Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A

Percentage Accurate: 77.3% → 99.5%

Time: 3.7s

Alternatives: 9

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / sin(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
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = (((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / Math.sin(x);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return (((8.0 / 3.0) * t_0) * t_0) / math.sin(x)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = (((8.0 / 3.0) * t_0) * t_0) / sin(x);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Initial Program: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / sin(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
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = (((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return (((8.0 / 3.0) * t_0) * t_0) / Math.sin(x);
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return (((8.0 / 3.0) * t_0) * t_0) / math.sin(x)

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x))
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = (((8.0 / 3.0) * t_0) * t_0) / sin(x);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot x\right)\\ \frac{t\_0 \cdot \frac{t\_0}{\sin x}}{0.375} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 x)))) (/ (* t_0 (/ t_0 (sin x))) 0.375)))

C

double code(double x) {
	double t_0 = sin((0.5 * x));
	return (t_0 * (t_0 / sin(x))) / 0.375;
}

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 = sin((0.5d0 * x))
    code = (t_0 * (t_0 / sin(x))) / 0.375d0
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((0.5 * x));
	return (t_0 * (t_0 / Math.sin(x))) / 0.375;
}

Python

def code(x):
	t_0 = math.sin((0.5 * x))
	return (t_0 * (t_0 / math.sin(x))) / 0.375

Julia

function code(x)
	t_0 = sin(Float64(0.5 * x))
	return Float64(Float64(t_0 * Float64(t_0 / sin(x))) / 0.375)
end

MATLAB

function tmp = code(x)
	t_0 = sin((0.5 * x));
	tmp = (t_0 * (t_0 / sin(x))) / 0.375;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.375), $MachinePrecision]]

Derivation

1. Initial program 77.3%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}{\sin x} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \sin \left(x \cdot \frac{1}{2}\right)} \]

   4. mult-flip-rev N/A

      \[\leadsto \color{blue}{\left(\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{\sin x}\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{\sin x}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sin x} \cdot \color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sin x} \cdot \frac{8}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sin x} \cdot \frac{8}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   10. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\frac{8}{3} \cdot \frac{1}{\sin x}\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   11. mult-flip-rev N/A

       \[\leadsto \left(\color{blue}{\frac{\frac{8}{3}}{\sin x}} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   12. lower-/.f64 99.2%

       \[\leadsto \left(\color{blue}{\frac{\frac{8}{3}}{\sin x}} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right) \]

   13. lift-/.f64 N/A

       \[\leadsto \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   14. metadata-eval 99.2%

       \[\leadsto \left(\frac{\color{blue}{2.6666666666666665}}{\sin x} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right) \]

   15. lift-*.f64 N/A

       \[\leadsto \left(\frac{\frac{8}{3}}{\sin x} \cdot \sin \color{blue}{\left(x \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   16. *-commutative N/A

       \[\leadsto \left(\frac{\frac{8}{3}}{\sin x} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   17. lower-*.f64 99.2%

       \[\leadsto \left(\frac{2.6666666666666665}{\sin x} \cdot \sin \color{blue}{\left(0.5 \cdot x\right)}\right) \cdot \sin \left(x \cdot 0.5\right) \]

   18. lift-*.f64 N/A

       \[\leadsto \left(\frac{\frac{8}{3}}{\sin x} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \sin \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]

   19. *-commutative N/A

       \[\leadsto \left(\frac{\frac{8}{3}}{\sin x} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot x\right)} \]

3. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(\frac{2.6666666666666665}{\sin x} \cdot \sin \left(0.5 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot x\right)} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{8}{3}}{\sin x}} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   3. mult-flip N/A

      \[\leadsto \left(\color{blue}{\left(\frac{8}{3} \cdot \frac{1}{\sin x}\right)} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{\sin x} \cdot \frac{8}{3}\right)} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{\sin x} \cdot \frac{8}{3}\right)} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\frac{1}{\sin x}} \cdot \frac{8}{3}\right) \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   7. metadata-eval 99.1%

      \[\leadsto \left(\left(\frac{1}{\sin x} \cdot \color{blue}{2.6666666666666665}\right) \cdot \sin \left(0.5 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot x\right) \]

5. Applied rewrites 99.1%

   \[\leadsto \left(\color{blue}{\left(\frac{1}{\sin x} \cdot 2.6666666666666665\right)} \cdot \sin \left(0.5 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot x\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sin x} \cdot \frac{8}{3}\right) \cdot \sin \left(\frac{1}{2} \cdot x\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{\sin x} \cdot \frac{8}{3}\right)} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   4. lift-sin.f64 N/A

      \[\leadsto \left(\frac{1}{\sin x} \cdot \left(\frac{8}{3} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot x\right)}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\frac{1}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \color{blue}{\left(x \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\frac{1}{\sin x} \cdot \color{blue}{\left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\frac{1}{\sin x} \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot x\right)} \cdot \frac{8}{3}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{\sin x} \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot x\right)} \cdot \frac{8}{3}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   11. lift-sin.f64 N/A

       \[\leadsto \left(\frac{1}{\sin x} \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot x\right)} \cdot \frac{8}{3}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right) \]

