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

Percentage Accurate: 76.7% → 99.2%

Time: 3.4s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

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: 76.7% accurate, 1.0× speedup

Math

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

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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \left(\frac{t\_0}{\sin x} \cdot t\_0\right) \cdot 2.6666666666666665 \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5)))) (* (* (/ t_0 (sin x)) t_0) 2.6666666666666665)))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	return ((t_0 / sin(x)) * t_0) * 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
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = ((t_0 / sin(x)) * t_0) * 2.6666666666666665d0
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	return ((t_0 / Math.sin(x)) * t_0) * 2.6666666666666665;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	return ((t_0 / math.sin(x)) * t_0) * 2.6666666666666665

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	return Float64(Float64(Float64(t_0 / sin(x)) * t_0) * 2.6666666666666665)
end

MATLAB

function tmp = code(x)
	t_0 = sin((x * 0.5));
	tmp = ((t_0 / sin(x)) * t_0) * 2.6666666666666665;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * 2.6666666666666665), $MachinePrecision]]

Derivation

1. Initial program 76.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lift-sin.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

5. Applied rewrites 99.2%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot x\right)\\ t\_0 \cdot \left(2.6666666666666665 \cdot \frac{t\_0}{\sin x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 x)))) (* t_0 (* 2.6666666666666665 (/ t_0 (sin x))))))

C

double code(double x) {
	double t_0 = sin((0.5 * x));
	return t_0 * (2.6666666666666665 * (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((0.5d0 * x))
    code = t_0 * (2.6666666666666665d0 * (t_0 / sin(x)))
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((0.5 * x));
	return t_0 * (2.6666666666666665 * (t_0 / Math.sin(x)));
}

Python

def code(x):
	t_0 = math.sin((0.5 * x))
	return t_0 * (2.6666666666666665 * (t_0 / math.sin(x)))

Julia

function code(x)
	t_0 = sin(Float64(0.5 * x))
	return Float64(t_0 * Float64(2.6666666666666665 * Float64(t_0 / sin(x))))
end

MATLAB

function tmp = code(x)
	t_0 = sin((0.5 * x));
	tmp = t_0 * (2.6666666666666665 * (t_0 / sin(x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 * N[(2.6666666666666665 * N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 N/A

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

3. Applied rewrites 99.2%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot x\right)\\ \frac{t\_0}{\sin x} \cdot \left(t\_0 \cdot 2.6666666666666665\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 x)))) (* (/ t_0 (sin x)) (* t_0 2.6666666666666665))))

C

double code(double x) {
	double t_0 = sin((0.5 * x));
	return (t_0 / sin(x)) * (t_0 * 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
    real(8) :: t_0
    t_0 = sin((0.5d0 * x))
    code = (t_0 / sin(x)) * (t_0 * 2.6666666666666665d0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((0.5 * x));
	return (t_0 / Math.sin(x)) * (t_0 * 2.6666666666666665);
}

Python

def code(x):
	t_0 = math.sin((0.5 * x))
	return (t_0 / math.sin(x)) * (t_0 * 2.6666666666666665)

Julia

function code(x)
	t_0 = sin(Float64(0.5 * x))
	return Float64(Float64(t_0 / sin(x)) * Float64(t_0 * 2.6666666666666665))
end

MATLAB

function tmp = code(x)
	t_0 = sin((0.5 * x));
	tmp = (t_0 / sin(x)) * (t_0 * 2.6666666666666665);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 2.6666666666666665), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lift-sin.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

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: 75.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-128}:\\ \;\;\;\;0.6666666666666666 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(x \cdot 0.5\right)}^{2} \cdot 2.6666666666666665}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-128)
   (* 0.6666666666666666 x)
   (/ (* (pow (sin (* x 0.5)) 2.0) 2.6666666666666665) (sin x))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-128) {
		tmp = 0.6666666666666666 * x;
	} else {
		tmp = (pow(sin((x * 0.5)), 2.0) * 2.6666666666666665) / sin(x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2d-128) then
        tmp = 0.6666666666666666d0 * x
    else
        tmp = ((sin((x * 0.5d0)) ** 2.0d0) * 2.6666666666666665d0) / sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e-128) {
		tmp = 0.6666666666666666 * x;
	} else {
		tmp = (Math.pow(Math.sin((x * 0.5)), 2.0) * 2.6666666666666665) / Math.sin(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e-128:
		tmp = 0.6666666666666666 * x
	else:
		tmp = (math.pow(math.sin((x * 0.5)), 2.0) * 2.6666666666666665) / math.sin(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-128)
		tmp = Float64(0.6666666666666666 * x);
	else
		tmp = Float64(Float64((sin(Float64(x * 0.5)) ^ 2.0) * 2.6666666666666665) / sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e-128)
		tmp = 0.6666666666666666 * x;
	else
		tmp = ((sin((x * 0.5)) ^ 2.0) * 2.6666666666666665) / sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-128], N[(0.6666666666666666 * x), $MachinePrecision], N[(N[(N[Power[N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * 2.6666666666666665), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000011e-128

