bug323 (missed optimization)

Percentage Accurate: 6.9% → 10.4%

Time: 4.0s

Alternatives: 11

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. div-add N/A

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

   5. *-lft-identity N/A

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

   6. frac-2neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. mult-flip N/A

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

   11. lift-PI.f64 N/A

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

   12. common-denominator N/A

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

   13. lower-/.f64 N/A

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

5. Applied rewrites 10.4%

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

Alternative 2: 10.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(fma(Float64(0.5 / asin(-1.0)), pi, 1.0) * asin(-1.0));
	else
		tmp = Float64(pi - Float64(Float64(Float64(pi * pi) - Float64(t_0 * pi)) / pi));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(0.5 / N[ArcSin[-1.0], $MachinePrecision]), $MachinePrecision] * Pi + 1.0), $MachinePrecision] * N[ArcSin[-1.0], $MachinePrecision]), $MachinePrecision], N[(Pi - N[(N[(N[(Pi * Pi), $MachinePrecision] - N[(t$95$0 * Pi), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. *-lft-identity N/A

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

      6. frac-2neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. mult-flip N/A

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

      11. lift-PI.f64 N/A

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

      12. common-denominator N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 10.4%

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

   7. Applied rewrites 10.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 7.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 7.7%

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

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

   1. Initial program 6.9%

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

      1. lift-acos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. acos-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-acos.f64 N/A

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

      8. lower--.f64 6.9

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

   3. Applied rewrites 6.9%

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

   5. Applied rewrites 6.9%

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

Alternative 3: 10.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(fma(Float64(0.5 / asin(-1.0)), pi, 1.0) * asin(-1.0));
	else
		tmp = Float64(pi - Float64(Float64(1.0 - Float64(t_0 / pi)) * pi));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(0.5 / N[ArcSin[-1.0], $MachinePrecision]), $MachinePrecision] * Pi + 1.0), $MachinePrecision] * N[ArcSin[-1.0], $MachinePrecision]), $MachinePrecision], N[(Pi - N[(N[(1.0 - N[(t$95$0 / Pi), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. *-lft-identity N/A

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

      6. frac-2neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. mult-flip N/A

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

      11. lift-PI.f64 N/A

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

      12. common-denominator N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 10.4%

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

   7. Applied rewrites 10.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 7.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 7.7%

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

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

   1. Initial program 6.9%

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

      1. lift-acos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. acos-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-acos.f64 N/A

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

      8. lower--.f64 6.9

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

   3. Applied rewrites 6.9%

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

   5. Applied rewrites 6.9%

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

   7. Applied rewrites 6.9%

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

Alternative 4: 10.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(x - 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(1.0 / t$95$0), $MachinePrecision] / 1.0), $MachinePrecision] * N[(Pi / N[(Pi / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / (-Pi)), $MachinePrecision]), $MachinePrecision] * Pi), $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. 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-/.f64 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f64 6.9

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

5. Applied rewrites 6.9%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. sub-to-fraction N/A

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

   4. sub-flip N/A

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

   5. *-lft-identity N/A

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

   6. metadata-eval N/A

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

   7. div-add N/A

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

   8. lift-/.f64 N/A

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

   9. mult-flip N/A

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

   10. *-commutative N/A

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

   11. *-lft-identity N/A

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

   12. times-frac N/A

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

   13. metadata-eval N/A

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

   14. distribute-neg-frac N/A

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

   15. lift-/.f64 N/A

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

7. Applied rewrites 10.3%

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

Alternative 5: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. mult-flip N/A

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

   9. lift-PI.f64 N/A

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

   10. +-commutative N/A

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

   11. div-add N/A

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

   12. lift-PI.f64 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. associate-/l* N/A

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

   16. *-inverses N/A

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

   17. lower-fma.f64 N/A

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

5. Applied rewrites 10.3%

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

Alternative 6: 10.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Sqrt[N[(N[(0.5 * Pi), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + N[ArcSin[N[(x - 1.0), $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. lift--.f64 N/A

