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

Percentage Accurate: 77.2% → 99.4%

Time: 3.7s

Alternatives: 13

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: 77.2% 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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x\_m}{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(-0.5 \cdot x\_m\right) \cdot \sin \left(0.5 \cdot x\_m\right)\right) \cdot \frac{-2.6666666666666665}{\sin x\_m}\\ \end{array} \end{array} \]

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 2e-16)
    (/ 1.0 (/ 1.0 (/ x_m 1.5)))
    (*
     (* (sin (* -0.5 x_m)) (sin (* 0.5 x_m)))
     (/ -2.6666666666666665 (sin x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2e-16) {
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	} else {
		tmp = (sin((-0.5 * x_m)) * sin((0.5 * x_m))) * (-2.6666666666666665 / sin(x_m));
	}
	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 <= 2d-16) then
        tmp = 1.0d0 / (1.0d0 / (x_m / 1.5d0))
    else
        tmp = (sin(((-0.5d0) * x_m)) * sin((0.5d0 * x_m))) * ((-2.6666666666666665d0) / sin(x_m))
    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 <= 2e-16) {
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	} else {
		tmp = (Math.sin((-0.5 * x_m)) * Math.sin((0.5 * x_m))) * (-2.6666666666666665 / Math.sin(x_m));
	}
	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 <= 2e-16:
		tmp = 1.0 / (1.0 / (x_m / 1.5))
	else:
		tmp = (math.sin((-0.5 * x_m)) * math.sin((0.5 * x_m))) * (-2.6666666666666665 / math.sin(x_m))
	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 <= 2e-16)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x_m / 1.5)));
	else
		tmp = Float64(Float64(sin(Float64(-0.5 * x_m)) * sin(Float64(0.5 * x_m))) * Float64(-2.6666666666666665 / sin(x_m)));
	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 <= 2e-16)
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	else
		tmp = (sin((-0.5 * x_m)) * sin((0.5 * x_m))) * (-2.6666666666666665 / sin(x_m));
	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, 2e-16], N[(1.0 / N[(1.0 / N[(x$95$m / 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[N[(-0.5 * x$95$m), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-2.6666666666666665 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e-16

   1. Initial program 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sqr-sin-a N/A

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

      7. sub-flip N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 52.3

         \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}} \]

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2

         \[\leadsto \frac{1}{\frac{1.5}{\color{blue}{x}}} \]

   8. Applied rewrites 51.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{3}{2}}{\color{blue}{x}}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

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

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

      4. lower-unsound-/.f64 51.3

         \[\leadsto \frac{1}{\frac{1}{\frac{x}{\color{blue}{1.5}}}} \]

   10. Applied rewrites 51.3%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{1.5}}}} \]

   if 2e-16 < x

   1. Initial program 77.2%

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. sqr-abs-rev N/A

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

      8. associate-*l* N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 99.2%

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

   5. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. frac-2neg N/A

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

      18. lift-neg.f64 N/A

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

      19. remove-double-neg N/A

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

   7. Applied rewrites 77.2%

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

Alternative 2: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-22}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x\_m}{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.6666666666666665 \cdot {\sin \left(0.5 \cdot x\_m\right)}^{2}}{\sin x\_m}\\ \end{array} \end{array} \]

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 1e-22)
    (/ 1.0 (/ 1.0 (/ x_m 1.5)))
    (/ (* 2.6666666666666665 (pow (sin (* 0.5 x_m)) 2.0)) (sin x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1e-22) {
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	} else {
		tmp = (2.6666666666666665 * pow(sin((0.5 * x_m)), 2.0)) / sin(x_m);
	}
	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 <= 1d-22) then
        tmp = 1.0d0 / (1.0d0 / (x_m / 1.5d0))
    else
        tmp = (2.6666666666666665d0 * (sin((0.5d0 * x_m)) ** 2.0d0)) / sin(x_m)
    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 <= 1e-22) {
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	} else {
		tmp = (2.6666666666666665 * Math.pow(Math.sin((0.5 * x_m)), 2.0)) / Math.sin(x_m);
	}
	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 <= 1e-22:
		tmp = 1.0 / (1.0 / (x_m / 1.5))
	else:
		tmp = (2.6666666666666665 * math.pow(math.sin((0.5 * x_m)), 2.0)) / math.sin(x_m)
	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 <= 1e-22)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x_m / 1.5)));
	else
		tmp = Float64(Float64(2.6666666666666665 * (sin(Float64(0.5 * x_m)) ^ 2.0)) / sin(x_m));
	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 <= 1e-22)
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	else
		tmp = (2.6666666666666665 * (sin((0.5 * x_m)) ^ 2.0)) / sin(x_m);
	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, 1e-22], N[(1.0 / N[(1.0 / N[(x$95$m / 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.6666666666666665 * N[Power[N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e-22