   1. Initial program 64.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 62.7

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

   4. Applied rewrites 62.7%

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

   if 2.00000000000000011e-128 < x

   1. Initial program 99.1%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 71.7%

      \[\leadsto \frac{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot x\right)\right)\right) \cdot 2.6666666666666665}}{\sin x} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-sin-a-rev N/A

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

      7. unpow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 5: 75.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0043:\\ \;\;\;\;\left(\mathsf{fma}\left(0.020833333333333332, x \cdot x, 0.25\right) \cdot x\right) \cdot 2.6666666666666665\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \cos x \cdot 0.5}{\sin x} \cdot 2.6666666666666665\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0043)
   (* (* (fma 0.020833333333333332 (* x x) 0.25) x) 2.6666666666666665)
   (* (/ (- 0.5 (* (cos x) 0.5)) (sin x)) 2.6666666666666665)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0043) {
		tmp = (fma(0.020833333333333332, (x * x), 0.25) * x) * 2.6666666666666665;
	} else {
		tmp = ((0.5 - (cos(x) * 0.5)) / sin(x)) * 2.6666666666666665;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0043)
		tmp = Float64(Float64(fma(0.020833333333333332, Float64(x * x), 0.25) * x) * 2.6666666666666665);
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(cos(x) * 0.5)) / sin(x)) * 2.6666666666666665);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0043], N[(N[(N[(0.020833333333333332 * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * 2.6666666666666665), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] * 2.6666666666666665), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0043

   1. Initial program 69.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-sin.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{1}{48} \cdot {x}^{2}\right)\right)} \cdot \frac{8}{3} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{4} + \frac{1}{48} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \cdot \frac{8}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} + \frac{1}{48} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \cdot \frac{8}{3} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{48} \cdot {x}^{2} + \frac{1}{4}\right) \cdot x\right) \cdot \frac{8}{3} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{48}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot \frac{8}{3} \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{48}, x \cdot x, \frac{1}{4}\right) \cdot x\right) \cdot \frac{8}{3} \]

      6. lift-*.f64 67.7

         \[\leadsto \left(\mathsf{fma}\left(0.020833333333333332, x \cdot x, 0.25\right) \cdot x\right) \cdot 2.6666666666666665 \]

   8. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.020833333333333332, x \cdot x, 0.25\right) \cdot x\right)} \cdot 2.6666666666666665 \]

   if 0.0043 < x

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

   3. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

   5. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 98.3

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

      16. lift-*.f64 N/A

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

      17. *-lft-identity 98.3

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

   7. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   9. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{0.5 - \cos x \cdot 0.5}{\sin x} \cdot 2.6666666666666665} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0043:\\ \;\;\;\;\left(\mathsf{fma}\left(0.020833333333333332, x \cdot x, 0.25\right) \cdot x\right) \cdot 2.6666666666666665\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 - 0.5 \cdot \cos x\right) \cdot 2.6666666666666665}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0043)
   (* (* (fma 0.020833333333333332 (* x x) 0.25) x) 2.6666666666666665)
   (/ (* (- 0.5 (* 0.5 (cos x))) 2.6666666666666665) (sin x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0043) {
		tmp = (fma(0.020833333333333332, (x * x), 0.25) * x) * 2.6666666666666665;
	} else {
		tmp = ((0.5 - (0.5 * cos(x))) * 2.6666666666666665) / sin(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0043)
		tmp = Float64(Float64(fma(0.020833333333333332, Float64(x * x), 0.25) * x) * 2.6666666666666665);
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(0.5 * cos(x))) * 2.6666666666666665) / sin(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0043], N[(N[(N[(0.020833333333333332 * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * 2.6666666666666665), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.6666666666666665), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0043