   12. lower-*.f64 99.1%

       \[\leadsto \left(\frac{1}{\sin x} \cdot \color{blue}{\left(\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665\right)}\right) \cdot \sin \left(0.5 \cdot x\right) \]

7. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(\frac{1}{\sin x} \cdot \left(\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665\right)\right)} \cdot \sin \left(0.5 \cdot x\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sin x} \cdot \left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot x\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{\sin x} \cdot \left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \sin \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\left(\frac{1}{\sin x} \cdot \left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{\sin x} \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{8}{3}\right)}\right) \]

   5. associate-*r* N/A

      \[\leadsto \sin \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\left(\left(\frac{1}{\sin x} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \frac{8}{3}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{\sin x} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right)\right) \cdot \frac{8}{3}} \]

   7. metadata-eval N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{\sin x} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{3}{8}}} \]

   8. mult-flip-rev N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{\sin x} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right)}{\frac{3}{8}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{\sin x} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right)}{\frac{3}{8}}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{\sin x} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right)}}{\frac{3}{8}} \]

   11. *-commutative N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \frac{1}{\sin x}\right)}}{\frac{3}{8}} \]

   12. lift-/.f64 N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\frac{1}{\sin x}}\right)}{\frac{3}{8}} \]

   13. mult-flip N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x}}}{\frac{3}{8}} \]

   14. lower-/.f64 99.5%

       \[\leadsto \frac{\sin \left(0.5 \cdot x\right) \cdot \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x}}}{0.375} \]

9. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right) \cdot \frac{\sin \left(0.5 \cdot x\right)}{\sin x}}{0.375}} \]
10. Add Preprocessing

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.6666666666666666, \left(\left|x\right| \cdot 0.05555555555555555\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot {\sin \left(0.5 \cdot \left|x\right|\right)}^{2}}{\sin \left(\left|x\right|\right)}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (copysign 1.0 x)
  (if (<= (fabs x) 1e-7)
    (fma
     (fabs x)
     0.6666666666666666
     (* (* (fabs x) 0.05555555555555555) (* (fabs x) (fabs x))))
    (/
     (* 2.6666666666666665 (pow (sin (* 0.5 (fabs x))) 2.0))
     (sin (fabs x))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-7) {
		tmp = fma(fabs(x), 0.6666666666666666, ((fabs(x) * 0.05555555555555555) * (fabs(x) * fabs(x))));
	} else {
		tmp = (2.6666666666666665 * pow(sin((0.5 * fabs(x))), 2.0)) / sin(fabs(x));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e-7)
		tmp = fma(abs(x), 0.6666666666666666, Float64(Float64(abs(x) * 0.05555555555555555) * Float64(abs(x) * abs(x))));
	else
		tmp = Float64(Float64(2.6666666666666665 * (sin(Float64(0.5 * abs(x))) ^ 2.0)) / sin(abs(x)));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1e-7], N[(N[Abs[x], $MachinePrecision] * 0.6666666666666666 + N[(N[(N[Abs[x], $MachinePrecision] * 0.05555555555555555), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.6666666666666665 * N[Power[N[Sin[N[(0.5 * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999995e-8

   1. Initial program 77.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \color{blue}{\left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\color{blue}{\frac{1}{18}} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \color{blue}{\left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)}\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\color{blue}{\frac{1}{180}} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \color{blue}{\frac{17}{30240} \cdot {x}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

      10. lower-pow.f64 51.0%

          \[\leadsto x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{2}\right)\right)\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \frac{1}{18}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 51.1%

      \[\leadsto x \cdot \left(0.6666666666666666 + {x}^{2} \cdot 0.05555555555555555\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \frac{1}{18}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + \color{blue}{{x}^{2} \cdot \frac{1}{18}}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto x \cdot \frac{2}{3} + \color{blue}{x \cdot \left({x}^{2} \cdot \frac{1}{18}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{2}{3}}, x \cdot \left({x}^{2} \cdot \frac{1}{18}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, x \cdot \left({x}^{2} \cdot \frac{1}{18}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, x \cdot \left(\frac{1}{18} \cdot {x}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, \left(x \cdot \frac{1}{18}\right) \cdot {x}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, \left(x \cdot \frac{1}{18}\right) \cdot {x}^{2}\right) \]

      9. lower-*.f64 51.1%

         \[\leadsto \mathsf{fma}\left(x, 0.6666666666666666, \left(x \cdot 0.05555555555555555\right) \cdot {x}^{2}\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, \left(x \cdot \frac{1}{18}\right) \cdot {x}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, \left(x \cdot \frac{1}{18}\right) \cdot \left(x \cdot x\right)\right) \]

