bug323 (missed optimization)

Percentage Accurate: 7.1% → 10.6%

Time: 9.0s

Alternatives: 11

Speedup: 0.7×

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := t\_0 \cdot 0 + {t\_0}^{2}\\ t_2 := \frac{2}{\mathsf{PI}\left(\right)}\\ \frac{\mathsf{fma}\left(\frac{-{t\_0}^{9}}{{t\_0}^{6}}, t\_2, t\_1\right)}{t\_1 \cdot t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x)))
        (t_1 (+ (* t_0 0.0) (pow t_0 2.0)))
        (t_2 (/ 2.0 (PI))))
   (/ (fma (/ (- (pow t_0 9.0)) (pow t_0 6.0)) t_2 t_1) (* t_1 t_2))))

Derivation

1. Initial program 6.8%

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

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-asin.f64 6.7

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

4. Applied rewrites 6.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. neg-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - \sin^{-1} \left(1 - x\right)\right)} + \mathsf{PI}\left(\right) \cdot \frac{1}{2} \]

   6. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)}} + \mathsf{PI}\left(\right) \cdot \frac{1}{2} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} \]

   9. div-inv N/A

      \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} \]

   10. clear-num N/A

       \[\leadsto \frac{{0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}}{0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)} + \color{blue}{\frac{1}{\frac{2}{\mathsf{PI}\left(\right)}}} \]

   11. frac-add N/A

       \[\leadsto \color{blue}{\frac{\left({0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \frac{2}{\mathsf{PI}\left(\right)} + \left(0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1}{\left(0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}}} \]

   12. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\left({0}^{3} - {\sin^{-1} \left(1 - x\right)}^{3}\right) \cdot \frac{2}{\mathsf{PI}\left(\right)} + \left(0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1}{\left(0 \cdot 0 + \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right) + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}}} \]

6. Applied rewrites 10.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0 - {\sin^{-1} \left(1 - x\right)}^{3}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(0 - \color{blue}{{\sin^{-1} \left(1 - x\right)}^{3}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(0 - {\sin^{-1} \left(1 - x\right)}^{\color{blue}{\left(9 - 6\right)}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(0 - {\sin^{-1} \left(1 - x\right)}^{\left(\color{blue}{3 \cdot 3} - 6\right)}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(0 - {\sin^{-1} \left(1 - x\right)}^{\left(3 \cdot 3 - \color{blue}{2 \cdot 3}\right)}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   5. pow-div N/A

      \[\leadsto \frac{\mathsf{fma}\left(0 - \color{blue}{\frac{{\sin^{-1} \left(1 - x\right)}^{\left(3 \cdot 3\right)}}{{\sin^{-1} \left(1 - x\right)}^{\left(2 \cdot 3\right)}}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   6. pow-pow N/A

      \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{\color{blue}{{\left({\sin^{-1} \left(1 - x\right)}^{3}\right)}^{3}}}{{\sin^{-1} \left(1 - x\right)}^{\left(2 \cdot 3\right)}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{{\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{3}\right)}}^{3}}{{\sin^{-1} \left(1 - x\right)}^{\left(2 \cdot 3\right)}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   8. pow-pow N/A

      \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{{\left({\sin^{-1} \left(1 - x\right)}^{3}\right)}^{3}}{\color{blue}{{\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3}}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{{\left({\sin^{-1} \left(1 - x\right)}^{3}\right)}^{3}}{{\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}}^{3}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   10. +-rgt-identity N/A

       \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{{\left({\sin^{-1} \left(1 - x\right)}^{3}\right)}^{3}}{\color{blue}{{\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + 0}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{{\left({\sin^{-1} \left(1 - x\right)}^{3}\right)}^{3}}{{\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + \color{blue}{{0}^{3}}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   12. mul0-lft N/A