   1. Initial program 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sqr-sin-a N/A

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

      7. sub-flip N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 52.3

         \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}} \]

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2

         \[\leadsto \frac{1}{\frac{1.5}{\color{blue}{x}}} \]

   8. Applied rewrites 51.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{3}{2}}{\color{blue}{x}}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

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

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

      4. lower-unsound-/.f64 51.3

         \[\leadsto \frac{1}{\frac{1}{\frac{x}{\color{blue}{1.5}}}} \]

   10. Applied rewrites 51.3%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{1.5}}}} \]

   if 1e-22 < x

   1. Initial program 77.2%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-*.f64 77.2

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

   4. Applied rewrites 77.2%

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

Alternative 3: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x\_m}{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;2.6666666666666665 \cdot \frac{{\sin \left(0.5 \cdot x\_m\right)}^{2}}{\sin x\_m}\\ \end{array} \end{array} \]

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-20)
    (/ 1.0 (/ 1.0 (/ x_m 1.5)))
    (* 2.6666666666666665 (/ (pow (sin (* 0.5 x_m)) 2.0) (sin x_m))))))

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-20) {
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	} else {
		tmp = 2.6666666666666665 * (pow(sin((0.5 * x_m)), 2.0) / sin(x_m));
	}
	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-20) then
        tmp = 1.0d0 / (1.0d0 / (x_m / 1.5d0))
    else
        tmp = 2.6666666666666665d0 * ((sin((0.5d0 * x_m)) ** 2.0d0) / sin(x_m))
    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-20) {
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	} else {
		tmp = 2.6666666666666665 * (Math.pow(Math.sin((0.5 * x_m)), 2.0) / Math.sin(x_m));
	}
	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-20:
		tmp = 1.0 / (1.0 / (x_m / 1.5))
	else:
		tmp = 2.6666666666666665 * (math.pow(math.sin((0.5 * x_m)), 2.0) / math.sin(x_m))
	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-20)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x_m / 1.5)));
	else
		tmp = Float64(2.6666666666666665 * Float64((sin(Float64(0.5 * x_m)) ^ 2.0) / sin(x_m)));
	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-20)
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	else
		tmp = 2.6666666666666665 * ((sin((0.5 * x_m)) ^ 2.0) / sin(x_m));
	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-20], N[(1.0 / N[(1.0 / N[(x$95$m / 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.6666666666666665 * N[(N[Power[N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999999e-20

   1. Initial program 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sqr-sin-a N/A

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

      7. sub-flip N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 52.3

         \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}} \]

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2

         \[\leadsto \frac{1}{\frac{1.5}{\color{blue}{x}}} \]

   8. Applied rewrites 51.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{3}{2}}{\color{blue}{x}}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

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

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

      4. lower-unsound-/.f64 51.3

         \[\leadsto \frac{1}{\frac{1}{\frac{x}{\color{blue}{1.5}}}} \]

   10. Applied rewrites 51.3%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{1.5}}}} \]

   if 4.9999999999999999e-20 < x

   1. Initial program 77.2%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 77.2

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

   4. Applied rewrites 77.2%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (sin((x_m * -0.5)) * ((sin((0.5 * x_m)) * -2.6666666666666665) / sin(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 * (sin((x_m * (-0.5d0))) * ((sin((0.5d0 * x_m)) * (-2.6666666666666665d0)) / sin(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 * (Math.sin((x_m * -0.5)) * ((Math.sin((0.5 * x_m)) * -2.6666666666666665) / Math.sin(x_m)));
}

Python

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(sin(Float64(x_m * -0.5)) * Float64(Float64(sin(Float64(0.5 * x_m)) * -2.6666666666666665) / sin(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 * (sin((x_m * -0.5)) * ((sin((0.5 * x_m)) * -2.6666666666666665) / sin(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[(N[Sin[N[(x$95$m * -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision] * -2.6666666666666665), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. sqr-abs-rev N/A