      12. lower-*.f64 51.1%

          \[\leadsto \mathsf{fma}\left(x, 0.6666666666666666, \left(x \cdot 0.05555555555555555\right) \cdot \left(x \cdot x\right)\right) \]

   9. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0.6666666666666666}, \left(x \cdot 0.05555555555555555\right) \cdot \left(x \cdot x\right)\right) \]

   if 9.9999999999999995e-8 < x

   1. Initial program 77.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot {\sin \left(\frac{1}{2} \cdot x\right)}^{2}}}{\sin x} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{8}{3} \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot x\right)}}^{2}}{\sin x} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}}{\sin x} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{8}{3} \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot x\right)}}^{2}}{\sin x} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{8}{3} \cdot {\sin \left(\frac{1}{2} \cdot x\right)}^{\color{blue}{2}}}{\sin x} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{8}{3} \cdot {\sin \left(\frac{1}{2} \cdot x\right)}^{2}}{\sin x} \]

      6. lower-*.f64 77.3%

         \[\leadsto \frac{2.6666666666666665 \cdot {\sin \left(0.5 \cdot x\right)}^{2}}{\sin x} \]

   4. Applied rewrites 77.3%

      \[\leadsto \frac{\color{blue}{2.6666666666666665 \cdot {\sin \left(0.5 \cdot x\right)}^{2}}}{\sin x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \left(-0.5 \cdot x\right) \cdot -2.6666666666666665\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (sin (* 0.5 x)) (sin x)) (* (sin (* -0.5 x)) -2.6666666666666665)))

C

double code(double x) {
	return (sin((0.5 * x)) / sin(x)) * (sin((-0.5 * x)) * -2.6666666666666665);
}

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 = (sin((0.5d0 * x)) / sin(x)) * (sin(((-0.5d0) * x)) * (-2.6666666666666665d0))
end function

Java

public static double code(double x) {
	return (Math.sin((0.5 * x)) / Math.sin(x)) * (Math.sin((-0.5 * x)) * -2.6666666666666665);
}

Python

def code(x):
	return (math.sin((0.5 * x)) / math.sin(x)) * (math.sin((-0.5 * x)) * -2.6666666666666665)

Julia

function code(x)
	return Float64(Float64(sin(Float64(0.5 * x)) / sin(x)) * Float64(sin(Float64(-0.5 * x)) * -2.6666666666666665))
end

MATLAB

function tmp = code(x)
	tmp = (sin((0.5 * x)) / sin(x)) * (sin((-0.5 * x)) * -2.6666666666666665);
end

Wolfram

code[x_] := N[(N[(N[Sin[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(-0.5 * x), $MachinePrecision]], $MachinePrecision] * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.3%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}{\sin x} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \]

   6. lower-/.f64 99.2%

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\sin x}} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\frac{1}{2} \cdot x\right)}}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \]

   9. lower-*.f64 99.2%

      \[\leadsto \frac{\sin \color{blue}{\left(0.5 \cdot x\right)}}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \]

   10. remove-double-neg N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin \left(x \cdot \frac{1}{2}\right) \cdot \frac{8}{3}}\right)\right)\right)\right) \]

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

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sin \left(x \cdot \frac{1}{2}\right)\right)\right) \cdot \frac{8}{3}}\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin \left(x \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\mathsf{neg}\left(\frac{8}{3}\right)\right)\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\sin \left(\frac{1}{2} \cdot x\right)}{\sin x} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\sin \left(x \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\mathsf{neg}\left(\frac{8}{3}\right)\right)\right)} \]

3. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \left(-0.5 \cdot x\right) \cdot -2.6666666666666665\right)} \]
4. Add Preprocessing

Alternative 4: 99.2% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right|, 0.6666666666666666, \left(\left|x\right| \cdot 0.05555555555555555\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(0.5 \cdot \left|x\right|\right)}^{2}}{\sin \left(\left|x\right|\right)}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (copysign 1.0 x)
  (if (<= (fabs x) 1e-7)
    (fma
     (fabs x)
     0.6666666666666666
     (* (* (fabs x) 0.05555555555555555) (* (fabs x) (fabs x))))
    (*
     2.6666666666666665
     (/ (pow (sin (* 0.5 (fabs x))) 2.0) (sin (fabs x)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-7) {
		tmp = fma(fabs(x), 0.6666666666666666, ((fabs(x) * 0.05555555555555555) * (fabs(x) * fabs(x))));
	} else {
		tmp = 2.6666666666666665 * (pow(sin((0.5 * fabs(x))), 2.0) / sin(fabs(x)));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e-7)
		tmp = fma(abs(x), 0.6666666666666666, Float64(Float64(abs(x) * 0.05555555555555555) * Float64(abs(x) * abs(x))));
	else
		tmp = Float64(2.6666666666666665 * Float64((sin(Float64(0.5 * abs(x))) ^ 2.0) / sin(abs(x))));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 1e-7], N[(N[Abs[x], $MachinePrecision] * 0.6666666666666666 + N[(N[(N[Abs[x], $MachinePrecision] * 0.05555555555555555), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[N[Sin[N[(0.5 * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999995e-8

