bug323 (missed optimization)

Percentage Accurate: 7.3% → 10.8%

Time: 3.4s

Alternatives: 10

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 7.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}\\ t_1 := \left(0.5 \cdot t\_0\right) \cdot t\_0\\ \left(1 - \frac{\sin^{-1} \left(1 - x\right)}{t\_1}\right) \cdot t\_1 \end{array} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.cbrt(Math.sqrt(((Math.PI * Math.PI) * Math.PI)));
	double t_1 = (0.5 * t_0) * t_0;
	return (1.0 - (Math.asin((1.0 - x)) / t_1)) * t_1;
}

Julia

function code(x)
	t_0 = cbrt(sqrt(Float64(Float64(pi * pi) * pi)))
	t_1 = Float64(Float64(0.5 * t_0) * t_0)
	return Float64(Float64(1.0 - Float64(asin(Float64(1.0 - x)) / t_1)) * t_1)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Sqrt[N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(1.0 - N[(N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
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-*.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--.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-/.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. lower-*.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. metadata-eval N/A

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

   12. mult-flip N/A

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

   13. lower-*.f64 N/A

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

   14. lower-PI.f64 N/A

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

   15. metadata-eval 7.3

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

3. Applied rewrites 7.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 5.5

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

5. Applied rewrites 5.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 5.5

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

7. Applied rewrites 5.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cbrt-cube N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. pow3 N/A

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

   8. sqrt-cbrt N/A

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

   9. lower-cbrt.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. pow3 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. lift-PI.f64 N/A

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

   17. lower-*.f64 10.7

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

9. Applied rewrites 10.7%

   \[\leadsto \left(1 - \frac{\sin^{-1} \left(1 - x\right)}{\left(0.5 \cdot \color{blue}{\sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}}\right) \cdot \sqrt{\pi}}\right) \cdot \left(\left(0.5 \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right) \]
10. Step-by-step derivation

    1. lift-sqrt.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. add-cbrt-cube N/A

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

    4. lift-PI.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-PI.f64 N/A

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

    7. pow3 N/A

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

    8. sqrt-cbrt N/A

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

    9. lower-cbrt.f64 N/A

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

    10. lower-sqrt.f64 N/A

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

    11. pow3 N/A

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

    12. lift-PI.f64 N/A

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

    13. lift-PI.f64 N/A

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

    14. lower-*.f64 N/A

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

    15. lift-PI.f64 N/A

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

    16. lift-PI.f64 N/A

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

    17. lower-*.f64 10.7

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

11. Applied rewrites 10.7%

    \[\leadsto \left(1 - \frac{\sin^{-1} \left(1 - x\right)}{\left(0.5 \cdot \sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}\right) \cdot \color{blue}{\sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}}}\right) \cdot \left(\left(0.5 \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right) \]
12. Step-by-step derivation

    1. lift-sqrt.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. add-cbrt-cube N/A

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

    4. lift-PI.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-PI.f64 N/A

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

    7. pow3 N/A

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

    8. sqrt-cbrt N/A

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

    9. lower-cbrt.f64 N/A

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

    10. lower-sqrt.f64 N/A

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

    11. pow3 N/A

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

    12. lift-PI.f64 N/A

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

    13. lift-PI.f64 N/A

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

    14. lower-*.f64 N/A

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

    15. lift-PI.f64 N/A

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

    16. lift-PI.f64 N/A

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

    17. lower-*.f64 10.7

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

13. Applied rewrites 10.7%

    \[\leadsto \left(1 - \frac{\sin^{-1} \left(1 - x\right)}{\left(0.5 \cdot \sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}\right) \cdot \sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}}\right) \cdot \left(\left(0.5 \cdot \color{blue}{\sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}}\right) \cdot \sqrt{\pi}\right) \]
14. Step-by-step derivation

    1. lift-sqrt.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. add-cbrt-cube N/A

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

    4. lift-PI.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-PI.f64 N/A

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

    7. pow3 N/A

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

    8. sqrt-cbrt N/A

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

    9. lower-cbrt.f64 N/A

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

    10. lower-sqrt.f64 N/A

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

    11. pow3 N/A

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

    12. lift-PI.f64 N/A

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

    13. lift-PI.f64 N/A

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

    14. lower-*.f64 N/A

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

    15. lift-PI.f64 N/A

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

    16. lift-PI.f64 N/A

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

    17. lower-*.f64 10.7

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

15. Applied rewrites 10.7%

    \[\leadsto \left(1 - \frac{\sin^{-1} \left(1 - x\right)}{\left(0.5 \cdot \sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}\right) \cdot \sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}}\right) \cdot \left(\left(0.5 \cdot \sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}\right) \cdot \color{blue}{\sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}}\right) \]
16. Add Preprocessing

Alternative 2: 10.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt(Math.PI);
	return ((1.0 - (Math.acos((x - 1.0)) * (t_0 * t_0))) * Math.sqrt(Math.PI)) * Math.sqrt(Math.PI);
}

Python

def code(x):
	t_0 = 1.0 / math.sqrt(math.pi)
	return ((1.0 - (math.acos((x - 1.0)) * (t_0 * t_0))) * math.sqrt(math.pi)) * math.sqrt(math.pi)

