bug323 (missed optimization)

Percentage Accurate: 6.7% → 10.3%

Time: 2.8s

Alternatives: 9

Speedup: 1.3×

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: 6.7% 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: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt(asin((1.0 - x)));
	double t_1 = t_0 * t_0;
	return fma(((((double) M_PI) / 2.0) * sqrt(((double) M_PI))), (sqrt(((double) M_PI)) / 2.0), (-t_1 * t_1)) / fma(t_0, t_0, (((double) M_PI) / 2.0));
}

Julia

function code(x)
	t_0 = sqrt(asin(Float64(1.0 - x)))
	t_1 = Float64(t_0 * t_0)
	return Float64(fma(Float64(Float64(pi / 2.0) * sqrt(pi)), Float64(sqrt(pi) / 2.0), Float64(Float64(-t_1) * t_1)) / fma(t_0, t_0, Float64(pi / 2.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, N[(N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] / 2.0), $MachinePrecision] + N[((-t$95$1) * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.7%

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

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-asin.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lift--.f64 N/A

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

   17. lower-+.f64 N/A

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

3. Applied rewrites 6.7%

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

   1. lift--.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. unpow1 N/A

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

   4. sqr-pow N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-asin.f64 N/A

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

   9. lift--.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-asin.f64 N/A

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

   13. lift--.f64 10.3

       \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

5. Applied rewrites 10.3%

   \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)} \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. unpow1 N/A

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

   4. sqr-pow N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-asin.f64 N/A

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

   9. lift--.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-asin.f64 N/A

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

   13. lift--.f64 10.3

       \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

7. Applied rewrites 10.3%

   \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. unpow1 N/A

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

   4. sqr-pow N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-asin.f64 N/A

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

   9. lift--.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-asin.f64 N/A

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

   13. lift--.f64 10.3

       \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}{\frac{\pi}{2} + {\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}} \]

9. Applied rewrites 10.3%

   \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}{\frac{\pi}{2} + \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}}} \]
10. Step-by-step derivation

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

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

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

    4. lift-*.f64 N/A

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

11. Applied rewrites 10.3%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. pow2 N/A

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

    5. lift-pow.f64 N/A

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

    6. lift--.f64 N/A

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

    7. lift-asin.f64 N/A

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

    8. pow1/2 N/A

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

    9. pow2 N/A

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

    10. lower-fma.f64 N/A

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

    11. lift-asin.f64 N/A

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

    12. lift--.f64 N/A

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

    13. lift-sqrt.f64 N/A

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

    14. lift-asin.f64 N/A

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

    15. lift--.f64 N/A

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

    16. lift-sqrt.f64 10.3

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

13. Applied rewrites 10.3%

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

Alternative 2: 10.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	double t_2 = t_1 * t_1;
	return fma(((((double) M_PI) / 2.0) * sqrt(((double) M_PI))), (sqrt(((double) M_PI)) / 2.0), (-t_2 * t_2)) / fma(0.5, ((double) M_PI), t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	t_2 = Float64(t_1 * t_1)
	return Float64(fma(Float64(Float64(pi / 2.0) * sqrt(pi)), Float64(sqrt(pi) / 2.0), Float64(Float64(-t_2) * t_2)) / fma(0.5, pi, t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] / 2.0), $MachinePrecision] + N[((-t$95$2) * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(0.5 * Pi + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 6.7%

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

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-asin.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lift--.f64 N/A

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

   17. lower-+.f64 N/A

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

3. Applied rewrites 6.7%

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

   1. lift--.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. unpow1 N/A

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

   4. sqr-pow N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-asin.f64 N/A

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

   9. lift--.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-asin.f64 N/A

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

   13. lift--.f64 10.3

       \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

5. Applied rewrites 10.3%

   \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)} \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. unpow1 N/A

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

   4. sqr-pow N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-asin.f64 N/A

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

   9. lift--.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-asin.f64 N/A

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

   13. lift--.f64 10.3

       \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

7. Applied rewrites 10.3%

   \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. unpow1 N/A

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

   4. sqr-pow N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-asin.f64 N/A

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

   9. lift--.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-asin.f64 N/A

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

   13. lift--.f64 10.3

       \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}{\frac{\pi}{2} + {\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}} \]

