bug323 (missed optimization)

Percentage Accurate: 7.0% → 10.6%

Time: 3.7s

Alternatives: 6

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fma(acos((x - 1.0)), asin((1.0 - x)), (((0.25 * sqrt(((double) M_PI))) * sqrt(((double) M_PI))) * ((double) M_PI)));
	return fma(((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)), (0.125 / t_0), (pow(asin((x - 1.0)), 3.0) / t_0));
}

Julia

function code(x)
	t_0 = fma(acos(Float64(x - 1.0)), asin(Float64(1.0 - x)), Float64(Float64(Float64(0.25 * sqrt(pi)) * sqrt(pi)) * pi))
	return fma(Float64(Float64(pi * pi) * pi), Float64(0.125 / t_0), Float64((asin(Float64(x - 1.0)) ^ 3.0) / t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[ArcCos[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(0.25 * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(0.125 / t$95$0), $MachinePrecision] + N[(N[Power[N[ArcSin[N[(x - 1.0), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.0%

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

   1. lift-acos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-negate-rev N/A

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

   4. acos-neg N/A

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

   5. lower--.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-acos.f64 N/A

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

   8. lower--.f64 7.0

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

3. Applied rewrites 7.0%

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

5. Applied rewrites 10.6%

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

   1. lift-*.f64 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 10.5

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

7. Applied rewrites 10.5%

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

   1. lift-*.f64 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 10.6

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

9. Applied rewrites 10.6%

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

Alternative 2: 10.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = fma(acos((x - 1.0)), asin((1.0 - x)), ((0.25 * ((double) M_PI)) * ((double) M_PI)));
	return fma(((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)), (0.125 / t_0), (pow(asin((x - 1.0)), 3.0) / t_0));
}

Julia

function code(x)
	t_0 = fma(acos(Float64(x - 1.0)), asin(Float64(1.0 - x)), Float64(Float64(0.25 * pi) * pi))
	return fma(Float64(Float64(pi * pi) * pi), Float64(0.125 / t_0), Float64((asin(Float64(x - 1.0)) ^ 3.0) / t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[ArcCos[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[(N[(0.25 * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(0.125 / t$95$0), $MachinePrecision] + N[(N[Power[N[ArcSin[N[(x - 1.0), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.0%

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

   1. lift-acos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-negate-rev N/A

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

   4. acos-neg N/A

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

   5. lower--.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-acos.f64 N/A

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

   8. lower--.f64 7.0

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

3. Applied rewrites 7.0%

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

5. Applied rewrites 10.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \pi, \frac{0.125}{\mathsf{fma}\left(\cos^{-1} \left(x - 1\right), \sin^{-1} \left(1 - x\right), \left(0.25 \cdot \pi\right) \cdot \pi\right)}, \frac{{\sin^{-1} \left(x - 1\right)}^{3}}{\mathsf{fma}\left(\cos^{-1} \left(x - 1\right), \sin^{-1} \left(1 - x\right), \left(0.25 \cdot \pi\right) \cdot \pi\right)}\right)} \]
6. Add Preprocessing

Alternative 3: 10.6% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/ (fma (asin (- x 1.0)) t_0 (* (* 0.25 PI) PI)) (fma PI 0.5 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(((double) M_PI), 0.5, t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(fma(asin(Float64(x - 1.0)), t_0, Float64(Float64(0.25 * pi) * pi)) / fma(pi, 0.5, 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[(N[(0.25 * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] / N[(Pi * 0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.0%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. flip-- N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   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. mult-flip N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. metadata-eval N/A

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

   11. mult-flip N/A

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

   12. lower-*.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-asin.f64 N/A

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

   17. lower-asin.f64 N/A

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

3. Applied rewrites 7.0%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-neg-out N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-asin.f64 N/A

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

   8. asin-neg-rev N/A

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

   9. lift--.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. lift--.f64 N/A

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

   12. lower-asin.f64 10.6

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

5. Applied rewrites 10.6%

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

Alternative 4: 10.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot 0.5 - \mathsf{fma}\left(0.5 \cdot \sqrt{\pi}, \sqrt{\pi}, -\cos^{-1} \left(1 - x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* PI 0.5) (fma (* 0.5 (sqrt PI)) (sqrt PI) (- (acos (- 1.0 x))))))

C

double code(double x) {
	return (((double) M_PI) * 0.5) - fma((0.5 * sqrt(((double) M_PI))), sqrt(((double) M_PI)), -acos((1.0 - x)));
}

Julia

function code(x)
	return Float64(Float64(pi * 0.5) - fma(Float64(0.5 * sqrt(pi)), sqrt(pi), Float64(-acos(Float64(1.0 - x)))))
end

Wolfram

code[x_] := N[(N[(Pi * 0.5), $MachinePrecision] - N[(N[(0.5 * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision] + (-N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. lower--.f64 N/A

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

   4. mult-flip N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-asin.f64 7.0

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

3. Applied rewrites 7.0%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. sub-negate-rev N/A

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

   9. lift--.f64 N/A

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

   10. acos-neg-rev N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-acos.f64 N/A

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

   13. sub-negate-rev N/A

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

   14. add-flip-rev N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-PI.f64 N/A

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

   18. add-sqr-sqrt N/A

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

   19. associate-*r* N/A

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

   20. sub-negate-rev N/A

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

   21. lift-PI.f64 N/A

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

   22. lift-acos.f64 N/A

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

   23. acos-neg-rev N/A

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

   24. lift--.f64 N/A

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

   25. sub-negate-rev N/A

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

   26. lift--.f64 N/A

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

5. Applied rewrites 10.5%

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

Alternative 5: 7.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(Pi * 0.5 + N[ArcSin[N[(N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. lift-acos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-negate-rev N/A

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

   4. acos-neg N/A

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

   5. lower--.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-acos.f64 N/A

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

   8. lower--.f64 7.0

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

3. Applied rewrites 7.0%

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

   1. lift--.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-acos.f64 N/A

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

   4. acos-neg-rev N/A

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

   7. lift--.f64 N/A

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

   8. acos-asin N/A

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

   9. lift-PI.f64 N/A

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

   10. mult-flip N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f64 N/A

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

   13. lift-asin.f64 N/A

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

   14. sub-flip N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. lift-asin.f64 N/A

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

   18. asin-neg-rev N/A

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

   19. lift--.f64 N/A

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

   20. sub-negate-rev N/A

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

   21. lift--.f64 N/A

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

   22. lower-asin.f64 7.0

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

5. Applied rewrites 7.0%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 7.4

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

8. Applied rewrites 7.4%

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

Alternative 6: 7.0% 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]

Derivation

1. Initial program 7.0%

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

Reproduce

herbie shell --seed 2025136 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))
  (acos (- 1.0 x)))