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

Percentage Accurate: 77.3% → 99.3%

Time: 8.1s

Alternatives: 8

Speedup: 1.5×

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: 77.3% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(0.0030864197530864196 \cdot {x\_m}^{4} - 0.4444444444444444\right) \cdot \frac{x\_m}{-0.05555555555555555 \cdot \left(x\_m \cdot x\_m\right) - 0.6666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\sin \left(0.5 \cdot x\_m\right)}^{2}}{\sin x\_m} \cdot 2.6666666666666665\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e-11)
    (*
     (- (* 0.0030864197530864196 (pow x_m 4.0)) 0.4444444444444444)
     (/ x_m (- (* -0.05555555555555555 (* x_m x_m)) 0.6666666666666666)))
    (* (/ (pow (sin (* 0.5 x_m)) 2.0) (sin x_m)) 2.6666666666666665))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5e-11) {
		tmp = ((0.0030864197530864196 * pow(x_m, 4.0)) - 0.4444444444444444) * (x_m / ((-0.05555555555555555 * (x_m * x_m)) - 0.6666666666666666));
	} else {
		tmp = (pow(sin((0.5 * x_m)), 2.0) / sin(x_m)) * 2.6666666666666665;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 5d-11) then
        tmp = ((0.0030864197530864196d0 * (x_m ** 4.0d0)) - 0.4444444444444444d0) * (x_m / (((-0.05555555555555555d0) * (x_m * x_m)) - 0.6666666666666666d0))
    else
        tmp = ((sin((0.5d0 * x_m)) ** 2.0d0) / sin(x_m)) * 2.6666666666666665d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 5e-11) {
		tmp = ((0.0030864197530864196 * Math.pow(x_m, 4.0)) - 0.4444444444444444) * (x_m / ((-0.05555555555555555 * (x_m * x_m)) - 0.6666666666666666));
	} else {
		tmp = (Math.pow(Math.sin((0.5 * x_m)), 2.0) / Math.sin(x_m)) * 2.6666666666666665;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 5e-11:
		tmp = ((0.0030864197530864196 * math.pow(x_m, 4.0)) - 0.4444444444444444) * (x_m / ((-0.05555555555555555 * (x_m * x_m)) - 0.6666666666666666))
	else:
		tmp = (math.pow(math.sin((0.5 * x_m)), 2.0) / math.sin(x_m)) * 2.6666666666666665
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 5e-11)
		tmp = Float64(Float64(Float64(0.0030864197530864196 * (x_m ^ 4.0)) - 0.4444444444444444) * Float64(x_m / Float64(Float64(-0.05555555555555555 * Float64(x_m * x_m)) - 0.6666666666666666)));
	else
		tmp = Float64(Float64((sin(Float64(0.5 * x_m)) ^ 2.0) / sin(x_m)) * 2.6666666666666665);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 5e-11)
		tmp = ((0.0030864197530864196 * (x_m ^ 4.0)) - 0.4444444444444444) * (x_m / ((-0.05555555555555555 * (x_m * x_m)) - 0.6666666666666666));
	else
		tmp = ((sin((0.5 * x_m)) ^ 2.0) / sin(x_m)) * 2.6666666666666665;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 5e-11], N[(N[(N[(0.0030864197530864196 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] - 0.4444444444444444), $MachinePrecision] * N[(x$95$m / N[(N[(-0.05555555555555555 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * 2.6666666666666665), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000018e-11

   1. Initial program 65.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.2

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

   5. Applied rewrites 71.2%

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

   7. Applied rewrites 70.9%

      \[\leadsto \left(0.0030864197530864196 \cdot {x}^{4} - 0.4444444444444444\right) \cdot \color{blue}{\frac{x}{-0.05555555555555555 \cdot \left(x \cdot x\right) - 0.6666666666666666}} \]

   if 5.00000000000000018e-11 < x

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 99.2

          \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x} \cdot \frac{8}{3} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.2

          \[\leadsto \frac{{\sin \color{blue}{\left(0.5 \cdot x\right)}}^{2}}{\sin x} \cdot \frac{8}{3} \]

      15. lift-/.f64 N/A

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

      16. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

Alternative 2: 57.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot 0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x\_m} \leq -0.02:\\ \;\;\;\;1.3333333333333333 \cdot {\sin \left(\mathsf{fma}\left(x\_m, 0.5, \mathsf{PI}\left(\right)\right)\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;1.3333333333333333 \cdot \left|t\_0\right|\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (*
    x_s
    (if (<= (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x_m)) -0.02)
      (* 1.3333333333333333 (pow (sin (fma x_m 0.5 (PI))) 1.0))
      (* 1.3333333333333333 (fabs t_0))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x)) < -0.0200000000000000004