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

   8. associate-*l* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

3. Applied rewrites 99.2%

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

5. Applied rewrites 99.2%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg2 N/A

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

   4. distribute-neg-frac N/A

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

   5. lower-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. metadata-eval 99.2

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

7. Applied rewrites 99.2%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * ((sin((0.5d0 * x_m)) / sin(x_m)) * (sin(((-0.5d0) * x_m)) * (-2.6666666666666665d0)))
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 * ((Math.sin((0.5 * x_m)) / Math.sin(x_m)) * (Math.sin((-0.5 * x_m)) * -2.6666666666666665));
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((sin((0.5 * x_m)) / sin(x_m)) * (sin((-0.5 * x_m)) * -2.6666666666666665));
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[(N[(N[Sin[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(-0.5 * x$95$m), $MachinePrecision]], $MachinePrecision] * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.2

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

   10. remove-double-neg N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

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

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

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

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

   15. lower-*.f64 N/A

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

3. Applied rewrites 99.2%

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

Alternative 6: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x\_m}{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin x\_m} \cdot -2.6666666666666665\right) \cdot \mathsf{fma}\left(\cos x\_m, 0.5, -0.5\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 8.2e-5)
    (/ 1.0 (/ 1.0 (/ x_m 1.5)))
    (* (* (/ 1.0 (sin x_m)) -2.6666666666666665) (fma (cos x_m) 0.5 -0.5)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 8.2e-5) {
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	} else {
		tmp = ((1.0 / sin(x_m)) * -2.6666666666666665) * fma(cos(x_m), 0.5, -0.5);
	}
	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 <= 8.2e-5)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x_m / 1.5)));
	else
		tmp = Float64(Float64(Float64(1.0 / sin(x_m)) * -2.6666666666666665) * fma(cos(x_m), 0.5, -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_] := N[(x$95$s * If[LessEqual[x$95$m, 8.2e-5], N[(1.0 / N[(1.0 / N[(x$95$m / 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * -2.6666666666666665), $MachinePrecision] * N[(N[Cos[x$95$m], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8.20000000000000009e-5

   1. Initial program 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sqr-sin-a N/A

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

      7. sub-flip N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 52.3

         \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}} \]

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2

         \[\leadsto \frac{1}{\frac{1.5}{\color{blue}{x}}} \]

   8. Applied rewrites 51.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{3}{2}}{\color{blue}{x}}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

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

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

      4. lower-unsound-/.f64 51.3

         \[\leadsto \frac{1}{\frac{1}{\frac{x}{\color{blue}{1.5}}}} \]

   10. Applied rewrites 51.3%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{1.5}}}} \]

   if 8.20000000000000009e-5 < x

   1. Initial program 77.2%

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

      1. remove-double-neg N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

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

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

      6. distribute-rgt-neg-out N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

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

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

   3. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin x} \cdot -2.6666666666666665\right) \cdot \mathsf{fma}\left(\cos x, 0.5, -0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x\_m}{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\cos x\_m\right) + 1\right) \cdot \frac{1.3333333333333333}{\sin x\_m}\\ \end{array} \end{array} \]

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 8.2e-5)
    (/ 1.0 (/ 1.0 (/ x_m 1.5)))
    (* (+ (- (cos x_m)) 1.0) (/ 1.3333333333333333 (sin x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 8.2e-5) {
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	} else {
		tmp = (-cos(x_m) + 1.0) * (1.3333333333333333 / sin(x_m));
	}
	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 <= 8.2d-5) then
        tmp = 1.0d0 / (1.0d0 / (x_m / 1.5d0))
    else
        tmp = (-cos(x_m) + 1.0d0) * (1.3333333333333333d0 / sin(x_m))
    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 <= 8.2e-5) {
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	} else {
		tmp = (-Math.cos(x_m) + 1.0) * (1.3333333333333333 / Math.sin(x_m));
	}
	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 <= 8.2e-5:
		tmp = 1.0 / (1.0 / (x_m / 1.5))
	else:
		tmp = (-math.cos(x_m) + 1.0) * (1.3333333333333333 / math.sin(x_m))
	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 <= 8.2e-5)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x_m / 1.5)));
	else
		tmp = Float64(Float64(Float64(-cos(x_m)) + 1.0) * Float64(1.3333333333333333 / sin(x_m)));
	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 <= 8.2e-5)
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	else
		tmp = (-cos(x_m) + 1.0) * (1.3333333333333333 / sin(x_m));
	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, 8.2e-5], N[(1.0 / N[(1.0 / N[(x$95$m / 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Cos[x$95$m], $MachinePrecision]) + 1.0), $MachinePrecision] * N[(1.3333333333333333 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8.20000000000000009e-5

