bug323 (missed optimization)

Percentage Accurate: 7.0% → 10.6%

Time: 3.8s

Alternatives: 11

Speedup: 0.7×

Specification

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

Math

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

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

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

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, 0.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = (((double) M_PI) * ((double) M_PI)) * ((double) M_PI);
	double t_1 = acos((1.0 - x));
	double t_2 = pow(t_1, 3.0);
	double t_3 = t_0 - t_2;
	double t_4 = fma((t_1 + ((double) M_PI)), t_1, (((double) M_PI) * ((double) M_PI)));
	return fma(t_3, ((1.0 / t_4) * ((t_2 - t_0) / t_4)), (((double) M_PI) * ((double) M_PI))) / (((double) M_PI) + (t_3 / fma(((double) M_PI), ((double) M_PI), fma(t_1, t_1, (((double) M_PI) * t_1)))));
}

Julia

function code(x)
	t_0 = Float64(Float64(pi * pi) * pi)
	t_1 = acos(Float64(1.0 - x))
	t_2 = t_1 ^ 3.0
	t_3 = Float64(t_0 - t_2)
	t_4 = fma(Float64(t_1 + pi), t_1, Float64(pi * pi))
	return Float64(fma(t_3, Float64(Float64(1.0 / t_4) * Float64(Float64(t_2 - t_0) / t_4)), Float64(pi * pi)) / Float64(pi + Float64(t_3 / fma(pi, pi, fma(t_1, t_1, Float64(pi * t_1))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 3.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 + Pi), $MachinePrecision] * t$95$1 + N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$3 * N[(N[(1.0 / t$95$4), $MachinePrecision] * N[(N[(t$95$2 - t$95$0), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] + N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / N[(Pi + N[(t$95$3 / N[(Pi * Pi + N[(t$95$1 * t$95$1 + N[(Pi * t$95$1), $MachinePrecision]), $MachinePrecision]), $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. 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. flip-- N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-unsound-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-unsound--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. lower-unsound-*.f64 N/A

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

   14. lower-acos.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-acos.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-unsound-+.f64 N/A

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

   19. lower-PI.f64 N/A

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

   20. lower-acos.f64 N/A

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

   21. lower--.f64 7.0%

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

3. Applied rewrites 7.0%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 10.6%

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

5. Applied rewrites 10.6%

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

   1. lift-acos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-negate-rev N/A

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

   4. lift--.f64 N/A

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

   5. acos-neg N/A

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

   6. lift-PI.f64 N/A

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

   7. flip3-- N/A

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

   8. lower-unsound-/.f64 N/A

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

   9. lower-unsound--.f64 N/A

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

   10. lower-unsound-pow.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. pow3 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-unsound-pow.f64 N/A

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

   16. lower-acos.f64 N/A

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

7. Applied rewrites 10.6%

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

   1. lift-acos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-negate-rev N/A

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

   4. lift--.f64 N/A

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

   5. acos-neg N/A

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

   6. lift-PI.f64 N/A

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

   7. flip3-- N/A

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

   8. lower-unsound-/.f64 N/A

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

   9. lower-unsound--.f64 N/A

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

   10. lower-unsound-pow.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. pow3 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-unsound-pow.f64 N/A

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

   16. lower-acos.f64 N/A

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

9. Applied rewrites 10.6%

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

    1. lift-acos.f64 N/A

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

    2. lift--.f64 N/A

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

    3. sub-negate-rev N/A

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

    4. lift--.f64 N/A

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

    5. acos-neg N/A

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

    6. lift-PI.f64 N/A

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

    7. flip3-- N/A

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

    8. lower-unsound-/.f64 N/A

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

    9. lower-unsound--.f64 N/A

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

    10. lower-unsound-pow.f32 N/A

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

    11. lower-pow.f32 N/A

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

    12. pow3 N/A

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

    13. lift-*.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. lower-unsound-pow.f64 N/A

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

    16. lower-acos.f64 N/A

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

11. Applied rewrites 10.6%

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

    1. lift-fma.f64 N/A

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

13. Applied rewrites 10.6%

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

Alternative 2: 10.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos^{-1} \left(1 - x\right) \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-\cos^{-1} -1, \cos^{-1} -1, \pi \cdot \pi\right)}{\pi + \cos^{-1} -1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{6}{\frac{\pi}{\sin^{-1} \left(1 - x\right)} \cdot 3}\right) \cdot \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (acos (- 1.0 x)) 0.0)
  (/ (fma (- (acos -1.0)) (acos -1.0) (* PI PI)) (+ PI (acos -1.0)))
  (* (- 1.0 (/ 6.0 (* (/ PI (asin (- 1.0 x))) 3.0))) (* 0.5 PI))))