   1. Initial program 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 99.0

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

      17. lift-/.f64 N/A

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

      18. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 36.5

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

      10. lift-sin.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sin.f64 36.5

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

   6. Applied rewrites 36.5%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 14.3%

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

       1. lift-sin.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow1 N/A

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. pow-prod-down N/A

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

       8. sqr-neg-rev N/A

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

       9. pow2 N/A

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

       10. pow-pow N/A

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

       11. metadata-eval N/A

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

       12. lower-pow.f64 N/A

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

       13. *-commutative N/A

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

       14. lift-*.f64 N/A

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

       15. sin-+PI-rev N/A

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

       16. lower-sin.f64 N/A

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-fma.f64 N/A

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

       20. lower-PI.f64 13.3

           \[\leadsto 1.3333333333333333 \cdot {\sin \left(\mathsf{fma}\left(x, 0.5, \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{1} \]

   11. Applied rewrites 13.3%

       \[\leadsto 1.3333333333333333 \cdot \color{blue}{{\sin \left(\mathsf{fma}\left(x, 0.5, \mathsf{PI}\left(\right)\right)\right)}^{1}} \]

   if -0.0200000000000000004 < (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x))

   1. Initial program 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 99.3

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

      17. lift-/.f64 N/A

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

      18. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 46.8

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

      10. lift-sin.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sin.f64 46.8

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

   6. Applied rewrites 46.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 70.7%

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

       1. lift-sin.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. rem-square-sqrt N/A

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

       5. sqrt-unprod N/A

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

       6. rem-sqrt-square N/A

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

       7. lower-fabs.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. lift-sin.f64 40.0

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 40.0

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

   11. Applied rewrites 40.0%

       \[\leadsto 1.3333333333333333 \cdot \color{blue}{\left|\sin \left(x \cdot 0.5\right)\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 57.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot 0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x\_m} \leq 10^{-6}:\\ \;\;\;\;1.3333333333333333 \cdot \sin \left(0.5 \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1.3333333333333333 \cdot \sqrt{\mathsf{fma}\left(-0.5, \cos x\_m, 0.5\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (*
    x_s
    (if (<= (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x_m)) 1e-6)
      (* 1.3333333333333333 (sin (* 0.5 x_m)))
      (* 1.3333333333333333 (sqrt (fma -0.5 (cos x_m) 0.5)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * 0.5));
	double tmp;
	if (((((8.0 / 3.0) * t_0) * t_0) / sin(x_m)) <= 1e-6) {
		tmp = 1.3333333333333333 * sin((0.5 * x_m));
	} else {
		tmp = 1.3333333333333333 * sqrt(fma(-0.5, cos(x_m), 0.5));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(x_m * 0.5))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x_m)) <= 1e-6)
		tmp = Float64(1.3333333333333333 * sin(Float64(0.5 * x_m)));
	else
		tmp = Float64(1.3333333333333333 * sqrt(fma(-0.5, cos(x_m), 0.5)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], 1e-6], N[(1.3333333333333333 * N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.3333333333333333 * N[Sqrt[N[(-0.5 * N[Cos[x$95$m], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x)) < 9.99999999999999955e-7

   1. Initial program 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 99.3

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

      17. lift-/.f64 N/A

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

      18. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 74.0%

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

   if 9.99999999999999955e-7 < (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x))

