bug323 (missed optimization)

Percentage Accurate: 6.8% → 10.3%

Time: 4.1s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.8% 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: 10.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(x - 1\right)\\ t_1 := \frac{\pi}{-2} + t\_0\\ t_2 := \sin^{-1} \left(1 - x\right)\\ t_3 := {t\_2}^{3} + {\left(\frac{\pi}{2}\right)}^{3}\\ \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.5 \cdot \pi, \cos^{-1} \left(1 - x\right), {t\_2}^{2}\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.25, t\_1, t\_3 \cdot {t\_0}^{2}\right)}{t\_3 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- x 1.0)))
        (t_1 (+ (/ PI -2.0) t_0))
        (t_2 (asin (- 1.0 x)))
        (t_3 (+ (pow t_2 3.0) (pow (/ PI 2.0) 3.0))))
   (/
    (fma
     (* (* (fma (* 0.5 PI) (acos (- 1.0 x)) (pow t_2 2.0)) (* PI PI)) 0.25)
     t_1
     (* t_3 (pow t_0 2.0)))
    (* t_3 t_1))))

C

double code(double x) {
	double t_0 = asin((x - 1.0));
	double t_1 = (((double) M_PI) / -2.0) + t_0;
	double t_2 = asin((1.0 - x));
	double t_3 = pow(t_2, 3.0) + pow((((double) M_PI) / 2.0), 3.0);
	return fma(((fma((0.5 * ((double) M_PI)), acos((1.0 - x)), pow(t_2, 2.0)) * (((double) M_PI) * ((double) M_PI))) * 0.25), t_1, (t_3 * pow(t_0, 2.0))) / (t_3 * t_1);
}

Julia

function code(x)
	t_0 = asin(Float64(x - 1.0))
	t_1 = Float64(Float64(pi / -2.0) + t_0)
	t_2 = asin(Float64(1.0 - x))
	t_3 = Float64((t_2 ^ 3.0) + (Float64(pi / 2.0) ^ 3.0))
	return Float64(fma(Float64(Float64(fma(Float64(0.5 * pi), acos(Float64(1.0 - x)), (t_2 ^ 2.0)) * Float64(pi * pi)) * 0.25), t_1, Float64(t_3 * (t_0 ^ 2.0))) / Float64(t_3 * t_1))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(x - 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi / -2.0), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$2, 3.0], $MachinePrecision] + N[Power[N[(Pi / 2.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(0.5 * Pi), $MachinePrecision] * N[ArcCos[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$1 + N[(t$95$3 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-acos.f64 N/A

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

   3. acos-asin N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-asin.f64 N/A

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

   8. lift--.f64 6.8

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

3. Applied rewrites 6.8%

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

5. Applied rewrites 10.3%

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

7. Applied rewrites 10.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \cos^{-1} \left(1 - x\right)\right) + {\sin^{-1} \left(x - 1\right)}^{2}\right)\right)}, \frac{\pi}{-2} + \sin^{-1} \left(x - 1\right), \left({\sin^{-1} \left(1 - x\right)}^{3} + {\left(\frac{\pi}{2}\right)}^{3}\right) \cdot {\sin^{-1} \left(x - 1\right)}^{2}\right)}{\left({\sin^{-1} \left(1 - x\right)}^{3} + {\left(\frac{\pi}{2}\right)}^{3}\right) \cdot \left(\frac{\pi}{-2} + \sin^{-1} \left(x - 1\right)\right)} \]

10. Applied rewrites 10.3%

    \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(0.5 \cdot \pi, \cos^{-1} \left(1 - x\right), {\sin^{-1} \left(1 - x\right)}^{2}\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.25}, \frac{\pi}{-2} + \sin^{-1} \left(x - 1\right), \left({\sin^{-1} \left(1 - x\right)}^{3} + {\left(\frac{\pi}{2}\right)}^{3}\right) \cdot {\sin^{-1} \left(x - 1\right)}^{2}\right)}{\left({\sin^{-1} \left(1 - x\right)}^{3} + {\left(\frac{\pi}{2}\right)}^{3}\right) \cdot \left(\frac{\pi}{-2} + \sin^{-1} \left(x - 1\right)\right)} \]
11. Add Preprocessing

Alternative 2: 10.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(x - 1\right)\\ \mathsf{fma}\left(\pi \cdot \pi, \frac{0.25}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}, \frac{{t\_0}^{2}}{\mathsf{fma}\left(-0.5, \pi, t\_0\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- x 1.0))))
   (fma
    (* PI PI)
    (/ 0.25 (fma PI 0.5 (asin (- 1.0 x))))
    (/ (pow t_0 2.0) (fma -0.5 PI t_0)))))

