bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.4%

Time: 2.9s

Alternatives: 9

Speedup: 1.0×

Specification

\[0 \leq x \land x \leq \frac{1}{2}\]

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (acos (- 1 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 - x), $MachinePrecision]], $MachinePrecision]

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (acos (- 1 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 - x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 10.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\\ t_1 := \frac{7}{4} \cdot \left(\pi \cdot \pi\right)\\ t_2 := \sin^{-1} \left(x - 1\right) - \frac{t\_0}{t\_1}\\ t_3 := \frac{t\_0 \cdot t\_0}{\left(t\_1 \cdot t\_1\right) \cdot t\_2}\\ t_4 := \frac{{\sin^{-1} \left(1 - x\right)}^{2}}{t\_2}\\ \frac{t\_4 \cdot t\_4 - t\_3 \cdot t\_3}{t\_4 + t\_3} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* 7/8 (* (* PI PI) PI)))
       (t_1 (* 7/4 (* PI PI)))
       (t_2 (- (asin (- x 1)) (/ t_0 t_1)))
       (t_3 (/ (* t_0 t_0) (* (* t_1 t_1) t_2)))
       (t_4 (/ (pow (asin (- 1 x)) 2) t_2)))
  (/ (- (* t_4 t_4) (* t_3 t_3)) (+ t_4 t_3))))

C

double code(double x) {
	double t_0 = 0.875 * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI));
	double t_1 = 1.75 * (((double) M_PI) * ((double) M_PI));
	double t_2 = asin((x - 1.0)) - (t_0 / t_1);
	double t_3 = (t_0 * t_0) / ((t_1 * t_1) * t_2);
	double t_4 = pow(asin((1.0 - x)), 2.0) / t_2;
	return ((t_4 * t_4) - (t_3 * t_3)) / (t_4 + t_3);
}

Java

public static double code(double x) {
	double t_0 = 0.875 * ((Math.PI * Math.PI) * Math.PI);
	double t_1 = 1.75 * (Math.PI * Math.PI);
	double t_2 = Math.asin((x - 1.0)) - (t_0 / t_1);
	double t_3 = (t_0 * t_0) / ((t_1 * t_1) * t_2);
	double t_4 = Math.pow(Math.asin((1.0 - x)), 2.0) / t_2;
	return ((t_4 * t_4) - (t_3 * t_3)) / (t_4 + t_3);
}

Python

def code(x):
	t_0 = 0.875 * ((math.pi * math.pi) * math.pi)
	t_1 = 1.75 * (math.pi * math.pi)
	t_2 = math.asin((x - 1.0)) - (t_0 / t_1)
	t_3 = (t_0 * t_0) / ((t_1 * t_1) * t_2)
	t_4 = math.pow(math.asin((1.0 - x)), 2.0) / t_2
	return ((t_4 * t_4) - (t_3 * t_3)) / (t_4 + t_3)

Julia

function code(x)
	t_0 = Float64(0.875 * Float64(Float64(pi * pi) * pi))
	t_1 = Float64(1.75 * Float64(pi * pi))
	t_2 = Float64(asin(Float64(x - 1.0)) - Float64(t_0 / t_1))
	t_3 = Float64(Float64(t_0 * t_0) / Float64(Float64(t_1 * t_1) * t_2))
	t_4 = Float64((asin(Float64(1.0 - x)) ^ 2.0) / t_2)
	return Float64(Float64(Float64(t_4 * t_4) - Float64(t_3 * t_3)) / Float64(t_4 + t_3))
end

MATLAB

function tmp = code(x)
	t_0 = 0.875 * ((pi * pi) * pi);
	t_1 = 1.75 * (pi * pi);
	t_2 = asin((x - 1.0)) - (t_0 / t_1);
	t_3 = (t_0 * t_0) / ((t_1 * t_1) * t_2);
	t_4 = (asin((1.0 - x)) ^ 2.0) / t_2;
	tmp = ((t_4 * t_4) - (t_3 * t_3)) / (t_4 + t_3);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(7/8 * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(7/4 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcSin[N[(x - 1), $MachinePrecision]], $MachinePrecision] - N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[ArcSin[N[(1 - x), $MachinePrecision]], $MachinePrecision], 2], $MachinePrecision] / t$95$2), $MachinePrecision]}, N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] - N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 6.9%

   \[\cos^{-1} \left(1 - x\right) \]
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-flip N/A