   1. Initial program 77.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \color{blue}{\left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\color{blue}{\frac{1}{18}} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \color{blue}{\left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)}\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\color{blue}{\frac{1}{180}} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \color{blue}{\frac{17}{30240} \cdot {x}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

      10. lower-pow.f64 51.0%

          \[\leadsto x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{2}\right)\right)\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \frac{1}{18}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 51.1%

      \[\leadsto x \cdot \left(0.6666666666666666 + {x}^{2} \cdot 0.05555555555555555\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \frac{1}{18}\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + \color{blue}{{x}^{2} \cdot \frac{1}{18}}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto x \cdot \frac{2}{3} + \color{blue}{x \cdot \left({x}^{2} \cdot \frac{1}{18}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{2}{3}}, x \cdot \left({x}^{2} \cdot \frac{1}{18}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, x \cdot \left({x}^{2} \cdot \frac{1}{18}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, x \cdot \left(\frac{1}{18} \cdot {x}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, \left(x \cdot \frac{1}{18}\right) \cdot {x}^{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, \left(x \cdot \frac{1}{18}\right) \cdot {x}^{2}\right) \]

      9. lower-*.f64 51.1%

         \[\leadsto \mathsf{fma}\left(x, 0.6666666666666666, \left(x \cdot 0.05555555555555555\right) \cdot {x}^{2}\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, \left(x \cdot \frac{1}{18}\right) \cdot {x}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{2}{3}, \left(x \cdot \frac{1}{18}\right) \cdot \left(x \cdot x\right)\right) \]

      12. lower-*.f64 51.1%

          \[\leadsto \mathsf{fma}\left(x, 0.6666666666666666, \left(x \cdot 0.05555555555555555\right) \cdot \left(x \cdot x\right)\right) \]

   9. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0.6666666666666666}, \left(x \cdot 0.05555555555555555\right) \cdot \left(x \cdot x\right)\right) \]

   if 9.9999999999999995e-8 < x

   1. Initial program 77.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{8}{3} \cdot \frac{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}{\sin x}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{8}{3} \cdot \frac{\color{blue}{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}}{\sin x} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{8}{3} \cdot \color{blue}{\frac{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}{\sin x}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{8}{3} \cdot \frac{\color{blue}{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}}{\sin x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{8}{3} \cdot \frac{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}{\color{blue}{\sin x}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{8}{3} \cdot \frac{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}{\sin \color{blue}{x}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{8}{3} \cdot \frac{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}{\sin x} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{8}{3} \cdot \frac{{\sin \left(\frac{1}{2} \cdot x\right)}^{2}}{\sin x} \]

      8. lower-sin.f64 77.3%

         \[\leadsto 2.6666666666666665 \cdot \frac{{\sin \left(0.5 \cdot x\right)}^{2}}{\sin x} \]

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{2.6666666666666665 \cdot \frac{{\sin \left(0.5 \cdot x\right)}^{2}}{\sin x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.00135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0005621693121693122, t\_0, 0.005555555555555556\right), t\_0, 0.05555555555555555\right) \cdot \left|x\right|, \left|x\right|, 0.6666666666666666\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin \left(\left|x\right|\right)}{\mathsf{fma}\left(\cos \left(\left|x\right|\right), -1.3333333333333333, 1.3333333333333333\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 0.00135)
      (*
       (fma
        (*
         (fma
          (fma 0.0005621693121693122 t_0 0.005555555555555556)
          t_0
          0.05555555555555555)
         (fabs x))
        (fabs x)
        0.6666666666666666)
       (fabs x))
      (/
       1.0
       (/
        (sin (fabs x))
        (fma (cos (fabs x)) -1.3333333333333333 1.3333333333333333)))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 0.00135) {
		tmp = fma((fma(fma(0.0005621693121693122, t_0, 0.005555555555555556), t_0, 0.05555555555555555) * fabs(x)), fabs(x), 0.6666666666666666) * fabs(x);
	} else {
		tmp = 1.0 / (sin(fabs(x)) / fma(cos(fabs(x)), -1.3333333333333333, 1.3333333333333333));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 0.00135)
		tmp = Float64(fma(Float64(fma(fma(0.0005621693121693122, t_0, 0.005555555555555556), t_0, 0.05555555555555555) * abs(x)), abs(x), 0.6666666666666666) * abs(x));
	else
		tmp = Float64(1.0 / Float64(sin(abs(x)) / fma(cos(abs(x)), -1.3333333333333333, 1.3333333333333333)));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 0.00135], N[(N[(N[(N[(N[(0.0005621693121693122 * t$95$0 + 0.005555555555555556), $MachinePrecision] * t$95$0 + 0.05555555555555555), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] / N[(N[Cos[N[Abs[x], $MachinePrecision]], $MachinePrecision] * -1.3333333333333333 + 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0013500000000000001