C

double code(double x) {
	double tmp;
	if (acos((1.0 - x)) <= 0.0) {
		tmp = fma(-acos(-1.0), acos(-1.0), (((double) M_PI) * ((double) M_PI))) / (((double) M_PI) + acos(-1.0));
	} else {
		tmp = (1.0 - (6.0 / ((((double) M_PI) / asin((1.0 - x))) * 3.0))) * (0.5 * ((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (acos(Float64(1.0 - x)) <= 0.0)
		tmp = Float64(fma(Float64(-acos(-1.0)), acos(-1.0), Float64(pi * pi)) / Float64(pi + acos(-1.0)));
	else
		tmp = Float64(Float64(1.0 - Float64(6.0 / Float64(Float64(pi / asin(Float64(1.0 - x))) * 3.0))) * Float64(0.5 * pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[((-N[ArcCos[-1.0], $MachinePrecision]) * N[ArcCos[-1.0], $MachinePrecision] + N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / N[(Pi + N[ArcCos[-1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(6.0 / N[(N[(Pi / N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. flip-- N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-unsound--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-unsound-*.f64 N/A

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

      14. lower-acos.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-acos.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-unsound-+.f64 N/A

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

      19. lower-PI.f64 N/A

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

      20. lower-acos.f64 N/A

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

      21. lower--.f64 7.0%

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

   3. Applied rewrites 7.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 10.6%

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

   5. Applied rewrites 10.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 8.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 7.7%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 7.7%

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

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

   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. sub-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound--.f64 N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-asin.f64 N/A

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

      8. mult-flip N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-PI.f64 N/A

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

      13. mult-flip N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-PI.f64 7.0%

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

   3. Applied rewrites 7.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 7.0%

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

   5. Applied rewrites 7.0%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. frac-times N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(1 - \frac{\color{blue}{6}}{\frac{\pi}{\sin^{-1} \left(1 - x\right)} \cdot 3}\right) \cdot \left(\frac{1}{2} \cdot \pi\right) \]

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \left(1 - \frac{\color{blue}{6}}{\frac{\pi}{\sin^{-1} \left(1 - x\right)} \cdot 3}\right) \cdot \left(\frac{1}{2} \cdot \pi\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(1 - \frac{6}{\color{blue}{\frac{\pi}{\sin^{-1} \left(1 - x\right)} \cdot 3}}\right) \cdot \left(\frac{1}{2} \cdot \pi\right) \]

      16. lower-/.f64 7.0%

          \[\leadsto \left(1 - \frac{6}{\color{blue}{\frac{\pi}{\sin^{-1} \left(1 - x\right)}} \cdot 3}\right) \cdot \left(0.5 \cdot \pi\right) \]

   7. Applied rewrites 7.0%

      \[\leadsto \left(1 - \color{blue}{\frac{6}{\frac{\pi}{\sin^{-1} \left(1 - x\right)} \cdot 3}}\right) \cdot \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 10.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(Pi - N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[((-t$95$0) * t$95$0 + N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / N[(Pi + 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. flip-- N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-unsound-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-unsound--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. lower-unsound-*.f64 N/A

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

   14. lower-acos.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-acos.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-unsound-+.f64 N/A

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

   19. lower-PI.f64 N/A

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

   20. lower-acos.f64 N/A

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

   21. lower--.f64 7.0%

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

3. Applied rewrites 7.0%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 10.6%

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

5. Applied rewrites 10.6%

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

   1. lift-acos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-negate-rev N/A

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

   4. lift--.f64 N/A

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

   5. acos-neg N/A

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

   6. lift-PI.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-acos.f64 10.6%

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

7. Applied rewrites 10.6%

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

   1. lift-acos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-negate-rev N/A

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

   4. lift--.f64 N/A

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

   5. acos-neg N/A

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

   6. lift-PI.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-acos.f64 10.6%

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

9. Applied rewrites 10.6%

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

    1. lift-acos.f64 N/A

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

    2. lift--.f64 N/A

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

    3. sub-negate-rev N/A

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

    4. lift--.f64 N/A

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

    5. acos-neg N/A

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

    6. lift-PI.f64 N/A

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

    7. lower--.f64 N/A

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

    8. lower-acos.f64 10.6%

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

11. Applied rewrites 10.6%

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

Alternative 4: 10.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(x - 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[((-t$95$0) * t$95$0 + N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] / N[(Pi + 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. flip-- N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-unsound-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-unsound--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. lower-unsound-*.f64 N/A