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

   4. +-commutative N/A

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

   5. sum-to-mult N/A

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

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

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

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

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

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

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-PI.f64 N/A

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

   14. asin-neg-rev N/A

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

   15. lift--.f64 N/A

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

   16. sub-negate-rev N/A

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

   17. lower-asin.f64 N/A

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

   18. lower--.f64 N/A

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

   19. asin-neg-rev N/A

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

   20. lift--.f64 N/A

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

   21. sub-negate-rev N/A

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

3. Applied rewrites 6.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-out-- N/A

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

   5. *-lft-identity N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. mult-flip N/A

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

   9. lift-PI.f64 N/A

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

   10. flip3-- N/A

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

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

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

5. Applied rewrites 10.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

7. Applied rewrites 10.4%

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

9. Applied rewrites 10.4%

   \[\leadsto \color{blue}{\frac{\frac{{\sin^{-1} \left(1 - x\right)}^{2}}{\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}} \cdot \frac{{\sin^{-1} \left(1 - x\right)}^{2}}{\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}} - \frac{\left(\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \left(\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)}{\left(\left(\frac{7}{4} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\frac{7}{4} \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}\right)} \cdot \frac{\left(\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \left(\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)}{\left(\left(\frac{7}{4} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\frac{7}{4} \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}\right)}}{\frac{{\sin^{-1} \left(1 - x\right)}^{2}}{\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}} + \frac{\left(\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \left(\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)}{\left(\left(\frac{7}{4} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\frac{7}{4} \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}\right)}}} \]
10. Add Preprocessing

Alternative 2: 10.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(\pi \cdot \pi\right) \cdot \pi\\ t_1 := \frac{7}{8} \cdot t\_0\\ t_2 := \frac{7}{4} \cdot \left(\pi \cdot \pi\right)\\ \frac{\frac{{\sin^{-1} \left(1 - x\right)}^{2} \cdot t\_2 - \frac{t\_1}{t\_2} \cdot t\_1}{t\_2}}{\sin^{-1} \left(x - 1\right) - \frac{t\_0 - \frac{1}{8} \cdot t\_0}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* PI PI) PI))
       (t_1 (* 7/8 t_0))
       (t_2 (* 7/4 (* PI PI))))
  (/
   (/ (- (* (pow (asin (- 1 x)) 2) t_2) (* (/ t_1 t_2) t_1)) t_2)
   (-
    (asin (- x 1))
    (/ (- t_0 (* 1/8 t_0)) (+ (* (* PI PI) 3/4) (* PI PI)))))))

C

double code(double x) {
	double t_0 = (((double) M_PI) * ((double) M_PI)) * ((double) M_PI);
	double t_1 = 0.875 * t_0;
	double t_2 = 1.75 * (((double) M_PI) * ((double) M_PI));
	return (((pow(asin((1.0 - x)), 2.0) * t_2) - ((t_1 / t_2) * t_1)) / t_2) / (asin((x - 1.0)) - ((t_0 - (0.125 * t_0)) / (((((double) M_PI) * ((double) M_PI)) * 0.75) + (((double) M_PI) * ((double) M_PI)))));
}

Java

public static double code(double x) {
	double t_0 = (Math.PI * Math.PI) * Math.PI;
	double t_1 = 0.875 * t_0;
	double t_2 = 1.75 * (Math.PI * Math.PI);
	return (((Math.pow(Math.asin((1.0 - x)), 2.0) * t_2) - ((t_1 / t_2) * t_1)) / t_2) / (Math.asin((x - 1.0)) - ((t_0 - (0.125 * t_0)) / (((Math.PI * Math.PI) * 0.75) + (Math.PI * Math.PI))));
}

Python

def code(x):
	t_0 = (math.pi * math.pi) * math.pi
	t_1 = 0.875 * t_0
	t_2 = 1.75 * (math.pi * math.pi)
	return (((math.pow(math.asin((1.0 - x)), 2.0) * t_2) - ((t_1 / t_2) * t_1)) / t_2) / (math.asin((x - 1.0)) - ((t_0 - (0.125 * t_0)) / (((math.pi * math.pi) * 0.75) + (math.pi * math.pi))))

Julia

function code(x)
	t_0 = Float64(Float64(pi * pi) * pi)
	t_1 = Float64(0.875 * t_0)
	t_2 = Float64(1.75 * Float64(pi * pi))
	return Float64(Float64(Float64(Float64((asin(Float64(1.0 - x)) ^ 2.0) * t_2) - Float64(Float64(t_1 / t_2) * t_1)) / t_2) / Float64(asin(Float64(x - 1.0)) - Float64(Float64(t_0 - Float64(0.125 * t_0)) / Float64(Float64(Float64(pi * pi) * 0.75) + Float64(pi * pi)))))
end