9. Applied rewrites 10.3%

   \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}{\frac{\pi}{2} + \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}}} \]
10. Step-by-step derivation

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

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

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

    4. lift-*.f64 N/A

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

11. Applied rewrites 10.3%

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

12. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lift-PI.f64 N/A

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

    3. lower-fma.f64 N/A

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

    4. lift-asin.f64 N/A

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

    5. lift--.f64 10.3

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

14. Applied rewrites 10.3%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{2} \cdot \sqrt{\pi}, \frac{\sqrt{\pi}}{2}, \left(-\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right) \cdot \left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)\right)}{\color{blue}{\mathsf{fma}\left(0.5, \pi, \sin^{-1} \left(1 - x\right)\right)}} \]
15. Add Preprocessing

Alternative 3: 10.3% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (sqrt t_0)))
   (/ (fma (* (- t_0) t_1) t_1 (* (/ PI 4.0) PI)) (+ (/ PI 2.0) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = sqrt(t_0);
	return fma((-t_0 * t_1), t_1, ((((double) M_PI) / 4.0) * ((double) M_PI))) / ((((double) M_PI) / 2.0) + t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = sqrt(t_0)
	return Float64(fma(Float64(Float64(-t_0) * t_1), t_1, Float64(Float64(pi / 4.0) * pi)) / Float64(Float64(pi / 2.0) + t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[(N[((-t$95$0) * t$95$1), $MachinePrecision] * t$95$1 + N[(N[(Pi / 4.0), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.7%

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

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-asin.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lift--.f64 N/A

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

   17. lower-+.f64 N/A

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

3. Applied rewrites 6.7%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-asin.f64 N/A

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

   6. pow2 N/A

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

   7. pow-to-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lift-asin.f64 N/A

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

   12. lift--.f64 6.7

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

5. Applied rewrites 6.7%

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

7. Applied rewrites 10.3%

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

Alternative 4: 10.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return fma(-t_0, t_0, ((((double) M_PI) * ((double) M_PI)) * 0.25)) / fma(0.5, ((double) M_PI), t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(fma(Float64(-t_0), t_0, Float64(Float64(pi * pi) * 0.25)) / fma(0.5, pi, t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[((-t$95$0) * t$95$0 + N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(0.5 * Pi + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.7%

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

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-asin.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lift--.f64 N/A

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

   17. lower-+.f64 N/A

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

3. Applied rewrites 6.7%

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

   1. lift--.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. unpow1 N/A

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

   4. sqr-pow N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-asin.f64 N/A

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

   9. lift--.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-asin.f64 N/A

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

   13. lift--.f64 10.3

       \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

5. Applied rewrites 10.3%

   \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)} \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. unpow1 N/A

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

   4. sqr-pow N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-asin.f64 N/A

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

   9. lift--.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-asin.f64 N/A

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

   13. lift--.f64 10.3

       \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

7. Applied rewrites 10.3%

   \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. unpow1 N/A

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

   4. sqr-pow N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-asin.f64 N/A

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

   9. lift--.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-asin.f64 N/A

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

   13. lift--.f64 10.3

       \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}{\frac{\pi}{2} + {\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}} \]

9. Applied rewrites 10.3%

   \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}{\frac{\pi}{2} + \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}}} \]
10. Step-by-step derivation