C

double code(double x) {
	double t_0 = asin((x - 1.0));
	return fma((((double) M_PI) * ((double) M_PI)), (0.25 / fma(((double) M_PI), 0.5, asin((1.0 - x)))), (pow(t_0, 2.0) / fma(-0.5, ((double) M_PI), t_0)));
}

Julia

function code(x)
	t_0 = asin(Float64(x - 1.0))
	return fma(Float64(pi * pi), Float64(0.25 / fma(pi, 0.5, asin(Float64(1.0 - x)))), Float64((t_0 ^ 2.0) / fma(-0.5, pi, t_0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(x - 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(Pi * Pi), $MachinePrecision] * N[(0.25 / N[(Pi * 0.5 + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[(-0.5 * Pi + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-acos.f64 N/A

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

   3. acos-asin N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-asin.f64 N/A

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

   8. lift--.f64 6.8

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

3. Applied rewrites 6.8%

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

4. Taylor expanded in x around 0

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

   1. acos-asin-rev N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-asin.f64 N/A

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

   6. lift--.f64 6.8

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

6. Applied rewrites 6.8%

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

8. Applied rewrites 10.3%

   \[\leadsto \mathsf{fma}\left(\pi \cdot \pi, \color{blue}{\frac{0.25}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(1 - x\right)\right)}}, \frac{{\sin^{-1} \left(x - 1\right)}^{2}}{\mathsf{fma}\left(-0.5, \pi, \sin^{-1} \left(x - 1\right)\right)}\right) \]
9. Add Preprocessing

Alternative 3: 10.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \frac{\mathsf{fma}\left(t\_0, \sin^{-1} \left(-1 + x\right), {\left(\frac{\pi}{2}\right)}^{2}\right)}{\frac{\pi}{2} + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (/ (fma t_0 (asin (+ -1.0 x)) (pow (/ PI 2.0) 2.0)) (+ (/ PI 2.0) t_0))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	return fma(t_0, asin((-1.0 + x)), pow((((double) M_PI) / 2.0), 2.0)) / ((((double) M_PI) / 2.0) + t_0);
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	return Float64(fma(t_0, asin(Float64(-1.0 + x)), (Float64(pi / 2.0) ^ 2.0)) / Float64(Float64(pi / 2.0) + t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * N[ArcSin[N[(-1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Power[N[(Pi / 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-acos.f64 N/A

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

   3. acos-asin N/A

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

   4. flip-- N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-asin.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lower-asin.f64 N/A

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

   16. lift--.f64 N/A

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

   17. lower-+.f64 N/A

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

3. Applied rewrites 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-asin.f64 N/A

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

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

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

5. Applied rewrites 10.3%

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

Alternative 4: 10.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin^{-1} \left(1 - x\right)}^{0.5}\\ \frac{\pi}{2} - t\_0 \cdot t\_0 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (asin (- 1.0 x)) 0.5))) (- (/ PI 2.0) (* t_0 t_0))))

C

double code(double x) {
	double t_0 = pow(asin((1.0 - x)), 0.5);
	return (((double) M_PI) / 2.0) - (t_0 * t_0);
}

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.asin((1.0 - x)), 0.5);
	return (Math.PI / 2.0) - (t_0 * t_0);
}

Python

def code(x):
	t_0 = math.pow(math.asin((1.0 - x)), 0.5)
	return (math.pi / 2.0) - (t_0 * t_0)

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x)) ^ 0.5
	return Float64(Float64(pi / 2.0) - Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = asin((1.0 - x)) ^ 0.5;
	tmp = (pi / 2.0) - (t_0 * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 0.5], $MachinePrecision]}, N[(N[(Pi / 2.0), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-acos.f64 N/A

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

   3. acos-asin N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-asin.f64 N/A

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

   8. lift--.f64 6.8

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

3. Applied rewrites 6.8%

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

   1. lift--.f64 N/A

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

   2. lift-asin.f64 N/A

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

   3. unpow1 N/A

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

   4. sqr-pow N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-asin.f64 N/A

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

   9. lift--.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-asin.f64 N/A

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

   13. lift--.f64 10.2

       \[\leadsto \frac{\pi}{2} - {\sin^{-1} \left(1 - x\right)}^{0.5} \cdot {\sin^{-1} \color{blue}{\left(1 - x\right)}}^{0.5} \]

5. Applied rewrites 10.2%

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

Alternative 5: 9.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\pi}{2}, \frac{\pi}{2}, \sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(-x\right)\right)}{\frac{\pi}{2} + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - t\_0 \cdot t\_0}{\frac{\pi}{2} + \sin^{-1} \left(\frac{1 - x \cdot x}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (asin (- 1.0 x))))
   (if (<= x 5.5e-17)
     (/
      (fma (/ PI 2.0) (/ PI 2.0) (* (asin (- x 1.0)) (asin (- x))))
      (+ (/ PI 2.0) t_0))
     (/
      (- (* (/ PI 2.0) (/ PI 2.0)) (* t_0 t_0))
      (+ (/ PI 2.0) (asin (/ (- 1.0 (* x x)) (+ x 1.0))))))))