MATLAB

function tmp = code(x)
	t_0 = (pi * pi) * pi;
	t_1 = 0.875 * t_0;
	t_2 = 1.75 * (pi * pi);
	tmp = ((((asin((1.0 - x)) ^ 2.0) * t_2) - ((t_1 / t_2) * t_1)) / t_2) / (asin((x - 1.0)) - ((t_0 - (0.125 * t_0)) / (((pi * pi) * 0.75) + (pi * pi))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(7/8 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(7/4 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[Power[N[ArcSin[N[(1 - x), $MachinePrecision]], $MachinePrecision], 2], $MachinePrecision] * t$95$2), $MachinePrecision] - N[(N[(t$95$1 / t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(N[ArcSin[N[(x - 1), $MachinePrecision]], $MachinePrecision] - N[(N[(t$95$0 - N[(1/8 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 3/4), $MachinePrecision] + N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 6.9%

   \[\cos^{-1} \left(1 - x\right) \]
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-flip N/A

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

   4. +-commutative N/A

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

   5. sum-to-mult N/A

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

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

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

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

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

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

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-PI.f64 N/A

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

   14. asin-neg-rev N/A

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

   15. lift--.f64 N/A

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

   16. sub-negate-rev N/A

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

   17. lower-asin.f64 N/A

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

   18. lower--.f64 N/A

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

   19. asin-neg-rev N/A

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

   20. lift--.f64 N/A

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

   21. sub-negate-rev N/A

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

3. Applied rewrites 6.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-out-- N/A

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

   5. *-lft-identity N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. mult-flip N/A

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

   9. lift-PI.f64 N/A

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

   10. flip3-- N/A

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

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

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

5. Applied rewrites 10.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

7. Applied rewrites 10.4%

   \[\leadsto \color{blue}{\frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

9. Applied rewrites 10.4%

   \[\leadsto \frac{\color{blue}{\frac{{\sin^{-1} \left(1 - x\right)}^{2} \cdot \left(\frac{7}{4} \cdot \left(\pi \cdot \pi\right)\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \left(\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]
10. Add Preprocessing

Alternative 3: 10.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sin^{-1} \left(x - 1\right)\\ t_1 := \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}\\ \frac{t\_0 \cdot t\_0 - t\_1 \cdot t\_1}{t\_0 - t\_1} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (asin (- x 1)))
       (t_1 (/ (* 7/8 (* (* PI PI) PI)) (* 7/4 (* PI PI)))))
  (/ (- (* t_0 t_0) (* t_1 t_1)) (- t_0 t_1))))

C

double code(double x) {
	double t_0 = asin((x - 1.0));
	double t_1 = (0.875 * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))) / (1.75 * (((double) M_PI) * ((double) M_PI)));
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
}

Java

public static double code(double x) {
	double t_0 = Math.asin((x - 1.0));
	double t_1 = (0.875 * ((Math.PI * Math.PI) * Math.PI)) / (1.75 * (Math.PI * Math.PI));
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
}

Python

def code(x):
	t_0 = math.asin((x - 1.0))
	t_1 = (0.875 * ((math.pi * math.pi) * math.pi)) / (1.75 * (math.pi * math.pi))
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)

Julia

function code(x)
	t_0 = asin(Float64(x - 1.0))
	t_1 = Float64(Float64(0.875 * Float64(Float64(pi * pi) * pi)) / Float64(1.75 * Float64(pi * pi)))
	return Float64(Float64(Float64(t_0 * t_0) - Float64(t_1 * t_1)) / Float64(t_0 - t_1))
end

MATLAB

function tmp = code(x)
	t_0 = asin((x - 1.0));
	t_1 = (0.875 * ((pi * pi) * pi)) / (1.75 * (pi * pi));
	tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(x - 1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(7/8 * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] / N[(7/4 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.9%

   \[\cos^{-1} \left(1 - x\right) \]
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-flip N/A

