neg.f64 6.9

      \[\leadsto \cos^{-1} \color{blue}{\left(-x\right)} \]

5. Applied rewrites 6.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(-x\right)} \]
6. Add Preprocessing

Alternative 10: 3.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (acos 1.0))

C

double code(double x) {
	return acos(1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = acos(1.0d0)
end function

Java

public static double code(double x) {
	return Math.acos(1.0);
}

Python

def code(x):
	return math.acos(1.0)

Julia

function code(x)
	return acos(1.0)
end

MATLAB

function tmp = code(x)
	tmp = acos(1.0);
end

Wolfram

code[x_] := N[ArcCos[1.0], $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\cos^{-1} \left(1 - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \cos^{-1} \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 2.0 * asin(sqrt((x / 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * asin(sqrt((x / 2.0d0)))
end function

Java

public static double code(double x) {
	return 2.0 * Math.asin(Math.sqrt((x / 2.0)));
}

Python

def code(x):
	return 2.0 * math.asin(math.sqrt((x / 2.0)))

Julia

function code(x)
	return Float64(2.0 * asin(sqrt(Float64(x / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * asin(sqrt((x / 2.0)));
end

Wolfram

code[x_] := N[(2.0 * N[ArcSin[N[Sqrt[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024343 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :alt
  (! :herbie-platform default (* 2 (asin (sqrt (/ x 2)))))

  (acos (- 1.0 x)))