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

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

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

    4. lift-*.f64 N/A

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

11. Applied rewrites 10.3%

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

12. Taylor expanded in x around 0

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

    1. lower-/.f64 N/A

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

14. Applied rewrites 10.3%

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

Alternative 5: 9.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;1 - x \leq 0.9999999999999362:\\ \;\;\;\;\frac{0.25 \cdot \left(\pi \cdot \pi\right) - t\_0 \cdot t\_0}{\mathsf{fma}\left(0.5, \pi, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= (- 1.0 x) 0.9999999999999362)
     (/ (- (* 0.25 (* PI PI)) (* t_0 t_0)) (fma 0.5 PI t_0))
     (acos (- x)))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if ((1.0 - x) <= 0.9999999999999362) {
		tmp = ((0.25 * (((double) M_PI) * ((double) M_PI))) - (t_0 * t_0)) / fma(0.5, ((double) M_PI), t_0);
	} else {
		tmp = acos(-x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (Float64(1.0 - x) <= 0.9999999999999362)
		tmp = Float64(Float64(Float64(0.25 * Float64(pi * pi)) - Float64(t_0 * t_0)) / fma(0.5, pi, t_0));
	else
		tmp = acos(Float64(-x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 - x), $MachinePrecision], 0.9999999999999362], N[(N[(N[(0.25 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * Pi + t$95$0), $MachinePrecision]), $MachinePrecision], N[ArcCos[(-x)], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.999999999999936162

   1. Initial program 6.7%

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

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

      10. lower-/.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-asin.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-asin.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-+.f64 N/A

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

   3. Applied rewrites 6.7%

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

      1. lift--.f64 N/A

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

      2. lift-asin.f64 N/A

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

      3. unpow1 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-asin.f64 N/A

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

      9. lift--.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-asin.f64 N/A

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

      13. lift--.f64 10.3

          \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   5. Applied rewrites 10.3%

      \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)} \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-asin.f64 N/A

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

      3. unpow1 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-asin.f64 N/A

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

      9. lift--.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-asin.f64 N/A

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

      13. lift--.f64 10.3

          \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   7. Applied rewrites 10.3%

      \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-asin.f64 N/A

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

      3. unpow1 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-asin.f64 N/A

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

      9. lift--.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-asin.f64 N/A

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

      13. lift--.f64 10.3

          \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}{\frac{\pi}{2} + {\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5}} \]

   9. Applied rewrites 10.3%

      \[\leadsto \frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right) \cdot \left({\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}\right)}{\frac{\pi}{2} + \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \left(1 - x\right)}^{0.5}}} \]

   10. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 6.7%

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

   if 0.999999999999936162 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 6.7%

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

   2. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.9

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

   4. Applied rewrites 6.9%

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

Alternative 6: 9.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (acos((1.0 - x)) <= 0.0) {
		tmp = acos(-x);
	} else {
		tmp = (0.5 * ((double) M_PI)) - asin((1.0 - x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.acos((1.0 - x)) <= 0.0) {
		tmp = Math.acos(-x);
	} else {
		tmp = (0.5 * Math.PI) - Math.asin((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.acos((1.0 - x)) <= 0.0:
		tmp = math.acos(-x)
	else:
		tmp = (0.5 * math.pi) - math.asin((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (acos(Float64(1.0 - x)) <= 0.0)
		tmp = acos(Float64(-x));
	else
		tmp = Float64(Float64(0.5 * pi) - asin(Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (acos((1.0 - x)) <= 0.0)
		tmp = acos(-x);
	else
		tmp = (0.5 * pi) - asin((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 0.0], N[ArcCos[(-x)], $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] - N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.7%

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

   2. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.9

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

   4. Applied rewrites 6.9%

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

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

   1. Initial program 6.7%

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

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

      7. lower-asin.f64 N/A

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

      8. lift--.f64 6.7

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in x around 0

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

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

      4. lift-asin.f64 N/A

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

      5. lift--.f64 6.7

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

   6. Applied rewrites 6.7%

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

Alternative 7: 9.2% accurate, 0.4× speedup

Math

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

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 6.7%

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

   2. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.9

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

   4. Applied rewrites 6.9%

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

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

   1. Initial program 6.7%

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

Alternative 8: 6.9% accurate, 1.3× 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 6.7%

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

2. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 6.9

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

4. Applied rewrites 6.9%

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

Alternative 9: 3.8% accurate, 1.5× 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 6.7%

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

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

Reproduce

herbie shell --seed 2025131 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))
  (acos (- 1.0 x)))