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

Percentage Accurate: 77.0% → 99.2%

Time: 3.0s

Alternatives: 7

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: 77.0% 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

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

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 77.0%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 99.2%

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

Alternative 2: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.020833333333333332, x \cdot x, 0.25\right) \cdot x\right) \cdot 2.6666666666666665\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(0.5 \cdot x\right)}^{2}}{\sin x} \cdot 2.6666666666666665\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-5)
   (* (* (fma 0.020833333333333332 (* x x) 0.25) x) 2.6666666666666665)
   (* (/ (pow (sin (* 0.5 x)) 2.0) (sin x)) 2.6666666666666665)))

C

double code(double x) {
	double tmp;
	if (x <= 2e-5) {
		tmp = (fma(0.020833333333333332, (x * x), 0.25) * x) * 2.6666666666666665;
	} else {
		tmp = (pow(sin((0.5 * x)), 2.0) / sin(x)) * 2.6666666666666665;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-5)
		tmp = Float64(Float64(fma(0.020833333333333332, Float64(x * x), 0.25) * x) * 2.6666666666666665);
	else
		tmp = Float64(Float64((sin(Float64(0.5 * x)) ^ 2.0) / sin(x)) * 2.6666666666666665);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e-5], N[(N[(N[(0.020833333333333332 * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * 2.6666666666666665), $MachinePrecision], N[(N[(N[Power[N[Sin[N[(0.5 * x), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] * 2.6666666666666665), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000016e-5

   1. Initial program 69.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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. associate-*l/ N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 36.9%

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

   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 67.2

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

   8. Applied rewrites 67.2%

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

   if 2.00000000000000016e-5 < 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

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

      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 99.0%

      \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665\right)} \]
   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. associate-*l/ N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 98.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

Alternative 3: 75.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.023:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.017361111111111112, x \cdot x, 0.16666666666666666\right), x \cdot x, 1.3333333333333333\right) \cdot \sin \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot x\right)\right)\right) \cdot \frac{2.6666666666666665}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.023)
   (*
    (fma
     (fma 0.017361111111111112 (* x x) 0.16666666666666666)
     (* x x)
     1.3333333333333333)
    (sin (* x 0.5)))
   (* (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 x))))) (/ 2.6666666666666665 (sin x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.023) {
		tmp = fma(fma(0.017361111111111112, (x * x), 0.16666666666666666), (x * x), 1.3333333333333333) * sin((x * 0.5));
	} else {
		tmp = (0.5 - (0.5 * cos((2.0 * (0.5 * x))))) * (2.6666666666666665 / sin(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.023)
		tmp = Float64(fma(fma(0.017361111111111112, Float64(x * x), 0.16666666666666666), Float64(x * x), 1.3333333333333333) * sin(Float64(x * 0.5)));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * x))))) * Float64(2.6666666666666665 / sin(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.023], N[(N[(N[(0.017361111111111112 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.3333333333333333), $MachinePrecision] * N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.6666666666666665 / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.023

   1. Initial program 69.4%

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

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}\right) + \color{blue}{\frac{4}{3}}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}, \color{blue}{{x}^{2}}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{5}{288} \cdot {x}^{2} + \frac{1}{6}, {\color{blue}{x}}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, {x}^{2}, \frac{1}{6}\right), {\color{blue}{x}}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, x \cdot x, \frac{1}{6}\right), {x}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, x \cdot x, \frac{1}{6}\right), {x}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, x \cdot x, \frac{1}{6}\right), x \cdot \color{blue}{x}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      9. lift-*.f64 67.3

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.017361111111111112, x \cdot x, 0.16666666666666666\right), x \cdot \color{blue}{x}, 1.3333333333333333\right) \cdot \sin \left(x \cdot 0.5\right) \]

   8. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.017361111111111112, x \cdot x, 0.16666666666666666\right), x \cdot x, 1.3333333333333333\right)} \cdot \sin \left(x \cdot 0.5\right) \]

   if 0.023 < 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. associate-*l* N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lift-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

   3. Applied rewrites 98.4%

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

Alternative 4: 75.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.023:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.017361111111111112, x \cdot x, 0.16666666666666666\right), x \cdot x, 1.3333333333333333\right) \cdot \sin \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - 0.5 \cdot \cos x}{\sin x} \cdot 2.6666666666666665\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.023)
   (*
    (fma
     (fma 0.017361111111111112 (* x x) 0.16666666666666666)
     (* x x)
     1.3333333333333333)
    (sin (* x 0.5)))
   (* (/ (- 0.5 (* 0.5 (cos x))) (sin x)) 2.6666666666666665)))

