bug323 (missed optimization)

Percentage Accurate: 7.1% → 10.6%

Time: 7.2s

Alternatives: 10

Speedup: 0.5×

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: 7.1% 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: 10.6% accurate, 0.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
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. acos-asin N/A

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

   5. flip-- N/A

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

   6. frac-sub N/A

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

   7. frac-sub N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 6.0%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f64 N/A

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

   7. add-sqr-sqrt N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

6. Applied rewrites 9.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 9.5

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

8. Applied rewrites 9.5%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. pow1/3 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-pow.f64 9.5

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

10. Applied rewrites 9.5%

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

Alternative 2: 10.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(t\_0, t\_0 - \frac{\mathsf{PI}\left(\right)}{-2}, {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right), \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_0\right), t\_0, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right) \cdot 2}, \sin^{-1} \left(1 - x\right) \cdot -1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (fma
    (fma t_0 (- t_0 (/ (PI) -2.0)) (pow (/ (PI) 2.0) 2.0))
    (/ (PI) (* (fma (fma 0.5 (PI) t_0) t_0 (* (* (PI) (PI)) 0.25)) 2.0))
    (* (asin (- 1.0 x)) -1.0))))

Derivation

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.0%

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

   1. lift-/.f64 N/A

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

6. Applied rewrites 9.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

8. Applied rewrites 9.5%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\cos^{-1} \left(1 - x\right), \cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}, {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right), \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \cos^{-1} \left(1 - x\right) \cdot \left(\cos^{-1} \left(1 - x\right) - \frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot 2}, \sin^{-1} \left(1 - x\right) \cdot -1\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

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

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

    5. metadata-eval N/A

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

    6. +-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. lower-PI.f64 N/A

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

    9. lower-acos.f64 N/A

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

    10. lower--.f64 N/A

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

    11. lower-acos.f64 N/A

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

    12. lower--.f64 N/A

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

    13. *-commutative N/A

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

    14. lower-*.f64 N/A

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

    15. unpow2 N/A

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

    16. lower-*.f64 N/A

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

    17. lower-PI.f64 N/A

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

    18. lower-PI.f64 9.5

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\cos^{-1} \left(1 - x\right), \cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}, {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right), \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right), \cos^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot 0.25\right) \cdot 2}, \sin^{-1} \left(1 - x\right) \cdot -1\right) \]

11. Applied rewrites 9.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\cos^{-1} \left(1 - x\right), \cos^{-1} \left(1 - x\right) - \frac{\mathsf{PI}\left(\right)}{-2}, {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right), \frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right), \cos^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)} \cdot 2}, \sin^{-1} \left(1 - x\right) \cdot -1\right) \]
12. Add Preprocessing

Alternative 3: 10.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(-0.125, {\mathsf{PI}\left(\right)}^{3}, {t\_0}^{3}\right), \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_0\right), t\_0, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)}, \mathsf{PI}\left(\right)\right) \cdot 0.5 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (*
    (fma
     (fma -0.125 (pow (PI) 3.0) (pow t_0 3.0))
     (/ 2.0 (fma (fma 0.5 (PI) t_0) t_0 (* (* (PI) (PI)) 0.25)))
     (PI))
    0.5)))

Derivation

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.0%

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-sqr-sqrt N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-sqrt.f64 9.5

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

6. Applied rewrites 9.5%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos^{-1} \left(1 - x\right)\right) + {\cos^{-1} \left(1 - x\right)}^{2}\right)\right) - 2 \cdot \left(\frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{3} - {\cos^{-1} \left(1 - x\right)}^{3}\right)}{\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos^{-1} \left(1 - x\right)\right) + {\cos^{-1} \left(1 - x\right)}^{2}\right)}} \]

9. Applied rewrites 9.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, {\mathsf{PI}\left(\right)}^{3}, {\cos^{-1} \left(1 - x\right)}^{3}\right), \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \cos^{-1} \left(1 - x\right)\right), \cos^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)}, \mathsf{PI}\left(\right)\right) \cdot 0.5} \]
10. Add Preprocessing

Alternative 4: 10.7% accurate, 0.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.0

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

4. Applied rewrites 6.0%

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

   1. unpow1 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. unpow1/2 N/A

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

   7. lower-sqrt.f64 9.5

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

6. Applied rewrites 9.5%

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

Alternative 5: 9.7% 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 53.6%

      \[\cos^{-1} \left(1 - x\right) \]
   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. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-asin.f64 53.6

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-asin.f64 N/A

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

      5. lower--.f64 53.6

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

   7. Applied rewrites 53.6%

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

   9. Applied rewrites 53.6%

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

Alternative 6: 10.7% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
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. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-asin.f64 6.0

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

4. Applied rewrites 6.0%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-asin.f64 N/A

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

   5. lower--.f64 6.0

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

7. Applied rewrites 6.0%

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

9. Applied rewrites 9.5%

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

Alternative 7: 9.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (acos (- 1.0 x)) 0.0) (acos (- x)) (- (* 0.5 (PI)) (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 53.6%

      \[\cos^{-1} \left(1 - x\right) \]
   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. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-asin.f64 53.6

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-asin.f64 N/A

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

      5. lower--.f64 53.6

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

   7. Applied rewrites 53.6%

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

Alternative 8: 9.7% 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 53.6%

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

Alternative 9: 7.0% 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 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.8

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

5. Applied rewrites 6.8%

   \[\leadsto \cos^{-1} \color{blue}{\left(-x\right)} \]
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 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
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 2024326 
(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)))