bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.4%

Time: 4.3s

Alternatives: 8

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(1 - x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin^{-1} 1, \sin^{-1} \left(-1 + x\right), \pi \cdot \frac{\pi}{4}\right)}{\frac{\pi}{2} + \sin^{-1} 1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \pi - \left(\frac{\pi}{2} - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0)
     (/
      (fma (asin 1.0) (asin (+ -1.0 x)) (* PI (/ PI 4.0)))
      (+ (/ PI 2.0) (asin 1.0)))
     (- (* 0.5 PI) (- (/ PI 2.0) t_0)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma(asin(1.0), asin((-1.0 + x)), (((double) M_PI) * (((double) M_PI) / 4.0))) / ((((double) M_PI) / 2.0) + asin(1.0));
	} else {
		tmp = (0.5 * ((double) M_PI)) - ((((double) M_PI) / 2.0) - t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(fma(asin(1.0), asin(Float64(-1.0 + x)), Float64(pi * Float64(pi / 4.0))) / Float64(Float64(pi / 2.0) + asin(1.0)));
	else
		tmp = Float64(Float64(0.5 * pi) - Float64(Float64(pi / 2.0) - t_0));
	end
	return 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[(N[(N[ArcSin[1.0], $MachinePrecision] * N[ArcSin[N[(-1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(Pi * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] + N[ArcSin[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] - N[(N[(Pi / 2.0), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.9%

      \[\cos^{-1} \left(1 - x\right) \]
   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 6.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-asin.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-asin.f64 N/A

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

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

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 10.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 8.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 8.1%

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

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

   1. Initial program 6.9%

      \[\cos^{-1} \left(1 - x\right) \]
   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 6.9

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

   3. Applied rewrites 6.9%

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

   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 6.9

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

   6. Applied rewrites 6.9%

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

      1. lift--.f64 N/A

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

      2. lift-asin.f64 N/A

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

      3. asin-acos N/A

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

      4. acos-asin-rev N/A

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

      5. lower--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. acos-asin-rev N/A

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

      9. lower-acos.f64 N/A

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

      10. lift--.f64 6.9

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

   8. Applied rewrites 6.9%

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

Alternative 2: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ t_1 := \sin^{-1} \left(x - 1\right) \cdot t\_0\\ \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot t\_0\right)}^{3}\right)}{\mathsf{fma}\left(0.0625, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi, t\_1 \cdot \left(t\_1 - \left(0.25 \cdot \pi\right) \cdot \pi\right)\right)}}{\frac{\pi}{2} + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))) (t_1 (* (asin (- x 1.0)) t_0)))
   (/
    (/
     (fma
      (/ (* (* PI PI) (* PI PI)) 16.0)
      (* PI (/ PI 4.0))
      (pow (* (asin (* (- 1.0 (/ 1.0 x)) x)) t_0) 3.0))
     (fma 0.0625 (* (* PI (* PI PI)) PI) (* t_1 (- t_1 (* (* 0.25 PI) PI)))))
    (+ (/ PI 2.0) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double t_1 = asin((x - 1.0)) * t_0;
	return (fma((((((double) M_PI) * ((double) M_PI)) * (((double) M_PI) * ((double) M_PI))) / 16.0), (((double) M_PI) * (((double) M_PI) / 4.0)), pow((asin(((1.0 - (1.0 / x)) * x)) * t_0), 3.0)) / fma(0.0625, ((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * ((double) M_PI)), (t_1 * (t_1 - ((0.25 * ((double) M_PI)) * ((double) M_PI)))))) / ((((double) M_PI) / 2.0) + t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	t_1 = Float64(asin(Float64(x - 1.0)) * t_0)
	return Float64(Float64(fma(Float64(Float64(Float64(pi * pi) * Float64(pi * pi)) / 16.0), Float64(pi * Float64(pi / 4.0)), (Float64(asin(Float64(Float64(1.0 - Float64(1.0 / x)) * x)) * t_0) ^ 3.0)) / fma(0.0625, Float64(Float64(pi * Float64(pi * pi)) * pi), Float64(t_1 * Float64(t_1 - Float64(Float64(0.25 * pi) * 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[(N[ArcSin[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision] * N[(Pi * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[ArcSin[N[(N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(0.0625 * N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] + N[(t$95$1 * N[(t$95$1 - N[(N[(0.25 * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.9%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.9%