C

double code(double x) {
	double tmp;
	if (x <= 0.023) {
		tmp = fma(fma(0.017361111111111112, (x * x), 0.16666666666666666), (x * x), 1.3333333333333333) * sin((x * 0.5));
	} else {
		tmp = ((0.5 - (0.5 * cos(x))) / sin(x)) * 2.6666666666666665;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.023)
		tmp = Float64(fma(fma(0.017361111111111112, Float64(x * x), 0.16666666666666666), Float64(x * x), 1.3333333333333333) * sin(Float64(x * 0.5)));
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(0.5 * cos(x))) / sin(x)) * 2.6666666666666665);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.023], N[(N[(N[(0.017361111111111112 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.3333333333333333), $MachinePrecision] * N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] * 2.6666666666666665), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.023

   1. Initial program 69.4%

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

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}\right) + \color{blue}{\frac{4}{3}}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}, \color{blue}{{x}^{2}}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{5}{288} \cdot {x}^{2} + \frac{1}{6}, {\color{blue}{x}}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, {x}^{2}, \frac{1}{6}\right), {\color{blue}{x}}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, x \cdot x, \frac{1}{6}\right), {x}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, x \cdot x, \frac{1}{6}\right), {x}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, x \cdot x, \frac{1}{6}\right), x \cdot \color{blue}{x}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      9. lift-*.f64 67.3

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.017361111111111112, x \cdot x, 0.16666666666666666\right), x \cdot \color{blue}{x}, 1.3333333333333333\right) \cdot \sin \left(x \cdot 0.5\right) \]

   8. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.017361111111111112, x \cdot x, 0.16666666666666666\right), x \cdot x, 1.3333333333333333\right)} \cdot \sin \left(x \cdot 0.5\right) \]

   if 0.023 < 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

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

      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 99.0%

      \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665\right)} \]
   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. associate-*l/ N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{x}}{\sin x} \cdot \frac{8}{3} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.4%

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

Alternative 5: 75.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.023:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.017361111111111112, x \cdot x, 0.16666666666666666\right), x \cdot x, 1.3333333333333333\right) \cdot \sin \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot \left(0.5 - \cos x \cdot 0.5\right)}{\sin x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.023)
   (*
    (fma
     (fma 0.017361111111111112 (* x x) 0.16666666666666666)
     (* x x)
     1.3333333333333333)
    (sin (* x 0.5)))
   (/ (* 2.6666666666666665 (- 0.5 (* (cos x) 0.5))) (sin x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.023) {
		tmp = fma(fma(0.017361111111111112, (x * x), 0.16666666666666666), (x * x), 1.3333333333333333) * sin((x * 0.5));
	} else {
		tmp = (2.6666666666666665 * (0.5 - (cos(x) * 0.5))) / sin(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.023)
		tmp = Float64(fma(fma(0.017361111111111112, Float64(x * x), 0.16666666666666666), Float64(x * x), 1.3333333333333333) * sin(Float64(x * 0.5)));
	else
		tmp = Float64(Float64(2.6666666666666665 * Float64(0.5 - Float64(cos(x) * 0.5))) / sin(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.023], N[(N[(N[(0.017361111111111112 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.3333333333333333), $MachinePrecision] * N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(2.6666666666666665 * N[(0.5 - N[(N[Cos[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.023

   1. Initial program 69.4%

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

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}\right)\right)} \cdot \sin \left(x \cdot \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}\right) + \color{blue}{\frac{4}{3}}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6} + \frac{5}{288} \cdot {x}^{2}, \color{blue}{{x}^{2}}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{5}{288} \cdot {x}^{2} + \frac{1}{6}, {\color{blue}{x}}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, {x}^{2}, \frac{1}{6}\right), {\color{blue}{x}}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, x \cdot x, \frac{1}{6}\right), {x}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, x \cdot x, \frac{1}{6}\right), {x}^{2}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{288}, x \cdot x, \frac{1}{6}\right), x \cdot \color{blue}{x}, \frac{4}{3}\right) \cdot \sin \left(x \cdot \frac{1}{2}\right) \]

      9. lift-*.f64 67.3

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.017361111111111112, x \cdot x, 0.16666666666666666\right), x \cdot \color{blue}{x}, 1.3333333333333333\right) \cdot \sin \left(x \cdot 0.5\right) \]

   8. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.017361111111111112, x \cdot x, 0.16666666666666666\right), x \cdot x, 1.3333333333333333\right)} \cdot \sin \left(x \cdot 0.5\right) \]

   if 0.023 < 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

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

      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 99.0%

      \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665\right)} \]
   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. associate-*l/ N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{8}{3} \cdot \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos x}{\sin x}} \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

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

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

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   8. Applied rewrites 98.4%

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

Alternative 6: 55.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 1.3333333333333333d0 * sin((x * 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

   \[\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \]
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 99.2%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\sin x} \cdot \left(\sin \left(0.5 \cdot x\right) \cdot 2.6666666666666665\right)} \]
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. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 55.0%

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

Alternative 7: 51.0% 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 77.0%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 51.0

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

4. Applied rewrites 51.0%

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

Reproduce

herbie shell --seed 2025114 
(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)))