bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.5%

Time: 5.4s

Alternatives: 11

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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: 10.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := {t\_0}^{3}\\ t_2 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_3 := {t\_2}^{3}\\ t_4 := t\_0 + t\_2\\ \frac{\mathsf{fma}\left(\frac{t\_3}{\mathsf{fma}\left(t\_2, \cos^{-1} \left(1 - x\right), {\sin^{-1} \left(\frac{1 - x \cdot x}{x + 1}\right)}^{2}\right)}, \frac{t\_3}{t\_4}, \frac{{\sin^{-1} \left(x - 1\right)}^{3} \cdot t\_1}{t\_1 + t\_3}\right)}{\mathsf{fma}\left(t\_0, t\_4, {t\_2}^{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x)))
        (t_1 (pow t_0 3.0))
        (t_2 (/ (PI) 2.0))
        (t_3 (pow t_2 3.0))
        (t_4 (+ t_0 t_2)))
   (/
    (fma
     (/
      t_3
      (fma
       t_2
       (acos (- 1.0 x))
       (pow (asin (/ (- 1.0 (* x x)) (+ x 1.0))) 2.0)))
     (/ t_3 t_4)
     (/ (* (pow (asin (- x 1.0)) 3.0) t_1) (+ t_1 t_3)))
    (fma t_0 t_4 (pow t_2 2.0)))))

Derivation

1. Initial program 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
2. 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. flip3-- N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-asin.f64 N/A

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

   11. +-commutative N/A

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

3. Applied rewrites 6.4%

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

5. Applied rewrites 10.3%

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

6. Taylor expanded in x around 0

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

   1. lower-asin.f64 N/A

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

   2. lower--.f64 10.3

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

8. Applied rewrites 10.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. +-commutative N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 10.3

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

10. Applied rewrites 10.3%

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

Alternative 2: 10.5% accurate, 0.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
2. 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. flip3-- N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-asin.f64 N/A

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

   11. +-commutative N/A

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

3. Applied rewrites 6.4%

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

5. Applied rewrites 10.3%

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

6. Taylor expanded in x around 0

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

   1. lower-asin.f64 N/A

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

   2. lower--.f64 10.3

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

8. Applied rewrites 10.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. +-commutative N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 10.3

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

10. Applied rewrites 10.3%

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

Alternative 3: 10.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_1 := {t\_0}^{3}\\ t_2 := \sin^{-1} \left(1 - x\right)\\ t_3 := t\_2 + t\_0\\ t_4 := {t\_2}^{3}\\ \frac{\mathsf{fma}\left(\frac{t\_1}{\mathsf{fma}\left(t\_0, \cos^{-1} \left(1 - x\right), {t\_2}^{2}\right)}, \frac{t\_1}{t\_3}, \frac{{\sin^{-1} \left(x - 1\right)}^{3} \cdot t\_4}{t\_4 + t\_1}\right)}{\mathsf{fma}\left(t\_2, t\_3, {t\_0}^{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (PI) 2.0))
        (t_1 (pow t_0 3.0))
        (t_2 (asin (- 1.0 x)))
        (t_3 (+ t_2 t_0))
        (t_4 (pow t_2 3.0)))
   (/
    (fma
     (/ t_1 (fma t_0 (acos (- 1.0 x)) (pow t_2 2.0)))
     (/ t_1 t_3)
     (/ (* (pow (asin (- x 1.0)) 3.0) t_4) (+ t_4 t_1)))
    (fma t_2 t_3 (pow t_0 2.0)))))

Derivation

1. Initial program 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
2. 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. flip3-- N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-asin.f64 N/A