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

5. Applied rewrites 10.4%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \color{blue}{\left(x \cdot \left(1 - \frac{1}{x}\right)\right)} \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot \color{blue}{x}\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   2. *-rgt-identity N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x} \cdot 1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x} \cdot \left(-1 \cdot -1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \left(\frac{1}{x} \cdot -1\right) \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   6. distribute-rgt-neg-out N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \left(\mathsf{neg}\left(\frac{1}{x} \cdot 1\right)\right) \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   7. *-rgt-identity N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

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

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 + \frac{1}{x} \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(-1 \cdot -1 + \frac{1}{x} \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   10. distribute-rgt-out N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(-1 \cdot \left(-1 + \frac{1}{x}\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   11. +-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(-1 \cdot \left(\frac{1}{x} + -1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(-1 \cdot \left(\frac{1}{x} + -1 \cdot 1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   13. fp-cancel-sign-sub N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(-1 \cdot \left(\frac{1}{x} - \left(\mathsf{neg}\left(-1\right)\right) \cdot 1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(-1 \cdot \left(\frac{1}{x} - 1 \cdot 1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(-1 \cdot \left(\frac{1}{x} - 1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   16. associate-*r* N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(-1 \cdot \color{blue}{\left(\left(\frac{1}{x} - 1\right) \cdot x\right)}\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   17. *-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(-1 \cdot \left(x \cdot \color{blue}{\left(\frac{1}{x} - 1\right)}\right)\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

8. Applied rewrites 10.4%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \color{blue}{\left(\left(1 - \frac{1}{x}\right) \cdot x\right)} \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \color{blue}{\left(x \cdot \left(1 - \frac{1}{x}\right)\right)} \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot \color{blue}{x}\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    2. *-rgt-identity N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(1 - \frac{1}{x} \cdot 1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    3. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(1 - \frac{1}{x} \cdot \left(-1 \cdot -1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    4. associate-*r* N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(1 - \left(\frac{1}{x} \cdot -1\right) \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    5. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(1 - \left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    6. distribute-rgt-neg-out N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(1 - \left(\mathsf{neg}\left(\frac{1}{x} \cdot 1\right)\right) \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    7. *-rgt-identity N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(1 - \left(\mathsf{neg}\left(\frac{1}{x}\right)\right) \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

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

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(1 + \frac{1}{x} \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    9. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(-1 \cdot -1 + \frac{1}{x} \cdot -1\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    10. distribute-rgt-out N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(-1 \cdot \left(-1 + \frac{1}{x}\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    11. +-commutative N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(-1 \cdot \left(\frac{1}{x} + -1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    12. metadata-eval N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(-1 \cdot \left(\frac{1}{x} + -1 \cdot 1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    13. fp-cancel-sign-sub N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(-1 \cdot \left(\frac{1}{x} - \left(\mathsf{neg}\left(-1\right)\right) \cdot 1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    14. metadata-eval N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(-1 \cdot \left(\frac{1}{x} - 1 \cdot 1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    15. metadata-eval N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(\left(-1 \cdot \left(\frac{1}{x} - 1\right)\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    16. associate-*r* N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 \cdot \color{blue}{\left(\left(\frac{1}{x} - 1\right) \cdot x\right)}\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

    17. *-commutative N/A

        \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 \cdot \left(x \cdot \color{blue}{\left(\frac{1}{x} - 1\right)}\right)\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

11. Applied rewrites 10.4%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \color{blue}{\left(\left(1 - \frac{1}{x}\right) \cdot x\right)} \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