   1. Initial program 77.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \color{blue}{\left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\color{blue}{\frac{1}{18}} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \color{blue}{\left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)}\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\color{blue}{\frac{1}{180}} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \color{blue}{\frac{17}{30240} \cdot {x}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

      10. lower-pow.f64 51.0%

          \[\leadsto x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{x} \]

      3. lower-*.f64 51.0%

         \[\leadsto \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{x} \]

   6. Applied rewrites 51.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0005621693121693122, x \cdot x, 0.005555555555555556\right), x \cdot x, 0.05555555555555555\right) \cdot x, x, 0.6666666666666666\right) \cdot \color{blue}{x} \]

   if 0.0013500000000000001 < x

   1. Initial program 77.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}{\sin x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}{\sin x} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \left(\color{blue}{\sin \left(x \cdot \frac{1}{2}\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}{\sin x} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(x \cdot \frac{1}{2}\right)}\right)}{\sin x} \]

      6. sqr-sin-a N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)}}{\sin x} \]

      7. sub-flip N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right)}}{\sin x} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{8}{3} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{8}{3} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\frac{8}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\frac{8}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{4}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{4}{3} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

   3. Applied rewrites 52.2%

      \[\leadsto \frac{\color{blue}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}{\sin x} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{4}{3} + \left(\frac{-1}{2} \cdot \cos x\right) \cdot \color{blue}{\frac{8}{3}}}{\sin x} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{4}{3} + \left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3}}{\sin x}} \]

      3. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\frac{4}{3} + \left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\frac{4}{3} + \left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3}}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{\frac{4}{3} + \left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3}}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{4}{3} + \left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3}}}} \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3} + \frac{4}{3}}}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3}} + \frac{4}{3}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\frac{-1}{2} \cdot \cos x\right)} \cdot \frac{8}{3} + \frac{4}{3}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\left(\cos x \cdot \frac{-1}{2}\right)} \cdot \frac{8}{3} + \frac{4}{3}}} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\cos x \cdot \left(\frac{-1}{2} \cdot \frac{8}{3}\right)} + \frac{4}{3}}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\sin x}{\cos x \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{8}{3}}\right) + \frac{4}{3}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\sin x}{\cos x \cdot \color{blue}{\frac{-4}{3}} + \frac{4}{3}}} \]

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\mathsf{fma}\left(\cos x, \mathsf{neg}\left(\frac{4}{3}\right), \frac{4}{3}\right)}}} \]

      16. metadata-eval 52.3%

          \[\leadsto \frac{1}{\frac{\sin x}{\mathsf{fma}\left(\cos x, \color{blue}{-1.3333333333333333}, 1.3333333333333333\right)}} \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\mathsf{fma}\left(\cos x, -1.3333333333333333, 1.3333333333333333\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.00135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0005621693121693122, t\_0, 0.005555555555555556\right), t\_0, 0.05555555555555555\right) \cdot \left|x\right|, \left|x\right|, 0.6666666666666666\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|x\right|\right) - 1}{-0.75 \cdot \sin \left(\left|x\right|\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 0.00135)
      (*
       (fma
        (*
         (fma
          (fma 0.0005621693121693122 t_0 0.005555555555555556)
          t_0
          0.05555555555555555)
         (fabs x))
        (fabs x)
        0.6666666666666666)
       (fabs x))
      (/ (- (cos (fabs x)) 1.0) (* -0.75 (sin (fabs x))))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 0.00135) {
		tmp = fma((fma(fma(0.0005621693121693122, t_0, 0.005555555555555556), t_0, 0.05555555555555555) * fabs(x)), fabs(x), 0.6666666666666666) * fabs(x);
	} else {
		tmp = (cos(fabs(x)) - 1.0) / (-0.75 * sin(fabs(x)));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 0.00135)
		tmp = Float64(fma(Float64(fma(fma(0.0005621693121693122, t_0, 0.005555555555555556), t_0, 0.05555555555555555) * abs(x)), abs(x), 0.6666666666666666) * abs(x));
	else
		tmp = Float64(Float64(cos(abs(x)) - 1.0) / Float64(-0.75 * sin(abs(x))));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 0.00135], N[(N[(N[(N[(N[(0.0005621693121693122 * t$95$0 + 0.005555555555555556), $MachinePrecision] * t$95$0 + 0.05555555555555555), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[Abs[x], $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(-0.75 * N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0013500000000000001