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

   11. +-commutative N/A

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

3. Applied rewrites 6.4%

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

5. Applied rewrites 10.3%

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

6. Taylor expanded in x around 0

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

   1. lower-asin.f64 N/A

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

   2. lower--.f64 10.3

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

8. Applied rewrites 10.3%

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

Alternative 4: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_1 := {t\_0}^{1.5}\\ t_2 := \sin^{-1} \left(1 - x\right)\\ t_3 := t\_2 + t\_0\\ \frac{\mathsf{fma}\left(t\_1, t\_1, \mathsf{fma}\left(-t\_3, {t\_2}^{2}, {t\_0}^{2} \cdot t\_2\right)\right)}{t\_3 \cdot t\_3} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (PI) 2.0))
        (t_1 (pow t_0 1.5))
        (t_2 (asin (- 1.0 x)))
        (t_3 (+ t_2 t_0)))
   (/
    (fma t_1 t_1 (fma (- t_3) (pow t_2 2.0) (* (pow t_0 2.0) t_2)))
    (* t_3 t_3))))

Derivation

1. Initial program 6.4%

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

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

   5. frac-sub N/A

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

   6. lower-/.f64 N/A

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

3. Applied rewrites 6.4%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. unpow3 N/A

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

   8. lift-pow.f64 N/A

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

   9. associate--l+ N/A

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

5. Applied rewrites 10.2%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. associate-+l+ N/A

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

7. Applied rewrites 10.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{1.5}, {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{1.5}, \mathsf{fma}\left(-\left(\sin^{-1} \left(1 - x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right), {\sin^{-1} \left(1 - x\right)}^{2}, {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\left(\sin^{-1} \left(1 - x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\sin^{-1} \left(1 - x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]
8. Add Preprocessing

Alternative 5: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_1 := \sin^{-1} \left(1 - x\right)\\ t_2 := t\_1 + t\_0\\ \frac{\mathsf{fma}\left(t\_0, t\_1 \cdot t\_0, {t\_0}^{3} - {t\_1}^{2} \cdot \left(t\_1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{t\_2 \cdot t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (PI) 2.0)) (t_1 (asin (- 1.0 x))) (t_2 (+ t_1 t_0)))
   (/
    (fma
     t_0
     (* t_1 t_0)
     (- (pow t_0 3.0) (* (pow t_1 2.0) (+ t_1 (* 0.5 (PI))))))
    (* t_2 t_2))))

Derivation

1. Initial program 6.4%

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

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

   5. frac-sub N/A

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

   6. lower-/.f64 N/A

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

3. Applied rewrites 6.4%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. unpow3 N/A

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

   8. lift-pow.f64 N/A

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

   9. associate--l+ N/A

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

5. Applied rewrites 10.2%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-asin.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-asin.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 10.2

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

8. Applied rewrites 10.2%

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

Alternative 6: 10.4% accurate, 0.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.4%

   \[\cos^{-1} \left(1 - x\right) \]
2. 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. flip3-- N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-asin.f64 N/A

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

   11. +-commutative N/A

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

3. Applied rewrites 6.4%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. cube-mult N/A

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

   4. unpow2 N/A

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

   5. lift-pow.f64 N/A

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

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

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

   7. lift-pow.f64 N/A

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

   8. unpow3 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. sqr-neg-rev N/A

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

   15. cube-unmult N/A

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

   16. lower-pow.f64 N/A

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

5. Applied rewrites 10.2%

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

Alternative 7: 10.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}{2} - \sin^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (cbrt (pow (PI) 3.0)) 2.0) (asin (- 1.0 x))))

Derivation

1. Initial program 6.4%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-asin.f64 6.4

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

3. Applied rewrites 6.4%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. pow1/3 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-pow.f64 10.1

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

5. Applied rewrites 10.1%

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

Alternative 8: 10.4% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.4%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-asin.f64 6.4

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

3. Applied rewrites 6.4%

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

   1. unpow1 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. unpow1/2 N/A

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

   7. lower-sqrt.f64 10.1

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

5. Applied rewrites 10.1%

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

Alternative 9: 9.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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. Taylor expanded in x around inf

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

      1. lower-*.f64 6.7

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

   4. Applied rewrites 6.7%

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

   6. Applied rewrites 6.7%

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

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

   1. Initial program 63.8%

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

Alternative 10: 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 6.4%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 7.0

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

4. Applied rewrites 7.0%

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

6. Applied rewrites 7.0%

   \[\leadsto \cos^{-1} \color{blue}{\left(-x\right)} \]
7. 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 6.4%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
5. 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 2024363 -o localize:costs -o setup:simplify -o generate:simplify
(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)))