Julia

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

MATLAB

function tmp = code(x)
	t_0 = 1.0 / sqrt(pi);
	tmp = ((1.0 - (acos((x - 1.0)) * (t_0 * t_0))) * sqrt(pi)) * sqrt(pi);
end

Wolfram

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

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
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 7.3

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

3. Applied rewrites 7.3%

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

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

   3. lift-PI.f64 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-sqrt.f64 7.3

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

5. Applied rewrites 7.3%

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

   1. lift-/.f64 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 5.4

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

7. Applied rewrites 5.4%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 10.7

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

9. Applied rewrites 10.7%

   \[\leadsto \left(\left(1 - \color{blue}{\cos^{-1} \left(x - 1\right) \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{1}{\sqrt{\pi}}\right)}\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi} \]
10. Add Preprocessing

Alternative 3: 10.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
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 7.3

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

3. Applied rewrites 7.3%

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

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

   3. lift-PI.f64 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-sqrt.f64 7.3

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

5. Applied rewrites 7.3%

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

   1. lift-/.f64 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 5.4

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

7. Applied rewrites 5.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. distribute-rgt-in N/A

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

   6. *-lft-identity N/A

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

   7. lift-sqrt.f64 N/A

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

   8. sqrt-fabs-rev N/A

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

   9. lift-sqrt.f64 N/A

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

   10. rem-sqrt-square-rev N/A

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

   11. sqrt-prod N/A

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

   12. lower-fma.f64 N/A

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

9. Applied rewrites 10.7%

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

Alternative 4: 10.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.3%

   \[\cos^{-1} \left(1 - x\right) \]
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 7.3

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

3. Applied rewrites 7.3%

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

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

   3. lift-PI.f64 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-sqrt.f64 7.3

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

5. Applied rewrites 7.3%

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

   1. lift-/.f64 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 5.4

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

7. Applied rewrites 5.4%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

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

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

   5. sub-flip N/A

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

   6. lift-sqrt.f64 N/A

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

   7. sqrt-fabs-rev N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-sqrt-square-rev N/A

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

   10. sqrt-prod N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. distribute-neg-frac2 N/A

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

9. Applied rewrites 10.8%

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

Alternative 5: 9.8% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (acos (- 1.0 x)) 0.0)
   (acos (- x))
   (* (+ (sqrt PI) (* (/ (acos (- x 1.0)) (- PI)) (sqrt PI))) (sqrt PI))))

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (acos((1.0 - x)) <= 0.0)
		tmp = acos(-x);
	else
		tmp = (sqrt(pi) + ((acos((x - 1.0)) / -pi) * sqrt(pi))) * sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 0.0], N[ArcCos[(-x)], $MachinePrecision], N[(N[(N[Sqrt[Pi], $MachinePrecision] + N[(N[(N[ArcCos[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[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 7.3%

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

   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 7.0

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

   4. Applied rewrites 7.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(-1 \cdot x\right)} \]
   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 7.0

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

   6. Applied rewrites 7.0%

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

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

   1. Initial program 7.3%

      \[\cos^{-1} \left(1 - x\right) \]
   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 7.3

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

   3. Applied rewrites 7.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lift-PI.f64 N/A

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

      4. add-sqr-sqrt N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-sqrt.f64 7.3

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

   5. Applied rewrites 7.3%

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

      1. lift-/.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 5.4

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

   7. Applied rewrites 5.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   9. Applied rewrites 7.3%

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

Alternative 6: 9.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.acos(-x)
	else:
		tmp = (1.0 - (math.acos((x - 1.0)) / math.pi)) * 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(acos(Float64(x - 1.0)) / pi)) * 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 - (acos((x - 1.0)) / pi)) * pi;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 7.3%

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

   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 7.0

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

   4. Applied rewrites 7.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(-1 \cdot x\right)} \]
   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 7.0

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

   6. Applied rewrites 7.0%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 7.3%

      \[\cos^{-1} \left(1 - x\right) \]
   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 7.3

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

   3. Applied rewrites 7.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 7.3

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

   5. Applied rewrites 7.3%

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

Alternative 7: 9.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 7.3%

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

   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 7.0

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

   4. Applied rewrites 7.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(-1 \cdot x\right)} \]
   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 7.0

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

   6. Applied rewrites 7.0%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 7.3%

      \[\cos^{-1} \left(1 - x\right) \]
   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 7.3

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

   3. Applied rewrites 7.3%

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

Alternative 8: 9.8% accurate, 0.7× speedup

Math

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

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 < 5.50000000000000001e-17

   1. Initial program 7.3%

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

   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 7.0

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

   4. Applied rewrites 7.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(-1 \cdot x\right)} \]
   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 7.0

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

   6. Applied rewrites 7.0%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 7.3%

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

Alternative 9: 7.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

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

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 7.0

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

4. Applied rewrites 7.0%

   \[\leadsto \cos^{-1} \color{blue}{\left(-1 \cdot x\right)} \]
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 7.0

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

6. Applied rewrites 7.0%

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

Alternative 10: 3.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

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

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 2025156 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))
  (acos (- 1.0 x)))