   1. Initial program 77.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \color{blue}{\left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\color{blue}{\frac{1}{18}} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \color{blue}{\left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)}\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\color{blue}{\frac{1}{180}} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \color{blue}{\frac{17}{30240} \cdot {x}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

      10. lower-pow.f64 51.0%

          \[\leadsto x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{x} \]

      3. lower-*.f64 51.0%

         \[\leadsto \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{x} \]

   6. Applied rewrites 51.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0005621693121693122, x \cdot x, 0.005555555555555556\right), x \cdot x, 0.05555555555555555\right) \cdot x, x, 0.6666666666666666\right) \cdot \color{blue}{x} \]

   if 0.0013500000000000001 < x

   1. Initial program 77.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}{\sin x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}{\sin x} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \left(\color{blue}{\sin \left(x \cdot \frac{1}{2}\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}{\sin x} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(x \cdot \frac{1}{2}\right)}\right)}{\sin x} \]

      6. sqr-sin-a N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)}}{\sin x} \]

      7. sub-flip N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right)}}{\sin x} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{8}{3} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{8}{3} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\frac{8}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\frac{8}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{4}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{4}{3} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

   3. Applied rewrites 52.2%

      \[\leadsto \frac{\color{blue}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}{\sin x} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{4}{3} + \left(\frac{-1}{2} \cdot \cos x\right) \cdot \color{blue}{\frac{8}{3}}}{\sin x} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{4}{3} + \left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3}}}{\sin x} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3} + \frac{4}{3}}}{\sin x} \]

      4. add-flip N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3} - \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}}{\sin x} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3} - \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}}{\sin x} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3}} - \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}{\sin x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \left(\frac{-1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}{\sin x} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}{\sin x} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{8}{3} \cdot \frac{-1}{2}\right) \cdot \cos x} - \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}{\sin x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{8}{3}} \cdot \frac{-1}{2}\right) \cdot \cos x - \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}{\sin x} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{-4}{3}} \cdot \cos x - \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}{\sin x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} \cdot \cos x - \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}{\sin x} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right) \cdot \cos x} - \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}{\sin x} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{-4}{3}} \cdot \cos x - \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}{\sin x} \]

      15. metadata-eval 52.2%

          \[\leadsto \frac{-1.3333333333333333 \cdot \cos x - \color{blue}{-1.3333333333333333}}{\sin x} \]

   5. Applied rewrites 52.2%

      \[\leadsto \frac{\color{blue}{-1.3333333333333333 \cdot \cos x - -1.3333333333333333}}{\sin x} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-4}{3} \cdot \cos x - \frac{-4}{3}}{\sin x}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-4}{3} \cdot \cos x - \frac{-4}{3}}}{\sin x} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{-4}{3} \cdot \cos x - \color{blue}{\frac{1}{\frac{-3}{4}}}}{\sin x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{-4}{3} \cdot \cos x - \frac{1}{\color{blue}{-2 \cdot \frac{3}{8}}}}{\sin x} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-4}{3} \cdot \cos x} - \frac{1}{-2 \cdot \frac{3}{8}}}{\sin x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos x \cdot \frac{-4}{3}} - \frac{1}{-2 \cdot \frac{3}{8}}}{\sin x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\cos x \cdot \color{blue}{\frac{1}{\frac{-3}{4}}} - \frac{1}{-2 \cdot \frac{3}{8}}}{\sin x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\cos x \cdot \frac{1}{\color{blue}{-2 \cdot \frac{3}{8}}} - \frac{1}{-2 \cdot \frac{3}{8}}}{\sin x} \]

      9. mult-flip-rev N/A

         \[\leadsto \frac{\color{blue}{\frac{\cos x}{-2 \cdot \frac{3}{8}}} - \frac{1}{-2 \cdot \frac{3}{8}}}{\sin x} \]

      10. div-sub N/A

          \[\leadsto \frac{\color{blue}{\frac{\cos x - 1}{-2 \cdot \frac{3}{8}}}}{\sin x} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\cos x - 1}}{-2 \cdot \frac{3}{8}}}{\sin x} \]

      12. div-flip-rev N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{\frac{-2 \cdot \frac{3}{8}}{\cos x - 1}}}}{\sin x} \]

      13. associate-*l/ N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-2}{\cos x - 1} \cdot \frac{3}{8}}}}{\sin x} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-2}{\cos x - 1}} \cdot \frac{3}{8}}}{\sin x} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-2}{\cos x - 1} \cdot \frac{3}{8}}}}{\sin x} \]

      16. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{1}{\left(\frac{-2}{\cos x - 1} \cdot \frac{3}{8}\right) \cdot \sin x}} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\frac{-2}{\cos x - 1} \cdot \frac{3}{8}\right)} \cdot \sin x} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\frac{-2}{\cos x - 1}} \cdot \frac{3}{8}\right) \cdot \sin x} \]