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 99.1

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

      17. lift-/.f64 N/A

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

      18. metadata-eval 99.1

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

   4. Applied rewrites 99.1%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 47.1

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

      10. lift-sin.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sin.f64 47.1

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

   6. Applied rewrites 47.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 11.6%

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

       1. lift-sin.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow1 N/A

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

       5. metadata-eval N/A

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

       6. sqrt-pow1 N/A

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

       7. pow2 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. sqr-sin-a N/A

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

       13. *-commutative N/A

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

       14. cos-neg-rev N/A

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

       15. lift-*.f64 N/A

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

       16. associate-*r* N/A

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

       17. metadata-eval N/A

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

       18. lift-*.f64 N/A

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

       19. cos-neg-rev N/A

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

       20. lift-cos.f64 N/A

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

       21. *-commutative N/A

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

       22. lift-*.f64 N/A

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

       23. lift--.f64 N/A

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

   11. Applied rewrites 21.9%

       \[\leadsto 1.3333333333333333 \cdot \color{blue}{\sqrt{\mathsf{fma}\left(-0.5, \cos x, 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \leq 10^{-6}:\\ \;\;\;\;1.3333333333333333 \cdot \sin \left(0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1.3333333333333333 \cdot \sqrt{\mathsf{fma}\left(-0.5, \cos x, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(x\_m \cdot 0.5\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{8}{3} \cdot t\_0\right) \cdot t\_0}{\sin x\_m} \leq -0.02:\\ \;\;\;\;1.3333333333333333 \cdot \sin \left(0.5 \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1.3333333333333333 \cdot \left|t\_0\right|\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* x_m 0.5))))
   (*
    x_s
    (if (<= (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x_m)) -0.02)
      (* 1.3333333333333333 (sin (* 0.5 x_m)))
      (* 1.3333333333333333 (fabs t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((x_m * 0.5));
	double tmp;
	if (((((8.0 / 3.0) * t_0) * t_0) / sin(x_m)) <= -0.02) {
		tmp = 1.3333333333333333 * sin((0.5 * x_m));
	} else {
		tmp = 1.3333333333333333 * fabs(t_0);
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x_m * 0.5d0))
    if (((((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x_m)) <= (-0.02d0)) then
        tmp = 1.3333333333333333d0 * sin((0.5d0 * x_m))
    else
        tmp = 1.3333333333333333d0 * abs(t_0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin((x_m * 0.5));
	double tmp;
	if (((((8.0 / 3.0) * t_0) * t_0) / Math.sin(x_m)) <= -0.02) {
		tmp = 1.3333333333333333 * Math.sin((0.5 * x_m));
	} else {
		tmp = 1.3333333333333333 * Math.abs(t_0);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((x_m * 0.5))
	tmp = 0
	if ((((8.0 / 3.0) * t_0) * t_0) / math.sin(x_m)) <= -0.02:
		tmp = 1.3333333333333333 * math.sin((0.5 * x_m))
	else:
		tmp = 1.3333333333333333 * math.fabs(t_0)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(x_m * 0.5))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x_m)) <= -0.02)
		tmp = Float64(1.3333333333333333 * sin(Float64(0.5 * x_m)));
	else
		tmp = Float64(1.3333333333333333 * abs(t_0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin((x_m * 0.5));
	tmp = 0.0;
	if (((((8.0 / 3.0) * t_0) * t_0) / sin(x_m)) <= -0.02)
		tmp = 1.3333333333333333 * sin((0.5 * x_m));
	else
		tmp = 1.3333333333333333 * abs(t_0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], -0.02], N[(1.3333333333333333 * N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.3333333333333333 * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x)) < -0.0200000000000000004

   1. Initial program 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 99.0

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

      17. lift-/.f64 N/A

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

      18. metadata-eval 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 14.3%

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

   if -0.0200000000000000004 < (/.f64 (*.f64 (*.f64 (/.f64 #s(literal 8 binary64) #s(literal 3 binary64)) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 (*.f64 x #s(literal 1/2 binary64)))) (sin.f64 x))

   1. Initial program 67.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 99.3

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

      17. lift-/.f64 N/A

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

      18. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 46.8

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

      10. lift-sin.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sin.f64 46.8

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

   6. Applied rewrites 46.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 70.7%

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

       1. lift-sin.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. rem-square-sqrt N/A

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

       5. sqrt-unprod N/A

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

       6. rem-sqrt-square N/A

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

       7. lower-fabs.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. lift-sin.f64 40.0

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 40.0

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

   11. Applied rewrites 40.0%

       \[\leadsto 1.3333333333333333 \cdot \color{blue}{\left|\sin \left(x \cdot 0.5\right)\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{8}{3} \cdot \sin \left(x \cdot 0.5\right)\right) \cdot \sin \left(x \cdot 0.5\right)}{\sin x} \leq -0.02:\\ \;\;\;\;1.3333333333333333 \cdot \sin \left(0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1.3333333333333333 \cdot \left|\sin \left(x \cdot 0.5\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot x\_m\right)\\ x\_s \cdot \left(\left(t\_0 \cdot \frac{2.6666666666666665}{\sin x\_m}\right) \cdot t\_0\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 x_m))))
   (* x_s (* (* t_0 (/ 2.6666666666666665 (sin x_m))) t_0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin((0.5 * x_m));
	return x_s * ((t_0 * (2.6666666666666665 / sin(x_m))) * t_0);
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    t_0 = sin((0.5d0 * x_m))
    code = x_s * ((t_0 * (2.6666666666666665d0 / sin(x_m))) * t_0)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin((0.5 * x_m));
	return x_s * ((t_0 * (2.6666666666666665 / Math.sin(x_m))) * t_0);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin((0.5 * x_m))
	return x_s * ((t_0 * (2.6666666666666665 / math.sin(x_m))) * t_0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = sin(Float64(0.5 * x_m))
	return Float64(x_s * Float64(Float64(t_0 * Float64(2.6666666666666665 / sin(x_m))) * t_0))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	t_0 = sin((0.5 * x_m));
	tmp = x_s * ((t_0 * (2.6666666666666665 / sin(x_m))) * t_0);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * N[(N[(t$95$0 * N[(2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 99.3

