Ian Simplification

Percentage Accurate: 6.6% → 8.1%

Time: 13.6s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

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

Initial Program: 6.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 8.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)\\ t_1 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_2 := t\_0 + t\_1\\ t\_0 - \left(\frac{{t\_1}^{2}}{t\_2} - \frac{{t\_0}^{2}}{t\_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (acos (sqrt (/ (- 1.0 x) 2.0))))
        (t_1 (/ (PI) 2.0))
        (t_2 (+ t_0 t_1)))
   (- t_0 (- (/ (pow t_1 2.0) t_2) (/ (pow t_0 2.0) t_2)))))

Derivation

1. Initial program 8.3%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2-rev N/A

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

   4. associate--r+ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-asin.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-acos.f64 8.3

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

4. Applied rewrites 8.3%

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

   1. lift-asin.f64 N/A

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

   2. asin-acos-rev N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-acos.f64 N/A

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

   6. flip-- N/A

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

   7. div-sub N/A

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

   8. lower--.f64 N/A

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

6. Applied rewrites 10.1%

   \[\leadsto \cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) - \color{blue}{\left(\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}} - \frac{{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right)}^{2}}{\cos^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}}\right)} \]
7. Add Preprocessing

Alternative 2: 8.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{2}\\ \cos^{-1} \left({\left({t\_0}^{0.25}\right)}^{2}\right) - \sin^{-1} \left(\sqrt{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) 2.0)))
   (- (acos (pow (pow t_0 0.25) 2.0)) (asin (sqrt t_0)))))

C

double code(double x) {
	double t_0 = (1.0 - x) / 2.0;
	return acos(pow(pow(t_0, 0.25), 2.0)) - asin(sqrt(t_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
    real(8) :: t_0
    t_0 = (1.0d0 - x) / 2.0d0
    code = acos(((t_0 ** 0.25d0) ** 2.0d0)) - asin(sqrt(t_0))
end function

Java

public static double code(double x) {
	double t_0 = (1.0 - x) / 2.0;
	return Math.acos(Math.pow(Math.pow(t_0, 0.25), 2.0)) - Math.asin(Math.sqrt(t_0));
}

Python

def code(x):
	t_0 = (1.0 - x) / 2.0
	return math.acos(math.pow(math.pow(t_0, 0.25), 2.0)) - math.asin(math.sqrt(t_0))

Julia

function code(x)
	t_0 = Float64(Float64(1.0 - x) / 2.0)
	return Float64(acos(((t_0 ^ 0.25) ^ 2.0)) - asin(sqrt(t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = (1.0 - x) / 2.0;
	tmp = acos(((t_0 ^ 0.25) ^ 2.0)) - asin(sqrt(t_0));
end

Wolfram

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

Derivation

1. Initial program 8.3%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2-rev N/A

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

   4. associate--r+ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-asin.f64 N/A

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

   8. acos-asin N/A

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

   9. lower--.f64 N/A

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

   10. lower-acos.f64 8.3

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

4. Applied rewrites 8.3%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval 10.0

       \[\leadsto \cos^{-1} \left({\left(\frac{1 - x}{2}\right)}^{0.25} \cdot {\left(\frac{1 - x}{2}\right)}^{\color{blue}{0.25}}\right) - \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

6. Applied rewrites 10.0%

   \[\leadsto \cos^{-1} \color{blue}{\left({\left(\frac{1 - x}{2}\right)}^{0.25} \cdot {\left(\frac{1 - x}{2}\right)}^{0.25}\right)} - \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. pow2 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left({\left({\left(\frac{1 - x}{2}\right)}^{\frac{1}{4}}\right)}^{2}\right)} - \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

   3. lower-pow.f64 10.0

      \[\leadsto \cos^{-1} \color{blue}{\left({\left({\left(\frac{1 - x}{2}\right)}^{0.25}\right)}^{2}\right)} - \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]

8. Applied rewrites 10.0%

   \[\leadsto \cos^{-1} \color{blue}{\left({\left({\left(\frac{1 - x}{2}\right)}^{0.25}\right)}^{2}\right)} - \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
9. Add Preprocessing

Alternative 3: 8.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (PI) 2.0)))
   (-
    t_0
    (-
     (+ (asin (sqrt (/ (- 1.0 x) 2.0))) t_0)
     (acos (sqrt (fma -0.5 x 0.5)))))))