      19. associate-*l/ N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot \frac{3}{8}}{\cos x - 1}} \cdot \sin x} \]

   7. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\frac{\cos x - 1}{-0.75 \cdot \sin x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.00135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0005621693121693122, t\_0, 0.005555555555555556\right), t\_0, 0.05555555555555555\right) \cdot \left|x\right|, \left|x\right|, 0.6666666666666666\right) \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos \left(\left|x\right|\right), -1.3333333333333333, 1.3333333333333333\right)}{\sin \left(\left|x\right|\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<= (fabs x) 0.00135)
      (*
       (fma
        (*
         (fma
          (fma 0.0005621693121693122 t_0 0.005555555555555556)
          t_0
          0.05555555555555555)
         (fabs x))
        (fabs x)
        0.6666666666666666)
       (fabs x))
      (/
       (fma (cos (fabs x)) -1.3333333333333333 1.3333333333333333)
       (sin (fabs x)))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 0.00135) {
		tmp = fma((fma(fma(0.0005621693121693122, t_0, 0.005555555555555556), t_0, 0.05555555555555555) * fabs(x)), fabs(x), 0.6666666666666666) * fabs(x);
	} else {
		tmp = fma(cos(fabs(x)), -1.3333333333333333, 1.3333333333333333) / sin(fabs(x));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 0.00135)
		tmp = Float64(fma(Float64(fma(fma(0.0005621693121693122, t_0, 0.005555555555555556), t_0, 0.05555555555555555) * abs(x)), abs(x), 0.6666666666666666) * abs(x));
	else
		tmp = Float64(fma(cos(abs(x)), -1.3333333333333333, 1.3333333333333333) / sin(abs(x)));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 0.00135], N[(N[(N[(N[(N[(0.0005621693121693122 * t$95$0 + 0.005555555555555556), $MachinePrecision] * t$95$0 + 0.05555555555555555), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[Abs[x], $MachinePrecision]], $MachinePrecision] * -1.3333333333333333 + 1.3333333333333333), $MachinePrecision] / N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0013500000000000001

   1. Initial program 77.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \color{blue}{\left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\color{blue}{\frac{1}{18}} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \color{blue}{\left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)}\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\color{blue}{\frac{1}{180}} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \color{blue}{\frac{17}{30240} \cdot {x}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

      10. lower-pow.f64 51.0%

          \[\leadsto x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{18} + {x}^{2} \cdot \left(\frac{1}{180} + \frac{17}{30240} \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{x} \]

      3. lower-*.f64 51.0%

         \[\leadsto \left(0.6666666666666666 + {x}^{2} \cdot \left(0.05555555555555555 + {x}^{2} \cdot \left(0.005555555555555556 + 0.0005621693121693122 \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{x} \]

   6. Applied rewrites 51.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0005621693121693122, x \cdot x, 0.005555555555555556\right), x \cdot x, 0.05555555555555555\right) \cdot x, x, 0.6666666666666666\right) \cdot \color{blue}{x} \]

   if 0.0013500000000000001 < x

   1. Initial program 77.3%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}{\sin x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}}{\sin x} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \left(\color{blue}{\sin \left(x \cdot \frac{1}{2}\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}{\sin x} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \left(\sin \left(x \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(x \cdot \frac{1}{2}\right)}\right)}{\sin x} \]

      6. sqr-sin-a N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)}}{\sin x} \]

      7. sub-flip N/A

         \[\leadsto \frac{\frac{8}{3} \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right)}}{\sin x} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{8}{3} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{8}{3} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\frac{8}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\frac{8}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{4}{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}{\sin x} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{4}{3} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(2 \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \frac{8}{3}}}{\sin x} \]

   3. Applied rewrites 52.2%

      \[\leadsto \frac{\color{blue}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}{\sin x} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{4}{3} + \left(\frac{-1}{2} \cdot \cos x\right) \cdot \color{blue}{\frac{8}{3}}}{\sin x} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{4}{3} + \left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3}}}{\sin x} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3} + \frac{4}{3}}}{\sin x} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \cos x\right) \cdot \frac{8}{3}} + \frac{4}{3}}{\sin x} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \cos x\right)} \cdot \frac{8}{3} + \frac{4}{3}}{\sin x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cos x \cdot \frac{-1}{2}\right)} \cdot \frac{8}{3} + \frac{4}{3}}{\sin x} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\cos x \cdot \left(\frac{-1}{2} \cdot \frac{8}{3}\right)} + \frac{4}{3}}{\sin x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\cos x \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{8}{3}}\right) + \frac{4}{3}}{\sin x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\cos x \cdot \color{blue}{\frac{-4}{3}} + \frac{4}{3}}{\sin x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\cos x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} + \frac{4}{3}}{\sin x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x, \mathsf{neg}\left(\frac{4}{3}\right), \frac{4}{3}\right)}}{\sin x} \]