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

   14. lower-acos.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-acos.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-unsound-+.f64 N/A

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

   19. lower-PI.f64 N/A

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

   20. lower-acos.f64 N/A

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

   21. lower--.f64 7.0%

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

3. Applied rewrites 7.0%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 10.6%

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

5. Applied rewrites 10.6%

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

Alternative 5: 10.5% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (*
 (fma (* 0.5 PI) (/ 2.0 PI) (* (asin (- x 1.0)) (/ 2.0 PI)))
 (* 0.5 PI)))

C

double code(double x) {
	return fma((0.5 * ((double) M_PI)), (2.0 / ((double) M_PI)), (asin((x - 1.0)) * (2.0 / ((double) M_PI)))) * (0.5 * ((double) M_PI));
}

Julia

function code(x)
	return Float64(fma(Float64(0.5 * pi), Float64(2.0 / pi), Float64(asin(Float64(x - 1.0)) * Float64(2.0 / pi))) * Float64(0.5 * pi))
end

Wolfram

code[x_] := N[(N[(N[(0.5 * Pi), $MachinePrecision] * N[(2.0 / Pi), $MachinePrecision] + N[(N[ArcSin[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * Pi), $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. sub-to-mult N/A

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

   4. lower-unsound-*.f64 N/A

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

   5. lower-unsound--.f64 N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. lower-asin.f64 N/A

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

   8. mult-flip N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-PI.f64 N/A

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

   13. mult-flip N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-PI.f64 7.0%

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

3. Applied rewrites 7.0%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. lift-PI.f64 N/A

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

   9. mult-flip N/A

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

   10. sub-to-fraction-rev N/A

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

   11. *-lft-identity N/A

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

   12. sub-flip N/A

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

   13. div-add N/A

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

5. Applied rewrites 10.5%

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

Alternative 6: 9.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{6}{\frac{\pi}{\sin^{-1} \left(1 - x\right)} \cdot 3}\right) \cdot \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 5.5e-17)
  (acos (- x))
  (* (- 1.0 (/ 6.0 (* (/ PI (asin (- 1.0 x))) 3.0))) (* 0.5 PI))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(-x);
	} else {
		tmp = (1.0 - (6.0 / ((((double) M_PI) / asin((1.0 - x))) * 3.0))) * (0.5 * ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.acos(-x);
	} else {
		tmp = (1.0 - (6.0 / ((Math.PI / Math.asin((1.0 - x))) * 3.0))) * (0.5 * Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.acos(-x)
	else:
		tmp = (1.0 - (6.0 / ((math.pi / math.asin((1.0 - x))) * 3.0))) * (0.5 * math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = acos(Float64(-x));
	else
		tmp = Float64(Float64(1.0 - Float64(6.0 / Float64(Float64(pi / asin(Float64(1.0 - x))) * 3.0))) * Float64(0.5 * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = acos(-x);
	else
		tmp = (1.0 - (6.0 / ((pi / asin((1.0 - x))) * 3.0))) * (0.5 * pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[(-x)], $MachinePrecision], N[(N[(1.0 - N[(6.0 / N[(N[(Pi / N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5e-17

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 6.9%

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

   4. Applied rewrites 6.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 6.9%

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

   6. Applied rewrites 6.9%

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

   if 5.5e-17 < x

   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. sub-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound--.f64 N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-asin.f64 N/A

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

      8. mult-flip N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-PI.f64 N/A

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

      13. mult-flip N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-PI.f64 7.0%

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

   3. Applied rewrites 7.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 7.0%

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

   5. Applied rewrites 7.0%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. frac-times N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(1 - \frac{\color{blue}{6}}{\frac{\pi}{\sin^{-1} \left(1 - x\right)} \cdot 3}\right) \cdot \left(\frac{1}{2} \cdot \pi\right) \]