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

   4. +-commutative N/A

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

   5. sum-to-mult N/A

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

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

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

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

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

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

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-PI.f64 N/A

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

   14. asin-neg-rev N/A

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

   15. lift--.f64 N/A

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

   16. sub-negate-rev N/A

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

   17. lower-asin.f64 N/A

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

   18. lower--.f64 N/A

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

   19. asin-neg-rev N/A

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

   20. lift--.f64 N/A

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

   21. sub-negate-rev N/A

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

3. Applied rewrites 6.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-out-- N/A

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

   5. *-lft-identity N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. mult-flip N/A

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

   9. lift-PI.f64 N/A

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

   10. flip3-- N/A

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

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

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

5. Applied rewrites 10.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

7. Applied rewrites 10.4%

   \[\leadsto \color{blue}{\frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. distribute-rgt1-in N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval 10.4%

      \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\color{blue}{\frac{7}{8}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4}} + \pi \cdot \pi} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\frac{3}{4} \cdot \left(\pi \cdot \pi\right)} + \pi \cdot \pi} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

   11. distribute-lft1-in N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\frac{3}{4} + 1\right) \cdot \left(\pi \cdot \pi\right)}} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\frac{3}{4} + 1\right) \cdot \left(\pi \cdot \pi\right)}} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

   13. metadata-eval 10.4%

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\frac{7}{4}} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

9. Applied rewrites 10.4%

   \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \color{blue}{\frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]
10. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\color{blue}{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \color{blue}{\frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

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

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\color{blue}{\left(\pi \cdot \pi\right) \cdot \pi + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    4. distribute-rgt1-in N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) + 1\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) + 1\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    6. metadata-eval N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\left(\color{blue}{\frac{-1}{8}} + 1\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    7. metadata-eval 10.4%

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\color{blue}{\frac{7}{8}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    8. lift-+.f64 N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    9. lift-*.f64 N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4}} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    10. *-commutative N/A

        \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\frac{3}{4} \cdot \left(\pi \cdot \pi\right)} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    11. distribute-lft1-in N/A

        \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\frac{3}{4} + 1\right) \cdot \left(\pi \cdot \pi\right)}}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    12. lower-*.f64 N/A

        \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\frac{3}{4} + 1\right) \cdot \left(\pi \cdot \pi\right)}}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    13. metadata-eval 10.4%

        \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\frac{7}{4}} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

11. Applied rewrites 10.4%

    \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \color{blue}{\frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]
12. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\color{blue}{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \color{blue}{\frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

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

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\color{blue}{\left(\pi \cdot \pi\right) \cdot \pi + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    4. distribute-rgt1-in N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) + 1\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) + 1\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    6. metadata-eval N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\color{blue}{\frac{-1}{8}} + 1\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    7. metadata-eval 10.4%

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\color{blue}{\frac{7}{8}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}} \]

    8. lift-+.f64 N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}} \]

    9. lift-*.f64 N/A

       \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4}} + \pi \cdot \pi}} \]

    10. *-commutative N/A

        \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\frac{3}{4} \cdot \left(\pi \cdot \pi\right)} + \pi \cdot \pi}} \]

    11. distribute-lft1-in N/A

        \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\frac{3}{4} + 1\right) \cdot \left(\pi \cdot \pi\right)}}} \]

    12. lower-*.f64 N/A

        \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\left(\frac{3}{4} + 1\right) \cdot \left(\pi \cdot \pi\right)}}} \]

    13. metadata-eval 10.4%

        \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\color{blue}{\frac{7}{4}} \cdot \left(\pi \cdot \pi\right)}} \]

13. Applied rewrites 10.4%

    \[\leadsto \frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} \cdot \frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}{\sin^{-1} \left(x - 1\right) - \color{blue}{\frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}}} \]
14. Add Preprocessing

Alternative 4: 10.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{7}{4} \cdot \left(\pi \cdot \pi\right)\\ \frac{t\_0 \cdot \sin^{-1} \left(x - 1\right) + \frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{t\_0} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* 7/4 (* PI PI))))
  (/ (+ (* t_0 (asin (- x 1))) (* 7/8 (* (* PI PI) PI))) t_0)))

C

double code(double x) {
	double t_0 = 1.75 * (((double) M_PI) * ((double) M_PI));
	return ((t_0 * asin((x - 1.0))) + (0.875 * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)))) / t_0;
}

Java

public static double code(double x) {
	double t_0 = 1.75 * (Math.PI * Math.PI);
	return ((t_0 * Math.asin((x - 1.0))) + (0.875 * ((Math.PI * Math.PI) * Math.PI))) / t_0;
}