   1. Initial program 69.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-sin.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{1}{48} \cdot {x}^{2}\right)\right)} \cdot \frac{8}{3} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{4} + \frac{1}{48} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \cdot \frac{8}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} + \frac{1}{48} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \cdot \frac{8}{3} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{48} \cdot {x}^{2} + \frac{1}{4}\right) \cdot x\right) \cdot \frac{8}{3} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{48}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot \frac{8}{3} \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{48}, x \cdot x, \frac{1}{4}\right) \cdot x\right) \cdot \frac{8}{3} \]

      6. lift-*.f64 67.7

         \[\leadsto \left(\mathsf{fma}\left(0.020833333333333332, x \cdot x, 0.25\right) \cdot x\right) \cdot 2.6666666666666665 \]

   8. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.020833333333333332, x \cdot x, 0.25\right) \cdot x\right)} \cdot 2.6666666666666665 \]

   if 0.0043 < x

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. associate-*l* N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   3. Applied rewrites 98.3%

      \[\leadsto \frac{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot x\right)\right)\right) \cdot 2.6666666666666665}}{\sin x} \]

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 98.3%

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

Alternative 7: 75.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0043:\\ \;\;\;\;\left(\mathsf{fma}\left(0.020833333333333332, x \cdot x, 0.25\right) \cdot x\right) \cdot 2.6666666666666665\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665}{\sin x} \cdot \left(0.5 - \cos x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0043)
   (* (* (fma 0.020833333333333332 (* x x) 0.25) x) 2.6666666666666665)
   (* (/ 2.6666666666666665 (sin x)) (- 0.5 (* (cos x) 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0043) {
		tmp = (fma(0.020833333333333332, (x * x), 0.25) * x) * 2.6666666666666665;
	} else {
		tmp = (2.6666666666666665 / sin(x)) * (0.5 - (cos(x) * 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0043)
		tmp = Float64(Float64(fma(0.020833333333333332, Float64(x * x), 0.25) * x) * 2.6666666666666665);
	else
		tmp = Float64(Float64(2.6666666666666665 / sin(x)) * Float64(0.5 - Float64(cos(x) * 0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0043], N[(N[(N[(0.020833333333333332 * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * 2.6666666666666665), $MachinePrecision], N[(N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0043

   1. Initial program 69.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-sin.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{4} + \frac{1}{48} \cdot {x}^{2}\right)\right)} \cdot \frac{8}{3} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{4} + \frac{1}{48} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \cdot \frac{8}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{4} + \frac{1}{48} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \cdot \frac{8}{3} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{48} \cdot {x}^{2} + \frac{1}{4}\right) \cdot x\right) \cdot \frac{8}{3} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{48}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot \frac{8}{3} \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{48}, x \cdot x, \frac{1}{4}\right) \cdot x\right) \cdot \frac{8}{3} \]

      6. lift-*.f64 67.7

         \[\leadsto \left(\mathsf{fma}\left(0.020833333333333332, x \cdot x, 0.25\right) \cdot x\right) \cdot 2.6666666666666665 \]

   8. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.020833333333333332, x \cdot x, 0.25\right) \cdot x\right)} \cdot 2.6666666666666665 \]

   if 0.0043 < x

   1. Initial program 99.0%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

   3. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

   5. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 98.3

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

      16. lift-*.f64 N/A

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

      17. *-lft-identity 98.3

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

   7. Applied rewrites 98.3%

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

Alternative 8: 57.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x \cdot 0.5\right)\\ \mathbf{if}\;\frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x} \leq 0.06:\\ \;\;\;\;\sin \left(0.5 \cdot x\right) \cdot 1.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(1.3333333333333333 \cdot x\right) \cdot \frac{{\left(0.5 - \cos \left(1 \cdot x\right) \cdot 0.5\right)}^{0.5}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sin (* x 0.5))))
   (if (<= (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x)) 0.06)
     (* (sin (* 0.5 x)) 1.3333333333333333)
     (*
      (* 1.3333333333333333 x)
      (/ (pow (- 0.5 (* (cos (* 1.0 x)) 0.5)) 0.5) x)))))

