bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.4%

Time: 8.0s

Alternatives: 9

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := \mathsf{fma}\left(\mathsf{PI}\left(\right) - \sin^{-1} \left(1 - x\right), t\_0, 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\\ \frac{\mathsf{fma}\left(t\_1 \cdot 4, 0.125 \cdot {\mathsf{PI}\left(\right)}^{3} - {t\_0}^{3}, -2 \cdot \left({t\_1}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}{-{\left(2 \cdot t\_1\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))
        (t_1 (fma (- (PI) (asin (- 1.0 x))) t_0 (* 0.25 (* (PI) (PI))))))
   (/
    (fma
     (* t_1 4.0)
     (- (* 0.125 (pow (PI) 3.0)) (pow t_0 3.0))
     (* -2.0 (* (pow t_1 2.0) (PI))))
    (- (pow (* 2.0 t_1) 2.0)))))

Derivation

1. Initial program 7.9%

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

4. Applied rewrites 7.9%

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

6. Applied rewrites 11.4%

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

8. Applied rewrites 11.4%

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

10. Applied rewrites 11.4%

    \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(4 \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) - \sin^{-1} \left(1 - x\right), \cos^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right), {\mathsf{PI}\left(\right)}^{3} \cdot 0.125 - {\cos^{-1} \left(1 - x\right)}^{3}, \left({\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) - \sin^{-1} \left(1 - x\right), \cos^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)\right)}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right)}{{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) - \sin^{-1} \left(1 - x\right), \cos^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right) \cdot 2\right)}^{2}}} \]

11. Final simplification 11.4%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) - \sin^{-1} \left(1 - x\right), \cos^{-1} \left(1 - x\right), 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 4, 0.125 \cdot {\mathsf{PI}\left(\right)}^{3} - {\cos^{-1} \left(1 - x\right)}^{3}, -2 \cdot \left({\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) - \sin^{-1} \left(1 - x\right), \cos^{-1} \left(1 - x\right), 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}{-{\left(2 \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) - \sin^{-1} \left(1 - x\right), \cos^{-1} \left(1 - x\right), 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}} \]
12. Add Preprocessing

Alternative 2: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ t_2 := \mathsf{fma}\left(t\_0, \mathsf{PI}\left(\right) - \sin^{-1} \left(1 - x\right), 0.25 \cdot t\_1\right)\\ t_3 := t\_2 \cdot 2\\ \mathsf{fma}\left(t\_2, \frac{\mathsf{PI}\left(\right)}{t\_3}, \frac{\left(\left(t\_1 \cdot \mathsf{PI}\left(\right)\right) \cdot 0.125 - {t\_0}^{3}\right) \cdot -2}{t\_3}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x)))
        (t_1 (* (PI) (PI)))
        (t_2 (fma t_0 (- (PI) (asin (- 1.0 x))) (* 0.25 t_1)))
        (t_3 (* t_2 2.0)))
   (fma
    t_2
    (/ (PI) t_3)
    (/ (* (- (* (* t_1 (PI)) 0.125) (pow t_0 3.0)) -2.0) t_3))))

Derivation

1. Initial program 7.9%

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

4. Applied rewrites 7.9%

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

6. Applied rewrites 11.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos^{-1} \left(1 - x\right), \mathsf{PI}\left(\right) - \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right), \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\cos^{-1} \left(1 - x\right), \mathsf{PI}\left(\right) - \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right) \cdot 2}, \frac{-2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot 0.125 - {\cos^{-1} \left(1 - x\right)}^{3}\right)}{\mathsf{fma}\left(\cos^{-1} \left(1 - x\right), \mathsf{PI}\left(\right) - \sin^{-1} \left(1 - x\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right) \cdot 2}\right)} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-*.f64 11.4

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

8. Applied rewrites 11.4%

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

9. Final simplification 11.4%

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

Alternative 3: 10.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\mathsf{PI}\left(\right)}\\ t_1 := \cos^{-1} \left(1 - x\right)\\ \frac{\mathsf{fma}\left(t\_0, 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), t\_1 - \mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \frac{-2}{\mathsf{PI}\left(\right)} \cdot {t\_1}^{2}\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_1\right) \cdot t\_0} \end{array} \end{array} \]