Python

def code(x):
	t_0 = 1.75 * (math.pi * math.pi)
	return ((t_0 * math.asin((x - 1.0))) + (0.875 * ((math.pi * math.pi) * math.pi))) / t_0

Julia

function code(x)
	t_0 = Float64(1.75 * Float64(pi * pi))
	return Float64(Float64(Float64(t_0 * asin(Float64(x - 1.0))) + Float64(0.875 * Float64(Float64(pi * pi) * pi))) / t_0)
end

MATLAB

function tmp = code(x)
	t_0 = 1.75 * (pi * pi);
	tmp = ((t_0 * asin((x - 1.0))) + (0.875 * ((pi * pi) * pi))) / t_0;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(7/4 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[ArcSin[N[(x - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(7/8 * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 6.9%

   \[\cos^{-1} \left(1 - x\right) \]
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-flip N/A

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

   4. +-commutative N/A

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

   5. sum-to-mult N/A

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

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

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

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

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

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

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-PI.f64 N/A

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

   14. asin-neg-rev N/A

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

   15. lift--.f64 N/A

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

   16. sub-negate-rev N/A

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

   17. lower-asin.f64 N/A

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

   18. lower--.f64 N/A

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

   19. asin-neg-rev N/A

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

   20. lift--.f64 N/A

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

   21. sub-negate-rev N/A

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

3. Applied rewrites 6.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-out-- N/A

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

   5. *-lft-identity N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. mult-flip N/A

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

   9. lift-PI.f64 N/A

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

   10. flip3-- N/A

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

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

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

5. Applied rewrites 10.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

7. Applied rewrites 10.4%

   \[\leadsto \color{blue}{\frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}} \]
8. Step-by-step derivation

9. Applied rewrites 10.4%

   \[\leadsto \color{blue}{\frac{\left(\frac{7}{4} \cdot \left(\pi \cdot \pi\right)\right) \cdot \sin^{-1} \left(x - 1\right) + \frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)}} \]
10. Add Preprocessing

Alternative 5: 10.4% accurate, 0.7× speedup

Math

\[\frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} + \sin^{-1} \left(x - 1\right) \]

FPCore

(FPCore (x)
  :precision binary64
  (+ (/ (* 7/8 (* (* PI PI) PI)) (* 7/4 (* PI PI))) (asin (- x 1))))

C

double code(double x) {
	return ((0.875 * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))) / (1.75 * (((double) M_PI) * ((double) M_PI)))) + asin((x - 1.0));
}

Java

public static double code(double x) {
	return ((0.875 * ((Math.PI * Math.PI) * Math.PI)) / (1.75 * (Math.PI * Math.PI))) + Math.asin((x - 1.0));
}

Python

def code(x):
	return ((0.875 * ((math.pi * math.pi) * math.pi)) / (1.75 * (math.pi * math.pi))) + math.asin((x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(0.875 * Float64(Float64(pi * pi) * pi)) / Float64(1.75 * Float64(pi * pi))) + asin(Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((0.875 * ((pi * pi) * pi)) / (1.75 * (pi * pi))) + asin((x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(7/8 * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] / N[(7/4 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[ArcSin[N[(x - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[\cos^{-1} \left(1 - x\right) \]
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-flip N/A

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

   4. +-commutative N/A

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

   5. sum-to-mult N/A

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

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

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

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

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

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

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-PI.f64 N/A

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

   14. asin-neg-rev N/A

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

   15. lift--.f64 N/A

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

   16. sub-negate-rev N/A

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

   17. lower-asin.f64 N/A

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

   18. lower--.f64 N/A

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

   19. asin-neg-rev N/A

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

   20. lift--.f64 N/A

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

   21. sub-negate-rev N/A

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

3. Applied rewrites 6.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-out-- N/A

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

   5. *-lft-identity N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. mult-flip N/A

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

   9. lift-PI.f64 N/A

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

   10. flip3-- N/A

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

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

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

5. Applied rewrites 10.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

7. Applied rewrites 10.4%

   \[\leadsto \color{blue}{\frac{\sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}{\sin^{-1} \left(x - 1\right) - \frac{\left(\pi \cdot \pi\right) \cdot \pi - \frac{1}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\left(\pi \cdot \pi\right) \cdot \frac{3}{4} + \pi \cdot \pi}}} \]
8. Step-by-step derivation