C

double code(double x) {
	double t_0 = asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = fma((((double) M_PI) / 2.0), (((double) M_PI) / 2.0), (asin((x - 1.0)) * asin(-x))) / ((((double) M_PI) / 2.0) + t_0);
	} else {
		tmp = (((((double) M_PI) / 2.0) * (((double) M_PI) / 2.0)) - (t_0 * t_0)) / ((((double) M_PI) / 2.0) + asin(((1.0 - (x * x)) / (x + 1.0))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = asin(Float64(1.0 - x))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(fma(Float64(pi / 2.0), Float64(pi / 2.0), Float64(asin(Float64(x - 1.0)) * asin(Float64(-x)))) / Float64(Float64(pi / 2.0) + t_0));
	else
		tmp = Float64(Float64(Float64(Float64(pi / 2.0) * Float64(pi / 2.0)) - Float64(t_0 * t_0)) / Float64(Float64(pi / 2.0) + asin(Float64(Float64(1.0 - Float64(x * x)) / Float64(x + 1.0)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(Pi / 2.0), $MachinePrecision] + N[(N[ArcSin[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * N[ArcSin[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] + N[ArcSin[N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. lift--.f64 N/A

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

      2. lift-acos.f64 N/A

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

      3. acos-asin N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-asin.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-asin.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-+.f64 N/A

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

   3. Applied rewrites 3.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 3.9

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.6

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

   8. Applied rewrites 6.6%

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

   10. Applied rewrites 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 62.1%

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

      1. lift--.f64 N/A

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

      2. lift-acos.f64 N/A

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

      3. acos-asin N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-asin.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-asin.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-+.f64 N/A

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

   3. Applied rewrites 62.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 62.1

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

   5. Applied rewrites 62.1%

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

Alternative 6: 9.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{2} + \sin^{-1} \left(1 - x\right)\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\pi}{2}, \frac{\pi}{2}, \sin^{-1} \left(x - 1\right) \cdot \sin^{-1} \left(-x\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\pi, \frac{\pi}{4}, -{\sin^{-1} \left(-1 + x\right)}^{2}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ PI 2.0) (asin (- 1.0 x)))))
   (if (<= x 5.5e-17)
     (/ (fma (/ PI 2.0) (/ PI 2.0) (* (asin (- x 1.0)) (asin (- x)))) t_0)
     (/ (fma PI (/ PI 4.0) (- (pow (asin (+ -1.0 x)) 2.0))) t_0))))

C

double code(double x) {
	double t_0 = (((double) M_PI) / 2.0) + asin((1.0 - x));
	double tmp;
	if (x <= 5.5e-17) {
		tmp = fma((((double) M_PI) / 2.0), (((double) M_PI) / 2.0), (asin((x - 1.0)) * asin(-x))) / t_0;
	} else {
		tmp = fma(((double) M_PI), (((double) M_PI) / 4.0), -pow(asin((-1.0 + x)), 2.0)) / t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(pi / 2.0) + asin(Float64(1.0 - x)))
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = Float64(fma(Float64(pi / 2.0), Float64(pi / 2.0), Float64(asin(Float64(x - 1.0)) * asin(Float64(-x)))) / t_0);
	else
		tmp = Float64(fma(pi, Float64(pi / 4.0), Float64(-(asin(Float64(-1.0 + x)) ^ 2.0))) / t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(Pi / 2.0), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.5e-17], N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(Pi / 2.0), $MachinePrecision] + N[(N[ArcSin[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * N[ArcSin[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(Pi * N[(Pi / 4.0), $MachinePrecision] + (-N[Power[N[ArcSin[N[(-1.0 + x), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

      1. lift--.f64 N/A

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

      2. lift-acos.f64 N/A

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

      3. acos-asin N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-asin.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-asin.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-+.f64 N/A

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

   3. Applied rewrites 3.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 3.9

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

   5. Applied rewrites 3.9%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.6

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

   8. Applied rewrites 6.6%

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

   10. Applied rewrites 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 62.1%

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

      1. lift--.f64 N/A

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

      2. lift-acos.f64 N/A

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

      3. acos-asin N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-asin.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-asin.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-+.f64 N/A

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

   3. Applied rewrites 62.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-asin.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-asin.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 62.0%

