bug323 (missed optimization)

Percentage Accurate: 7.1% → 10.6%

Time: 3.7s

Alternatives: 9

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 10.6% accurate, 0.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.9%

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

   1. lift--.f64 N/A

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

   2. lift-acos.f64 N/A

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

   3. acos-asin N/A

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

   4. flip-- N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-asin.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lift--.f64 N/A

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

   17. lower-+.f64 N/A

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

4. Applied rewrites 7.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-asin.f64 N/A

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

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

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

6. Applied rewrites 11.0%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. inv-pow N/A

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

   5. lower-pow.f64 11.0

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

9. Applied rewrites 11.0%

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

    1. lift--.f64 N/A

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

    2. lift-asin.f64 N/A

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

    3. asin-acos N/A

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

    4. lift-/.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. acos-asin-rev N/A

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

    7. lower--.f64 N/A

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

    8. acos-asin-rev N/A

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

    9. lower-acos.f64 N/A

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

    10. lift--.f64 11.1

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

11. Applied rewrites 11.1%

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

Alternative 2: 10.6% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/
    (fma t_0 (asin (* (- 1.0 (pow x -1.0)) x)) (/ (* (PI) (PI)) 4.0))
    (+ (/ (PI) 2.0) t_0))))

Derivation

1. Initial program 7.9%

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

   1. lift--.f64 N/A

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

   2. lift-acos.f64 N/A

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

   3. acos-asin N/A

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

   4. flip-- N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-asin.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lift--.f64 N/A

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

   17. lower-+.f64 N/A

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

4. Applied rewrites 7.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-asin.f64 N/A

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

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

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

6. Applied rewrites 11.0%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. inv-pow N/A

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

   5. lower-pow.f64 11.0

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

9. Applied rewrites 11.0%

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

    1. lift-pow.f64 N/A

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

    2. unpow2 N/A

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

    3. lift-PI.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. frac-times N/A

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

    8. pow2 N/A

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

    9. metadata-eval N/A

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

    10. metadata-eval N/A

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

    11. lower-/.f64 N/A

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

    12. pow2 N/A

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

    13. lift-*.f64 N/A

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

    14. lift-PI.f64 N/A

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

    15. lift-PI.f64 N/A

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

    16. metadata-eval 11.0

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

11. Applied rewrites 11.0%

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

Alternative 3: 10.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{2}\\ \mathbf{if}\;x \leq 2.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin^{-1} \left(1 - x\right), \sin^{-1} \left(-1 + x\right), \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4}\right)}{t\_0 + \sin^{-1} 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(\cos^{-1} \left(x - 1\right) + -0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (PI) 2.0)))
   (if (<= x 2.1e-12)
     (/
      (fma (asin (- 1.0 x)) (asin (+ -1.0 x)) (/ (* (PI) (PI)) 4.0))
      (+ t_0 (asin 1.0)))
     (- t_0 (+ (acos (- x 1.0)) (* -0.5 (PI)))))))

Derivation

1. Split input into 2 regimes

2. if x < 2.09999999999999994e-12

   1. Initial program 4.5%

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

      1. lift--.f64 N/A

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

      2. lift-acos.f64 N/A

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

      3. acos-asin N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-asin.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-asin.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-+.f64 N/A

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

   4. Applied rewrites 4.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-asin.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-asin.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 7.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 7.8%

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

       1. lift-pow.f64 N/A

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

       2. unpow2 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. frac-times N/A

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

       8. pow2 N/A

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

       9. metadata-eval N/A

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

       10. metadata-eval N/A

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

       11. lower-/.f64 N/A

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

       12. pow2 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. lift-PI.f64 N/A

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

       16. metadata-eval 7.8

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

   11. Applied rewrites 7.8%

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

   if 2.09999999999999994e-12 < x

   1. Initial program 76.7%

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

      1. lift--.f64 N/A

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

      2. lift-acos.f64 N/A

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

      3. acos-asin 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-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-asin.f64 N/A

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

      8. lift--.f64 76.7

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

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-asin.f64 N/A

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

      5. lift--.f64 76.7

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

   7. Applied rewrites 76.7%

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

   9. Applied rewrites 76.0%

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

   11. Applied rewrites 76.8%

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

Alternative 4: 10.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/ (fma t_0 (asin (+ -1.0 x)) (/ (* (PI) (PI)) 4.0)) (+ (/ (PI) 2.0) t_0))))

Derivation

1. Initial program 7.9%

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

   1. lift--.f64 N/A

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

   2. lift-acos.f64 N/A

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

   3. acos-asin N/A

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

   4. flip-- N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-asin.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lift--.f64 N/A

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

   17. lower-+.f64 N/A

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

4. Applied rewrites 7.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-asin.f64 N/A

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

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

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

6. Applied rewrites 11.0%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. unpow2 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. metadata-eval 11.0

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

8. Applied rewrites 11.0%

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

Alternative 5: 9.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{2} - \left(\cos^{-1} \left(x - 1\right) + -0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17)
   (acos (- x))
   (- (/ (PI) 2.0) (+ (acos (- x 1.0)) (* -0.5 (PI))))))

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.4

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

   5. Applied rewrites 6.4%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 64.0%

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

      1. lift--.f64 N/A

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

      2. lift-acos.f64 N/A

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

      3. acos-asin 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-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-asin.f64 N/A

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

      8. lift--.f64 63.9

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-asin.f64 N/A

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

      5. lift--.f64 63.9

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

   7. Applied rewrites 63.9%

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

   9. Applied rewrites 63.8%

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

   11. Applied rewrites 64.0%

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

Alternative 6: 9.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{PI}\left(\right) - \left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(1 - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17)
   (acos (- x))
   (- (* 0.5 (PI)) (- (/ (PI) 2.0) (acos (- 1.0 x))))))

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.4

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

   5. Applied rewrites 6.4%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 64.0%

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

      1. lift--.f64 N/A

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

      2. lift-acos.f64 N/A

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

      3. acos-asin 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-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-asin.f64 N/A

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

      8. lift--.f64 63.9

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-asin.f64 N/A

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

      5. lift--.f64 63.9

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

   7. Applied rewrites 63.9%

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

      1. lift--.f64 N/A

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

      2. lift-asin.f64 N/A

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

      3. asin-acos N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. acos-asin-rev N/A

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

      7. lower--.f64 N/A

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

      8. acos-asin-rev N/A

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

      9. lower-acos.f64 N/A

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

      10. lift--.f64 64.0

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

   9. Applied rewrites 64.0%

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

Alternative 7: 9.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.6e-17) (acos (- x)) (acos (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = acos(-x);
	} else {
		tmp = acos((1.0 - x));
	}
	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) :: tmp
    if (x <= 5.6d-17) then
        tmp = acos(-x)
    else
        tmp = acos((1.0d0 - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.acos(-x);
	} else {
		tmp = Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.acos(-x)
	else:
		tmp = math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = acos(Float64(-x));
	else
		tmp = acos(Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = acos(-x);
	else
		tmp = acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[ArcCos[(-x)], $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 3.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.4

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

   5. Applied rewrites 6.4%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 64.0%

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

Alternative 8: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 7.9%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 6.9

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

5. Applied rewrites 6.9%

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

Alternative 9: 3.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 7.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 3.8%

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

Developer Target 1: 100.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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 2025051 
(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)))