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

   17. lift-/.f64 N/A

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

   18. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

Alternative 6: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000112:\\ \;\;\;\;\left(0.0030864197530864196 \cdot {x\_m}^{4} - 0.4444444444444444\right) \cdot \frac{x\_m}{-0.05555555555555555 \cdot \left(x\_m \cdot x\_m\right) - 0.6666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x\_m, -0.5, 0.5\right)}{\sin x\_m} \cdot 2.6666666666666665\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.000112)
    (*
     (- (* 0.0030864197530864196 (pow x_m 4.0)) 0.4444444444444444)
     (/ x_m (- (* -0.05555555555555555 (* x_m x_m)) 0.6666666666666666)))
    (* (/ (fma (cos x_m) -0.5 0.5) (sin x_m)) 2.6666666666666665))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.000112) {
		tmp = ((0.0030864197530864196 * pow(x_m, 4.0)) - 0.4444444444444444) * (x_m / ((-0.05555555555555555 * (x_m * x_m)) - 0.6666666666666666));
	} else {
		tmp = (fma(cos(x_m), -0.5, 0.5) / sin(x_m)) * 2.6666666666666665;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.000112)
		tmp = Float64(Float64(Float64(0.0030864197530864196 * (x_m ^ 4.0)) - 0.4444444444444444) * Float64(x_m / Float64(Float64(-0.05555555555555555 * Float64(x_m * x_m)) - 0.6666666666666666)));
	else
		tmp = Float64(Float64(fma(cos(x_m), -0.5, 0.5) / sin(x_m)) * 2.6666666666666665);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.000112], N[(N[(N[(0.0030864197530864196 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] - 0.4444444444444444), $MachinePrecision] * N[(x$95$m / N[(N[(-0.05555555555555555 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x$95$m], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * 2.6666666666666665), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.11999999999999998e-4

   1. Initial program 65.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 71.1%

      \[\leadsto \left(0.0030864197530864196 \cdot {x}^{4} - 0.4444444444444444\right) \cdot \color{blue}{\frac{x}{-0.05555555555555555 \cdot \left(x \cdot x\right) - 0.6666666666666666}} \]

   if 1.11999999999999998e-4 < x

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 99.2

          \[\leadsto \frac{\color{blue}{{\sin \left(x \cdot 0.5\right)}^{2}}}{\sin x} \cdot \frac{8}{3} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.2

          \[\leadsto \frac{{\sin \color{blue}{\left(0.5 \cdot x\right)}}^{2}}{\sin x} \cdot \frac{8}{3} \]

      15. lift-/.f64 N/A

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

      16. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

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

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

      7. cos-neg-rev N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lift-cos.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-lft-identity N/A

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

      19. metadata-eval 98.2

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

   6. Applied rewrites 98.2%

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

Alternative 7: 54.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(1.3333333333333333 \cdot \sin \left(0.5 \cdot x\_m\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* 1.3333333333333333 (sin (* 0.5 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (1.3333333333333333 * sin((0.5 * x_m)));
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (1.3333333333333333d0 * sin((0.5d0 * x_m)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (1.3333333333333333 * Math.sin((0.5 * x_m)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (1.3333333333333333 * math.sin((0.5 * x_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(1.3333333333333333 * sin(Float64(0.5 * x_m))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (1.3333333333333333 * sin((0.5 * x_m)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(1.3333333333333333 * N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 99.3

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

   17. lift-/.f64 N/A

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

   18. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 58.1%

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

8. Final simplification 58.1%

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

Alternative 8: 50.6% accurate, 57.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(0.6666666666666666 \cdot x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* 0.6666666666666666 x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (0.6666666666666666 * x_m);
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (0.6666666666666666d0 * x_m)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (0.6666666666666666 * x_m);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (0.6666666666666666 * x_m)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(0.6666666666666666 * x_m))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (0.6666666666666666 * x_m);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(0.6666666666666666 * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.3%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 53.9

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

5. Applied rewrites 53.9%

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

Developer Target 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sin((x * 0.5d0))
    code = ((8.0d0 * t_0) / 3.0d0) / (sin(x) / t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024352 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (* 8 (sin (* x 1/2))) 3) (/ (sin x) (sin (* x 1/2)))))

  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))