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \left(1 - \frac{\color{blue}{6}}{\frac{\pi}{\sin^{-1} \left(1 - x\right)} \cdot 3}\right) \cdot \left(\frac{1}{2} \cdot \pi\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(1 - \frac{6}{\color{blue}{\frac{\pi}{\sin^{-1} \left(1 - x\right)} \cdot 3}}\right) \cdot \left(\frac{1}{2} \cdot \pi\right) \]

      16. lower-/.f64 7.0%

          \[\leadsto \left(1 - \frac{6}{\color{blue}{\frac{\pi}{\sin^{-1} \left(1 - x\right)}} \cdot 3}\right) \cdot \left(0.5 \cdot \pi\right) \]

   7. Applied rewrites 7.0%

      \[\leadsto \left(1 - \color{blue}{\frac{6}{\frac{\pi}{\sin^{-1} \left(1 - x\right)} \cdot 3}}\right) \cdot \left(0.5 \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 9.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi, 0.5, \left(0.5 \cdot \pi\right) \cdot \left(\sin^{-1} \left(x - 1\right) \cdot \frac{2}{\pi}\right)\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 5.5e-17)
  (acos (- x))
  (fma PI 0.5 (* (* 0.5 PI) (* (asin (- x 1.0)) (/ 2.0 PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(-x);
	} else {
		tmp = fma(((double) M_PI), 0.5, ((0.5 * ((double) M_PI)) * (asin((x - 1.0)) * (2.0 / ((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = acos(Float64(-x));
	else
		tmp = fma(pi, 0.5, Float64(Float64(0.5 * pi) * Float64(asin(Float64(x - 1.0)) * Float64(2.0 / pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[(-x)], $MachinePrecision], N[(Pi * 0.5 + N[(N[(0.5 * Pi), $MachinePrecision] * N[(N[ArcSin[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5e-17

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 6.9%

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

   4. Applied rewrites 6.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 6.9%

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

   6. Applied rewrites 6.9%

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

   if 5.5e-17 < x

   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. acos-asin N/A

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

      8. lift-PI.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lift-asin.f64 N/A

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

      12. lift--.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. sub-to-mult-rev N/A

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

      16. lift-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

   5. Applied rewrites 7.0%

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

Alternative 8: 9.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\pi - \left(\pi - \cos^{-1} \left(1 - x\right)\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 5.5e-17) (acos (- x)) (- PI (- PI (acos (- 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(-x);
	} else {
		tmp = ((double) M_PI) - (((double) M_PI) - acos((1.0 - x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.acos(-x);
	} else {
		tmp = Math.PI - (Math.PI - Math.acos((1.0 - x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.acos(-x)
	else:
		tmp = math.pi - (math.pi - math.acos((1.0 - x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = acos(Float64(-x));
	else
		tmp = Float64(pi - Float64(pi - acos(Float64(1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.5e-17)
		tmp = acos(-x);
	else
		tmp = pi - (pi - acos((1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[(-x)], $MachinePrecision], N[(Pi - N[(Pi - N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5e-17

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 6.9%

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

   4. Applied rewrites 6.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 6.9%

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

   6. Applied rewrites 6.9%

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

   if 5.5e-17 < x

   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-acos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. acos-neg N/A

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

      5. lift-PI.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-acos.f64 N/A

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

      8. lift--.f64 7.0%

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

   5. Applied rewrites 7.0%

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

Alternative 9: 9.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 5.5e-17) (acos (- x)) (acos (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-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 <= 5.5d-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 <= 5.5e-17) {
		tmp = Math.acos(-x);
	} else {
		tmp = Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.acos(-x)
	else:
		tmp = math.acos((1.0 - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-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 <= 5.5e-17)
		tmp = acos(-x);
	else
		tmp = acos((1.0 - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[(-x)], $MachinePrecision], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5e-17

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 6.9%

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

   4. Applied rewrites 6.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 6.9%

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

   6. Applied rewrites 6.9%

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

   if 5.5e-17 < x

   1. Initial program 7.0%

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

Alternative 10: 6.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 6.9%

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

4. Applied rewrites 6.9%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 6.9%

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

6. Applied rewrites 6.9%

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

Alternative 11: 3.8% accurate, 1.5× speedup

Math

\[\cos^{-1} 1 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

code[x_] := N[ArcCos[1.0], $MachinePrecision]

Derivation

1. Initial program 7.0%

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

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 2025212 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))
  (acos (- 1.0 x)))