Derivation

1. Initial program 8.3%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lift-asin.f64 N/A

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

   4. asin-acos N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-+r- N/A

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

   8. lower--.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-acos.f64 10.0

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

4. Applied rewrites 10.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 10.0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - \left(\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - \cos^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, x, 0.5\right)}}\right)\right) \]

7. Applied rewrites 10.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{2} - \left(\left(\sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - \cos^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, x, 0.5\right)}}\right)\right) \]
8. Add Preprocessing

Alternative 4: 8.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (PI) 2.0)))
   (fma (- t_0 (acos (sqrt (fma -0.5 x 0.5)))) -2.0 t_0)))

Derivation

1. Initial program 8.3%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 8.3

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

4. Applied rewrites 8.3%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 8.3

      \[\leadsto \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, x, 0.5\right)}}\right), -2, \frac{\mathsf{PI}\left(\right)}{2}\right) \]

7. Applied rewrites 8.3%

   \[\leadsto \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, x, 0.5\right)}}\right), -2, \frac{\mathsf{PI}\left(\right)}{2}\right) \]
8. Step-by-step derivation

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-acos.f64 10.0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}, -2, \frac{\mathsf{PI}\left(\right)}{2}\right) \]

9. Applied rewrites 10.0%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right)}, -2, \frac{\mathsf{PI}\left(\right)}{2}\right) \]
10. Add Preprocessing

Alternative 5: 8.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (PI) 0.5)))
   (fma (- t_0 (acos (* (sqrt (- 1.0 x)) (sqrt 0.5)))) -2.0 t_0)))

Derivation

1. Initial program 8.3%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-asin.f64 N/A

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

   2. asin-acos N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-acos.f64 10.0

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

4. Applied rewrites 10.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. cancel-sign-sub N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

7. Applied rewrites 10.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 0.5 - \cos^{-1} \left(\sqrt{1 - x} \cdot \sqrt{0.5}\right), -2, \mathsf{PI}\left(\right) \cdot 0.5\right)} \]
8. Add Preprocessing

Alternative 6: 6.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\left(\frac{0.5}{x} - 0.5\right) \cdot x}\right), -2, \frac{\mathsf{PI}\left(\right)}{2}\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 8.3%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 8.3

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

4. Applied rewrites 8.3%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. remove-double-neg N/A

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

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

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

   6. distribute-lft-neg-in N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-in N/A

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

   9. +-commutative N/A

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

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

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

   11. lower-*.f64 N/A

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

7. Applied rewrites 8.4%

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

Alternative 7: 6.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-0.5, x, 0.5\right)}\right), -2, \frac{\mathsf{PI}\left(\right)}{2}\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 8.3%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 8.3

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

4. Applied rewrites 8.3%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 8.3

      \[\leadsto \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, x, 0.5\right)}}\right), -2, \frac{\mathsf{PI}\left(\right)}{2}\right) \]

7. Applied rewrites 8.3%

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

Alternative 8: 4.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin^{-1} \left(\sqrt{0.5}\right), -2, \frac{\mathsf{PI}\left(\right)}{2}\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 8.3%

   \[\frac{\mathsf{PI}\left(\right)}{2} - 2 \cdot \sin^{-1} \left(\sqrt{\frac{1 - x}{2}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 8.3

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

4. Applied rewrites 8.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 4.2%

   \[\leadsto \mathsf{fma}\left(\sin^{-1} \left(\sqrt{\color{blue}{0.5}}\right), -2, \frac{\mathsf{PI}\left(\right)}{2}\right) \]
8. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (asin x))

C

double code(double x) {
	return asin(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 = asin(x)
end function

Java

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

Python

def code(x):
	return math.asin(x)

Julia

function code(x)
	return asin(x)
end

MATLAB

function tmp = code(x)
	tmp = asin(x);
end

Wolfram

code[x_] := N[ArcSin[x], $MachinePrecision]

Reproduce

herbie shell --seed 2024357 
(FPCore (x)
  :name "Ian Simplification"
  :precision binary64

  :alt
  (! :herbie-platform default (asin x))

  (- (/ (PI) 2.0) (* 2.0 (asin (sqrt (/ (- 1.0 x) 2.0))))))