FPCore

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

Derivation

1. Initial program 7.9%

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

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

   6. flip-- N/A

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

   7. frac-sub N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 7.9%

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

6. Applied rewrites 11.3%

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

7. Final simplification 11.3%

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

Alternative 4: 10.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \sin^{-1} \left(-1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), 0.5 \cdot \mathsf{PI}\left(\right)\right) - \cos^{-1} \left(x - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17)
   (fma (/ 2.0 (PI)) (* 0.25 (* (PI) (PI))) (asin (- 1.0)))
   (- (fma 0.5 (PI) (* 0.5 (PI))) (acos (- x 1.0)))))

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.9%

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

      1. lift-acos.f64 N/A

         \[\leadsto \color{blue}{\cos^{-1} 1} \]

      2. acos-asin N/A

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

      3. sub-neg N/A

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

   7. Applied rewrites 7.5%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 61.5%

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

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-asin.f64 N/A

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

      13. difference-of-squares N/A

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

      14. acos-asin N/A

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

      15. lift-acos.f64 N/A

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

   4. Applied rewrites 61.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity 61.5

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

      9. lift-acos.f64 N/A

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

      10. acos-asin N/A

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

      11. lift-asin.f64 N/A

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

      12. sub-neg N/A

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

   6. Applied rewrites 61.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-asin.f64 N/A

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

      3. asin-acos N/A

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

      4. lift-PI.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. rem-3cbrt-lft N/A

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

      8. lift-cbrt.f64 N/A

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

      9. lift-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

   8. Applied rewrites 61.6%

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

4. Final simplification 11.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \sin^{-1} \left(-1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), 0.5 \cdot \mathsf{PI}\left(\right)\right) - \cos^{-1} \left(x - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 10.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \sin^{-1} \left(x - 1\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (/ 2.0 (PI)) (* 0.25 (* (PI) (PI))) (asin (- x 1.0))))

Derivation

1. Initial program 7.9%

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

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-asin.f64 N/A

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

   13. difference-of-squares N/A

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

   14. acos-asin N/A

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

   15. lift-acos.f64 N/A

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

4. Applied rewrites 7.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num-rev N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-inverses N/A

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

   8. *-rgt-identity 7.9

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

   9. lift-acos.f64 N/A

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

   10. acos-asin N/A

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

   11. lift-asin.f64 N/A

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

   12. sub-neg N/A

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

6. Applied rewrites 11.3%

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

7. Final simplification 11.3%

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

Alternative 6: 9.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), 0.5 \cdot \mathsf{PI}\left(\right)\right) - \cos^{-1} \left(x - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17)
   (acos (- x))
   (- (fma 0.5 (PI) (* 0.5 (PI))) (acos (- x 1.0)))))

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   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 6.6

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

   5. Applied rewrites 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 61.5%

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

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-asin.f64 N/A

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

      13. difference-of-squares N/A

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

      14. acos-asin N/A

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

      15. lift-acos.f64 N/A

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

   4. Applied rewrites 61.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity 61.5

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

      9. lift-acos.f64 N/A

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

      10. acos-asin N/A

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

      11. lift-asin.f64 N/A

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

      12. sub-neg N/A

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

   6. Applied rewrites 61.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-asin.f64 N/A

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

      3. asin-acos N/A

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

      4. lift-PI.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. rem-3cbrt-lft N/A

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

      8. lift-cbrt.f64 N/A

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

      9. lift-cbrt.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

   8. Applied rewrites 61.6%

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

4. Final simplification 10.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), 0.5 \cdot \mathsf{PI}\left(\right)\right) - \cos^{-1} \left(x - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 9.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17) (acos (- x)) (acos (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(-x);
	} else {
		tmp = acos((1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.5d-17) then
        tmp = acos(-x)
    else
        tmp = acos((1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.acos(-x);
	} else {
		tmp = Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.acos(-x)
	else:
		tmp = math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = acos(Float64(-x));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = acos(-x);
	else
		tmp = acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[(-x)], $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   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 6.6

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

   5. Applied rewrites 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 61.5%

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

Alternative 8: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[\cos^{-1} \left(1 - x\right) \]
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 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 9: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[\cos^{-1} \left(1 - x\right) \]
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 2024298 
(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)))