      12. metadata-eval 52.3%

          \[\leadsto \frac{\mathsf{fma}\left(\cos x, \color{blue}{-1.3333333333333333}, 1.3333333333333333\right)}{\sin x} \]

   5. Applied rewrites 52.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x, -1.3333333333333333, 1.3333333333333333\right)}}{\sin x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 55.2% accurate, 3.0× speedup

Math

\[1.3333333333333333 \cdot \sin \left(0.5 \cdot x\right) \]

FPCore

(FPCore (x) :precision binary64 (* 1.3333333333333333 (sin (* 0.5 x))))

C

double code(double x) {
	return 1.3333333333333333 * sin((0.5 * 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 = 1.3333333333333333d0 * sin((0.5d0 * x))
end function

Java

public static double code(double x) {
	return 1.3333333333333333 * Math.sin((0.5 * x));
}

Python

def code(x):
	return 1.3333333333333333 * math.sin((0.5 * x))

Julia

function code(x)
	return Float64(1.3333333333333333 * sin(Float64(0.5 * x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.3333333333333333 * sin((0.5 * x));
end

Wolfram

code[x_] := N[(1.3333333333333333 * N[Sin[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.3%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}}{\sin x} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \cdot \sin \left(x \cdot \frac{1}{2}\right)} \]

   4. mult-flip-rev N/A

      \[\leadsto \color{blue}{\left(\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{\sin x}\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{\sin x}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sin x} \cdot \left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sin x} \cdot \color{blue}{\left(\frac{8}{3} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sin x} \cdot \frac{8}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sin x} \cdot \frac{8}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   10. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\frac{8}{3} \cdot \frac{1}{\sin x}\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   11. mult-flip-rev N/A

       \[\leadsto \left(\color{blue}{\frac{\frac{8}{3}}{\sin x}} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   12. lower-/.f64 99.2%

       \[\leadsto \left(\color{blue}{\frac{\frac{8}{3}}{\sin x}} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right) \]

   13. lift-/.f64 N/A

       \[\leadsto \left(\frac{\color{blue}{\frac{8}{3}}}{\sin x} \cdot \sin \left(x \cdot \frac{1}{2}\right)\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   14. metadata-eval 99.2%

       \[\leadsto \left(\frac{\color{blue}{2.6666666666666665}}{\sin x} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right) \]

   15. lift-*.f64 N/A

       \[\leadsto \left(\frac{\frac{8}{3}}{\sin x} \cdot \sin \color{blue}{\left(x \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   16. *-commutative N/A

       \[\leadsto \left(\frac{\frac{8}{3}}{\sin x} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

   17. lower-*.f64 99.2%

       \[\leadsto \left(\frac{2.6666666666666665}{\sin x} \cdot \sin \color{blue}{\left(0.5 \cdot x\right)}\right) \cdot \sin \left(x \cdot 0.5\right) \]

   18. lift-*.f64 N/A

       \[\leadsto \left(\frac{\frac{8}{3}}{\sin x} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \sin \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]

   19. *-commutative N/A

       \[\leadsto \left(\frac{\frac{8}{3}}{\sin x} \cdot \sin \left(\frac{1}{2} \cdot x\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot x\right)} \]

3. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\left(\frac{2.6666666666666665}{\sin x} \cdot \sin \left(0.5 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot x\right)} \]

4. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{4}{3}} \cdot \sin \left(0.5 \cdot x\right) \]
5. Step-by-step derivation

6. Applied rewrites 55.2%

   \[\leadsto \color{blue}{1.3333333333333333} \cdot \sin \left(0.5 \cdot x\right) \]
7. Add Preprocessing

Alternative 9: 51.3% accurate, 29.0× speedup

Math

\[0.6666666666666666 \cdot x \]

FPCore

(FPCore (x) :precision binary64 (* 0.6666666666666666 x))

C

double code(double x) {
	return 0.6666666666666666 * 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 = 0.6666666666666666d0 * x
end function

Java

public static double code(double x) {
	return 0.6666666666666666 * x;
}

Python

def code(x):
	return 0.6666666666666666 * x

Julia

function code(x)
	return Float64(0.6666666666666666 * x)
end

MATLAB

function tmp = code(x)
	tmp = 0.6666666666666666 * x;
end

Wolfram

code[x_] := N[(0.6666666666666666 * x), $MachinePrecision]

Derivation

1. Initial program 77.3%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{2}{3} \cdot x} \]
3. Step-by-step derivation

   1. lower-*.f64 51.3%

      \[\leadsto 0.6666666666666666 \cdot \color{blue}{x} \]

4. Applied rewrites 51.3%

   \[\leadsto \color{blue}{0.6666666666666666 \cdot x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64
  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))