12. Taylor expanded in x around 0

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\color{blue}{\left(\frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{4} + {\sin^{-1} \left(1 - x\right)}^{4}\right) - \frac{1}{4} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(x - 1\right)\right)\right)}}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

14. Applied rewrites 10.4%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(\left(1 - \frac{1}{x}\right) \cdot x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\color{blue}{\mathsf{fma}\left(0.0625, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi, \left(\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(1 - x\right)\right) \cdot \left(\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(1 - x\right) - \left(0.25 \cdot \pi\right) \cdot \pi\right)\right)}}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]
15. Add Preprocessing

Alternative 3: 10.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-asin.f64 N/A

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

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

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

5. Applied rewrites 10.4%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. *-rgt-identity N/A

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

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

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

   9. metadata-eval N/A

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

   10. distribute-rgt-out N/A

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

   11. +-commutative N/A

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

   12. metadata-eval N/A

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

   13. fp-cancel-sign-sub N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. associate-*r* N/A

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

   17. *-commutative N/A

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

8. Applied rewrites 10.4%

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

Alternative 4: 10.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

   \[\cos^{-1} \left(1 - x\right) \]
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 6.9%

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

5. Applied rewrites 10.4%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16}, \pi \cdot \frac{\pi}{4}, {\left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)}^{3}\right)}{\frac{\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)}{16} + \left({\sin^{-1} \left(1 - x\right)}^{4} - \left(\pi \cdot \frac{\pi}{4}\right) \cdot \left(\sin^{-1} \left(-1 + x\right) \cdot \sin^{-1} \left(1 - x\right)\right)\right)}}}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot {\mathsf{PI}\left(\right)}^{6} + {\sin^{-1} \left(1 - x\right)}^{3} \cdot {\sin^{-1} \left(x - 1\right)}^{3}}{\left(\sin^{-1} \left(1 - x\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{4} + {\sin^{-1} \left(1 - x\right)}^{4}\right) - \frac{1}{4} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(x - 1\right)\right)\right)\right)}} \]

8. Applied rewrites 10.4%

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

Alternative 5: 9.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = acos(-x);
	} else {
		tmp = (0.5 * ((double) M_PI)) - ((((double) M_PI) / 2.0) - acos((1.0 - x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.6e-17) {
		tmp = Math.acos(-x);
	} else {
		tmp = (0.5 * Math.PI) - ((Math.PI / 2.0) - Math.acos((1.0 - x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.6e-17:
		tmp = math.acos(-x)
	else:
		tmp = (0.5 * math.pi) - ((math.pi / 2.0) - math.acos((1.0 - x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.6e-17)
		tmp = acos(Float64(-x));
	else
		tmp = Float64(Float64(0.5 * pi) - Float64(Float64(pi / 2.0) - acos(Float64(1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.6e-17)
		tmp = acos(-x);
	else
		tmp = (0.5 * pi) - ((pi / 2.0) - acos((1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.6e-17], N[ArcCos[(-x)], $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] - N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 6.9%

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

   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 5.5999999999999998e-17 < x

   1. Initial program 6.9%

      \[\cos^{-1} \left(1 - x\right) \]
   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 6.9

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

   3. Applied rewrites 6.9%

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

   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 6.9

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

   6. Applied rewrites 6.9%

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

      1. lift--.f64 N/A

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

      2. lift-asin.f64 N/A

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

      3. asin-acos N/A

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

      4. acos-asin-rev N/A

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

      5. lower--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. acos-asin-rev N/A

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

      9. lower-acos.f64 N/A

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

      10. lift--.f64 6.9

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

   8. Applied rewrites 6.9%

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

Alternative 6: 9.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.6d-17) then
        tmp = acos(-x)
    else
        tmp = acos((1.0d0 - x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 6.9%

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

   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 5.5999999999999998e-17 < x

   1. Initial program 6.9%

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

Alternative 7: 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 6.9%

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

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 8: 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 6.9%

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

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 2025139 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))
  (acos (- 1.0 x)))