       \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{{\left({\sin^{-1} \left(1 - x\right)}^{3}\right)}^{3}}{{\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + {\color{blue}{\left(0 \cdot \sin^{-1} \left(1 - x\right)\right)}}^{3}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{{\left({\sin^{-1} \left(1 - x\right)}^{3}\right)}^{3}}{{\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + {\color{blue}{\left(0 \cdot \sin^{-1} \left(1 - x\right)\right)}}^{3}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   14. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(0 - \color{blue}{\frac{{\left({\sin^{-1} \left(1 - x\right)}^{3}\right)}^{3}}{{\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + {\left(0 \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   15. lift-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{{\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{3}\right)}}^{3}}{{\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + {\left(0 \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   16. pow-pow N/A

       \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{\color{blue}{{\sin^{-1} \left(1 - x\right)}^{\left(3 \cdot 3\right)}}}{{\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + {\left(0 \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   17. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{\color{blue}{{\sin^{-1} \left(1 - x\right)}^{\left(3 \cdot 3\right)}}}{{\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + {\left(0 \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

   18. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(0 - \frac{{\sin^{-1} \left(1 - x\right)}^{\color{blue}{9}}}{{\left({\sin^{-1} \left(1 - x\right)}^{2}\right)}^{3} + {\left(0 \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

8. Applied rewrites 10.4%

   \[\leadsto \frac{\mathsf{fma}\left(0 - \color{blue}{\frac{{\sin^{-1} \left(1 - x\right)}^{9}}{{\sin^{-1} \left(1 - x\right)}^{6}}}, \frac{2}{\mathsf{PI}\left(\right)}, \left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot 1\right)}{\left(0 + \left({\sin^{-1} \left(1 - x\right)}^{2} + 0 \cdot \sin^{-1} \left(1 - x\right)\right)\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]

9. Final simplification 10.4%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{-{\sin^{-1} \left(1 - x\right)}^{9}}{{\sin^{-1} \left(1 - x\right)}^{6}}, \frac{2}{\mathsf{PI}\left(\right)}, \sin^{-1} \left(1 - x\right) \cdot 0 + {\sin^{-1} \left(1 - x\right)}^{2}\right)}{\left(\sin^{-1} \left(1 - x\right) \cdot 0 + {\sin^{-1} \left(1 - x\right)}^{2}\right) \cdot \frac{2}{\mathsf{PI}\left(\right)}} \]
10. Add Preprocessing

Alternative 2: 10.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\mathsf{PI}\left(\right)}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_1\right) \cdot 2\\ t_3 := \sqrt{\mathsf{PI}\left(\right)}\\ \frac{t\_2 - \mathsf{fma}\left(t\_3 \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, t\_1\right), t\_3, \left(-\cos^{-1} \left(1 - x\right)\right) \cdot \mathsf{fma}\left(2, t\_1, \mathsf{PI}\left(\right)\right)\right) \cdot t\_0}{t\_2 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 2.0 (PI)))
        (t_1 (asin (- 1.0 x)))
        (t_2 (* (fma 0.5 (PI) t_1) 2.0))
        (t_3 (sqrt (PI))))
   (/
    (-
     t_2
     (*
      (fma
       (* t_3 (fma (PI) 0.5 t_1))
       t_3
       (* (- (acos (- 1.0 x))) (fma 2.0 t_1 (PI))))
      t_0))
    (* t_2 t_0))))

Derivation

1. Initial program 6.8%

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

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

   4. asin-acos N/A

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

   5. acos-asin N/A

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

   6. flip-- N/A

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

   7. frac-sub N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 6.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

6. Applied rewrites 10.4%

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

7. Final simplification 10.4%

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

Alternative 3: 9.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-2, t\_0, \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}, 2\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0)
     (acos (- x))
     (* (* 0.25 (fma -2.0 (/ (fma -2.0 t_0 (PI)) (PI)) 2.0)) (PI)))))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.8%

      \[\cos^{-1} \left(1 - x\right) \]
   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 6.6