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

   3. sub-negate-rev N/A

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

   4. acos-neg N/A

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

   5. lower--.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-acos.f64 N/A

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

   8. lower--.f64 6.9

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

3. Applied rewrites 6.9%

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

5. Applied rewrites 6.9%

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

7. Applied rewrites 10.3%

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

Alternative 7: 9.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (- 1.0 x))))
   (if (<= t_0 0.0) (acos (- x)) (- PI (* (- 1.0 (/ t_0 PI)) PI)))))

C

double code(double x) {
	double t_0 = acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = acos(-x);
	} else {
		tmp = ((double) M_PI) - ((1.0 - (t_0 / ((double) M_PI))) * ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.acos((1.0 - x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.acos(-x);
	} else {
		tmp = Math.PI - ((1.0 - (t_0 / Math.PI)) * Math.PI);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.acos((1.0 - x))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.acos(-x)
	else:
		tmp = math.pi - ((1.0 - (t_0 / math.pi)) * math.pi)
	return tmp

Julia

function code(x)
	t_0 = acos(Float64(1.0 - x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = acos(Float64(-x));
	else
		tmp = Float64(pi - Float64(Float64(1.0 - Float64(t_0 / pi)) * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = acos((1.0 - x));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = acos(-x);
	else
		tmp = pi - ((1.0 - (t_0 / pi)) * pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[ArcCos[(-x)], $MachinePrecision], N[(Pi - N[(N[(1.0 - N[(t$95$0 / Pi), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 6.9

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

   4. Applied rewrites 6.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 6.9

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

   6. Applied rewrites 6.9%

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

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

   1. Initial program 6.9%

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

      1. lift-acos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. acos-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-acos.f64 N/A

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

      8. lower--.f64 6.9

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

   3. Applied rewrites 6.9%

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

   5. Applied rewrites 6.9%

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

   7. Applied rewrites 6.9%

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

Alternative 8: 9.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.5999999999999998e-17

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 6.9

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

   4. Applied rewrites 6.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 6.9

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

   6. Applied rewrites 6.9%

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

   if 5.5999999999999998e-17 < x

   1. Initial program 6.9%

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

      1. lift-acos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. acos-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-acos.f64 N/A

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

      8. lower--.f64 6.9

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

   3. Applied rewrites 6.9%

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

      1. lift-acos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. acos-neg N/A

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

      5. lift-PI.f64 N/A

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

      6. acos-asin N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. asin-neg-rev N/A

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

      10. lift-asin.f64 N/A

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

      11. add-flip-rev N/A

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

      12. +-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lift-PI.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower--.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

   5. Applied rewrites 6.9%

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

Alternative 9: 9.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \leq 0.9999999999999996:\\ \;\;\;\;\cos^{-1} \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 x) 0.9999999999999996) (acos (- 1.0 x)) (acos (- x))))

C

double code(double x) {
	double tmp;
	if ((1.0 - x) <= 0.9999999999999996) {
		tmp = acos((1.0 - x));
	} else {
		tmp = acos(-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 ((1.0d0 - x) <= 0.9999999999999996d0) then
        tmp = acos((1.0d0 - x))
    else
        tmp = acos(-x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((1.0 - x) <= 0.9999999999999996) {
		tmp = Math.acos((1.0 - x));
	} else {
		tmp = Math.acos(-x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 - x) <= 0.9999999999999996:
		tmp = math.acos((1.0 - x))
	else:
		tmp = math.acos(-x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - x) <= 0.9999999999999996)
		tmp = acos(Float64(1.0 - x));
	else
		tmp = acos(Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 - x) <= 0.9999999999999996)
		tmp = acos((1.0 - x));
	else
		tmp = acos(-x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - x), $MachinePrecision], 0.9999999999999996], N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], N[ArcCos[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) x) < 0.99999999999999956

   1. Initial program 6.9%

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

   if 0.99999999999999956 < (-.f64 #s(literal 1 binary64) x)

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 6.9

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

   4. Applied rewrites 6.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 6.9

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

   6. Applied rewrites 6.9%

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

Alternative 10: 6.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 6.9

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

4. Applied rewrites 6.9%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 6.9

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

6. Applied rewrites 6.9%

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

Alternative 11: 3.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

function code(x)
	return acos(1.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 3.8%

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

Reproduce

herbie shell --seed 2025155 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))
  (acos (- 1.0 x)))