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

Alternative 7: 9.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\pi, \frac{\pi}{4}, -{\sin^{-1} \left(-1 + x\right)}^{2}\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17)
   (acos (- x))
   (/
    (fma PI (/ PI 4.0) (- (pow (asin (+ -1.0 x)) 2.0)))
    (+ (/ PI 2.0) (asin (- 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(-x);
	} else {
		tmp = fma(((double) M_PI), (((double) M_PI) / 4.0), -pow(asin((-1.0 + x)), 2.0)) / ((((double) M_PI) / 2.0) + asin((1.0 - x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = acos(Float64(-x));
	else
		tmp = Float64(fma(pi, Float64(pi / 4.0), Float64(-(asin(Float64(-1.0 + x)) ^ 2.0))) / Float64(Float64(pi / 2.0) + asin(Float64(1.0 - x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[(-x)], $MachinePrecision], N[(N[(Pi * N[(Pi / 4.0), $MachinePrecision] + (-N[Power[N[ArcSin[N[(-1.0 + x), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] / N[(N[(Pi / 2.0), $MachinePrecision] + N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

   2. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.6

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

   4. Applied rewrites 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 62.1%

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

      1. lift--.f64 N/A

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

      2. lift-acos.f64 N/A

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

      3. acos-asin N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-asin.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-asin.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-+.f64 N/A

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

   3. Applied rewrites 62.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-asin.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-asin.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 62.0%

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

Alternative 8: 9.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\cos^{-1} \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.25, \pi \cdot \pi, {\sin^{-1} \left(1 - x\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, \pi, \sin^{-1} \left(x - 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.5e-17)
   (acos (- x))
   (/
    (fma -0.25 (* PI PI) (pow (asin (- 1.0 x)) 2.0))
    (fma -0.5 PI (asin (- x 1.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 5.5e-17) {
		tmp = acos(-x);
	} else {
		tmp = fma(-0.25, (((double) M_PI) * ((double) M_PI)), pow(asin((1.0 - x)), 2.0)) / fma(-0.5, ((double) M_PI), asin((x - 1.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.5e-17)
		tmp = acos(Float64(-x));
	else
		tmp = Float64(fma(-0.25, Float64(pi * pi), (asin(Float64(1.0 - x)) ^ 2.0)) / fma(-0.5, pi, asin(Float64(x - 1.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.5e-17], N[ArcCos[(-x)], $MachinePrecision], N[(N[(-0.25 * N[(Pi * Pi), $MachinePrecision] + N[Power[N[ArcSin[N[(1.0 - x), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(-0.5 * Pi + N[ArcSin[N[(x - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

   2. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.6

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

   4. Applied rewrites 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 62.1%

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

      1. lift--.f64 N/A

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

      2. lift-acos.f64 N/A

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

      3. acos-asin N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-asin.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower-asin.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lower-+.f64 N/A

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

   3. Applied rewrites 62.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 13.6

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

   8. Applied rewrites 13.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(\frac{1 - {x}^{2}}{1 + x}\right)}{\sin^{-1} \left(1 - x\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)}} \]

   11. Applied rewrites 62.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.25, \pi \cdot \pi, {\sin^{-1} \left(1 - x\right)}^{2}\right)}{\mathsf{fma}\left(-0.5, \pi, \sin^{-1} \left(x - 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 9.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

   2. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.6

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

   4. Applied rewrites 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 62.1%

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

      1. lift--.f64 N/A

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

      2. lift-acos.f64 N/A

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

      3. acos-asin N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-asin.f64 N/A

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

      8. lift--.f64 62.0

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in x around 0

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

      1. acos-asin-rev N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-asin.f64 N/A

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

      6. lift--.f64 62.0

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

   6. Applied rewrites 62.0%

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

Alternative 10: 9.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.5d-17) then
        tmp = acos(-x)
    else
        tmp = acos((1.0d0 - x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000001e-17

   1. Initial program 3.9%

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

   2. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.6

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

   4. Applied rewrites 6.6%

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

   if 5.50000000000000001e-17 < x

   1. Initial program 62.1%

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

Alternative 11: 6.9% accurate, 1.0× 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 6.8%

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

2. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 6.9

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

4. Applied rewrites 6.9%

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

Alternative 12: 3.8% accurate, 1.0× 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 6.8%

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

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

Developer Target 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sin^{-1} \left(\sqrt{\frac{x}{2}}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (asin (sqrt (/ x 2.0)))))

C

double code(double x) {
	return 2.0 * asin(sqrt((x / 2.0)));
}

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 = 2.0d0 * asin(sqrt((x / 2.0d0)))
end function

Java

public static double code(double x) {
	return 2.0 * Math.asin(Math.sqrt((x / 2.0)));
}

Python

def code(x):
	return 2.0 * math.asin(math.sqrt((x / 2.0)))

Julia

function code(x)
	return Float64(2.0 * asin(sqrt(Float64(x / 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * asin(sqrt((x / 2.0)));
end

Wolfram

code[x_] := N[(2.0 * N[ArcSin[N[Sqrt[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025103 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :alt
  (! :herbie-platform default (* 2 (asin (sqrt (/ x 2)))))

  (acos (- 1.0 x)))