9. Applied rewrites 10.4%

   \[\leadsto \color{blue}{\frac{\frac{7}{8} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{\frac{7}{4} \cdot \left(\pi \cdot \pi\right)} + \sin^{-1} \left(x - 1\right)} \]
10. Add Preprocessing

Alternative 6: 9.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{8924260225606735}{162259276829213363391578010288128}:\\ \;\;\;\;\cos^{-1} 0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-3 \cdot \cos^{-1} \left(1 - x\right)\right) \cdot \pi}{-3}}{\pi}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 8924260225606735/162259276829213363391578010288128)
  (acos 0)
  (/ (/ (* (* -3 (acos (- 1 x))) PI) -3) PI)))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(0.0);
	} else {
		tmp = (((-3.0 * acos((1.0 - x))) * ((double) M_PI)) / -3.0) / ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = Math.acos(0.0);
	} else {
		tmp = (((-3.0 * Math.acos((1.0 - x))) * Math.PI) / -3.0) / Math.PI;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.acos(0.0)
	else:
		tmp = (((-3.0 * math.acos((1.0 - x))) * math.pi) / -3.0) / math.pi
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 8924260225606735/162259276829213363391578010288128], N[ArcCos[0], $MachinePrecision], N[(N[(N[(N[(-3 * N[ArcCos[N[(1 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] / -3), $MachinePrecision] / Pi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5e-17

   1. Initial program 6.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \cos^{-1} \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 3.8%

      \[\leadsto \cos^{-1} \color{blue}{1} \]

   5. Taylor expanded in undef-var around zero

      \[\leadsto \cos^{-1} 0 \]
   6. Step-by-step derivation

   7. Applied rewrites 6.9%

      \[\leadsto \cos^{-1} 0 \]

   if 5.5e-17 < x

   1. Initial program 6.9%

      \[\cos^{-1} \left(1 - x\right) \]
   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. sub-to-mult N/A

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

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

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

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

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

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

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

      9. lower-acos.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-PI.f64 6.9%

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

   3. Applied rewrites 6.9%

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

      1. lift-acos.f64 N/A

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

      2. acos-asin N/A

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

      3. lift-asin.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 6.9%

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

   5. Applied rewrites 6.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. frac-sub N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 6.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 6.9%

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

Alternative 7: 9.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{8924260225606735}{162259276829213363391578010288128}:\\ \;\;\;\;\cos^{-1} 0\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= x 8924260225606735/162259276829213363391578010288128)
  (acos 0)
  (acos (- 1 x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(0.0);
	} 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(0.0d0)
    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(0.0);
	} else {
		tmp = Math.acos((1.0 - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.5e-17:
		tmp = math.acos(0.0)
	else:
		tmp = math.acos((1.0 - x))
	return tmp

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 8924260225606735/162259276829213363391578010288128], N[ArcCos[0], $MachinePrecision], N[ArcCos[N[(1 - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5e-17

   1. Initial program 6.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \cos^{-1} \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 3.8%

      \[\leadsto \cos^{-1} \color{blue}{1} \]

   5. Taylor expanded in undef-var around zero

      \[\leadsto \cos^{-1} 0 \]
   6. Step-by-step derivation

   7. Applied rewrites 6.9%

      \[\leadsto \cos^{-1} 0 \]

   if 5.5e-17 < x

   1. Initial program 6.9%

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

Alternative 8: 6.9% accurate, 1.0× speedup

Math

\[\cos^{-1} 0 \]

FPCore

(FPCore (x)
  :precision binary64
  (acos 0))

C

double code(double x) {
	return acos(0.0);
}

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(0.0d0)
end function

Java

public static double code(double x) {
	return Math.acos(0.0);
}

Python

def code(x):
	return math.acos(0.0)

Julia

function code(x)
	return acos(0.0)
end

MATLAB

function tmp = code(x)
	tmp = acos(0.0);
end

Wolfram

code[x_] := N[ArcCos[0], $MachinePrecision]

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around 0

   \[\leadsto \cos^{-1} \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]

5. Taylor expanded in undef-var around zero

   \[\leadsto \cos^{-1} 0 \]
6. Step-by-step derivation

7. Applied rewrites 6.9%

   \[\leadsto \cos^{-1} 0 \]
8. Add Preprocessing

Alternative 9: 3.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (acos 1))

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], $MachinePrecision]

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around 0

   \[\leadsto \cos^{-1} \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 3.8%

   \[\leadsto \cos^{-1} \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0 x) (<= x 1/2))
  (acos (- 1 x)))