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

   5. Applied rewrites 6.6%

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

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

   1. Initial program 66.1%

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

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

      4. asin-acos N/A

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

      5. acos-asin N/A

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

      6. flip-- N/A

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

      7. frac-sub N/A

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

      8. frac-sub N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-2, \cos^{-1} \left(1 - x\right), \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}, 2\right) \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \mathsf{fma}\left(-2, \frac{\mathsf{fma}\left(-2, \cos^{-1} \left(1 - x\right), \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}, 2\right)\right) \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 9.7% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.8%

      \[\cos^{-1} \left(1 - x\right) \]
   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 6.6

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

   5. Applied rewrites 6.6%

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

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

   1. Initial program 66.1%

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

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-asin.f64 66.1

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

   4. Applied rewrites 66.1%

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

      1. lift-asin.f64 N/A

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

      2. asin-acos N/A

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

      3. lift-acos.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. sub-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 66.1

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

   6. Applied rewrites 66.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lift-neg.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 66.1

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

   8. Applied rewrites 66.1%

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

Alternative 5: 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 3.8%

      \[\cos^{-1} \left(1 - x\right) \]
   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 6.6

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

   5. Applied rewrites 6.6%

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

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

   1. Initial program 66.1%

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

Alternative 6: 9.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;1 - x \leq 0.9999999999999999:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot t\_0, 0.5, -\sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (if (<= (- 1.0 x) 0.9999999999999999)
     (fma (* t_0 t_0) 0.5 (- (asin (- 1.0 x))))
     (acos (- x)))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.999999999999999889

   1. Initial program 66.1%

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

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-asin.f64 66.1

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

   4. Applied rewrites 66.1%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-*.f64 66.3

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

   6. Applied rewrites 66.3%

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

   if 0.999999999999999889 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 3.8%

      \[\cos^{-1} \left(1 - x\right) \]
   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 6.6

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

   5. Applied rewrites 6.6%

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

Alternative 7: 9.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;1 - x \leq 0.9999999999999999:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0 \cdot 0.5, -\sin^{-1} \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (if (<= (- 1.0 x) 0.9999999999999999)
     (fma t_0 (* t_0 0.5) (- (asin (- 1.0 x))))
     (acos (- x)))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.999999999999999889

   1. Initial program 66.1%

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

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

      4. div-inv N/A

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

      5. add-sqr-sqrt N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-asin.f64 66.3

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

   4. Applied rewrites 66.3%

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

   if 0.999999999999999889 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 3.8%

      \[\cos^{-1} \left(1 - x\right) \]
   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 6.6

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

   5. Applied rewrites 6.6%

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

Alternative 8: 10.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (fma (PI) 0.5 (- (fma (* t_0 t_0) 0.5 (- (acos (- 1.0 x))))))))

Derivation

1. Initial program 6.8%

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

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-asin.f64 6.7

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

4. Applied rewrites 6.7%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-acos.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. sub-neg N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 6.8

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

6. Applied rewrites 6.8%

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

   1. rem-square-sqrt N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 10.4

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

8. Applied rewrites 10.4%

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

Alternative 9: 10.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (fma (PI) 0.5 (- (fma (* t_0 0.5) t_0 (- (acos (- 1.0 x))))))))

Derivation

1. Initial program 6.8%

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

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

   4. div-inv N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-asin.f64 6.7

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

4. Applied rewrites 6.7%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-acos.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. sub-neg N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. rem-square-sqrt N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. associate-*r* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-neg.f64 10.4

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

6. Applied rewrites 10.4%

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

7. Final simplification 10.4%

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

Alternative 10: 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 6.8%

   \[\cos^{-1} \left(1 - x\right) \]
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 7.0

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

5. Applied rewrites 7.0%

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

Alternative 11: 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 6.8%

   \[\cos^{-1} \left(1 - x\right) \]
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 2024259 
(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)))