C

double code(double x) {
	double t_0 = sin((x * 0.5));
	double tmp;
	if (((((8.0 / 3.0) * t_0) * t_0) / sin(x)) <= 0.06) {
		tmp = sin((0.5 * x)) * 1.3333333333333333;
	} else {
		tmp = (1.3333333333333333 * x) * (pow((0.5 - (cos((1.0 * x)) * 0.5)), 0.5) / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x * 0.5d0))
    if (((((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x)) <= 0.06d0) then
        tmp = sin((0.5d0 * x)) * 1.3333333333333333d0
    else
        tmp = (1.3333333333333333d0 * x) * (((0.5d0 - (cos((1.0d0 * x)) * 0.5d0)) ** 0.5d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sin((x * 0.5));
	double tmp;
	if (((((8.0 / 3.0) * t_0) * t_0) / Math.sin(x)) <= 0.06) {
		tmp = Math.sin((0.5 * x)) * 1.3333333333333333;
	} else {
		tmp = (1.3333333333333333 * x) * (Math.pow((0.5 - (Math.cos((1.0 * x)) * 0.5)), 0.5) / x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sin((x * 0.5))
	tmp = 0
	if ((((8.0 / 3.0) * t_0) * t_0) / math.sin(x)) <= 0.06:
		tmp = math.sin((0.5 * x)) * 1.3333333333333333
	else:
		tmp = (1.3333333333333333 * x) * (math.pow((0.5 - (math.cos((1.0 * x)) * 0.5)), 0.5) / x)
	return tmp

Julia

function code(x)
	t_0 = sin(Float64(x * 0.5))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x)) <= 0.06)
		tmp = Float64(sin(Float64(0.5 * x)) * 1.3333333333333333);
	else
		tmp = Float64(Float64(1.3333333333333333 * x) * Float64((Float64(0.5 - Float64(cos(Float64(1.0 * x)) * 0.5)) ^ 0.5) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sin((x * 0.5));
	tmp = 0.0;
	if (((((8.0 / 3.0) * t_0) * t_0) / sin(x)) <= 0.06)
		tmp = sin((0.5 * x)) * 1.3333333333333333;
	else
		tmp = (1.3333333333333333 * x) * (((0.5 - (cos((1.0 * x)) * 0.5)) ^ 0.5) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision], 0.06], N[(N[Sin[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] * 1.3333333333333333), $MachinePrecision], N[(N[(1.3333333333333333 * x), $MachinePrecision] * N[(N[Power[N[(0.5 - N[(N[Cos[N[(1.0 * x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x)) < 0.059999999999999998

   1. Initial program 69.7%

      \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
   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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \sin \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\frac{4}{3}} \]
   5. Step-by-step derivation

   6. Applied rewrites 69.2%

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

   if 0.059999999999999998 < (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x))

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(\frac{4}{3} \cdot x\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\sin x} \]
   3. Step-by-step derivation

      1. lower-*.f64 3.4

         \[\leadsto \frac{\left(1.3333333333333333 \cdot \color{blue}{x}\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   4. Applied rewrites 3.4%

      \[\leadsto \frac{\color{blue}{\left(1.3333333333333333 \cdot x\right)} \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\frac{4}{3} \cdot x\right) \cdot \sin \left(x \cdot \frac{1}{2}\right)}{\color{blue}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 11.7%

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

      1. metadata-eval 11.7

         \[\leadsto \frac{\left(1.3333333333333333 \cdot x\right) \cdot \sin \left(x \cdot 0.5\right)}{x} \]

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 11.7%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. unpow1 N/A

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

       4. metadata-eval N/A

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

       5. pow-unpow N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. sqr-sin-a-rev N/A

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

       9. lower-pow.f64 N/A

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

       10. lower--.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-cos.f64 N/A

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

       14. associate-*r* N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 21.0

           \[\leadsto \left(1.3333333333333333 \cdot x\right) \cdot \frac{{\left(0.5 - \cos \color{blue}{\left(1 \cdot x\right)} \cdot 0.5\right)}^{0.5}}{x} \]

   11. Applied rewrites 21.0%

       \[\leadsto \left(1.3333333333333333 \cdot x\right) \cdot \frac{\color{blue}{{\left(0.5 - \cos \left(1 \cdot x\right) \cdot 0.5\right)}^{0.5}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 55.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(0.5 \cdot x\right) \cdot 1.3333333333333333 \end{array} \]

FPCore

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

C

double code(double x) {
	return sin((0.5 * x)) * 1.3333333333333333;
}

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)) * 1.3333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 N/A

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

3. Applied rewrites 99.2%

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

4. Taylor expanded in x around 0

   \[\leadsto \sin \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\frac{4}{3}} \]
5. Step-by-step derivation

6. Applied rewrites 55.4%

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

Alternative 10: 51.2% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.6666666666666666 \cdot x \end{array} \]

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 76.7%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 51.2

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

4. Applied rewrites 51.2%

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

Reproduce

herbie shell --seed 2025119 
(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)))