   1. Initial program 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sqr-sin-a N/A

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

      7. sub-flip N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 52.3

         \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}} \]

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2

         \[\leadsto \frac{1}{\frac{1.5}{\color{blue}{x}}} \]

   8. Applied rewrites 51.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{3}{2}}{\color{blue}{x}}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

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

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

      4. lower-unsound-/.f64 51.3

         \[\leadsto \frac{1}{\frac{1}{\frac{x}{\color{blue}{1.5}}}} \]

   10. Applied rewrites 51.3%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{1.5}}}} \]

   if 8.20000000000000009e-5 < x

   1. Initial program 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sqr-sin-a N/A

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

      7. sub-flip N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 52.3

         \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}} \]

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 52.4

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

   5. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{\mathsf{fma}\left(\cos x, -1.3333333333333333, 1.3333333333333333\right)}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. sub-negate N/A

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

      15. metadata-eval N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lift-cos.f64 N/A

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

      19. lower--.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-cos.f64 N/A

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

      22. associate-*r* N/A

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

   7. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\frac{-1}{\sin x} \cdot \left(-1.3333333333333333 - -1.3333333333333333 \cdot \cos x\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-/.f64 N/A

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

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

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

      19. lift-/.f64 N/A

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

      20. mult-flip N/A

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

      21. lift-/.f64 N/A

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

   9. Applied rewrites 52.4%

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

Alternative 8: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x\_m}{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x\_m, -1.3333333333333333, 1.3333333333333333\right)}{\sin x\_m}\\ \end{array} \end{array} \]

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 8.2e-5)
    (/ 1.0 (/ 1.0 (/ x_m 1.5)))
    (/ (fma (cos x_m) -1.3333333333333333 1.3333333333333333) (sin x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 8.2e-5) {
		tmp = 1.0 / (1.0 / (x_m / 1.5));
	} else {
		tmp = fma(cos(x_m), -1.3333333333333333, 1.3333333333333333) / sin(x_m);
	}
	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 <= 8.2e-5)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x_m / 1.5)));
	else
		tmp = Float64(fma(cos(x_m), -1.3333333333333333, 1.3333333333333333) / sin(x_m));
	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, 8.2e-5], N[(1.0 / N[(1.0 / N[(x$95$m / 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x$95$m], $MachinePrecision] * -1.3333333333333333 + 1.3333333333333333), $MachinePrecision] / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8.20000000000000009e-5

   1. Initial program 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sqr-sin-a N/A

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

      7. sub-flip N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 52.3

         \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}} \]

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2

         \[\leadsto \frac{1}{\frac{1.5}{\color{blue}{x}}} \]

   8. Applied rewrites 51.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{3}{2}}{\color{blue}{x}}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

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

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

      4. lower-unsound-/.f64 51.3

         \[\leadsto \frac{1}{\frac{1}{\frac{x}{\color{blue}{1.5}}}} \]

   10. Applied rewrites 51.3%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{1.5}}}} \]

   if 8.20000000000000009e-5 < x

   1. Initial program 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sqr-sin-a N/A

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

      7. sub-flip N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-+.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

   3. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 52.5

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

   5. Applied rewrites 52.5%

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

Alternative 9: 55.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

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 * (sin((x_m * (-0.5d0))) * (-1.3333333333333333d0))
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 * (Math.sin((x_m * -0.5)) * -1.3333333333333333);
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (sin((x_m * -0.5)) * -1.3333333333333333);
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[(N[Sin[N[(x$95$m * -0.5), $MachinePrecision]], $MachinePrecision] * -1.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. sqr-abs-rev N/A

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

   8. associate-*l* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

3. Applied rewrites 99.2%

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 55.2%

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

Alternative 10: 51.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{1}{\frac{1.5 + -0.125 \cdot {x\_m}^{2}}{x\_m}} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ 1.0 (/ (+ 1.5 (* -0.125 (pow x_m 2.0))) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (1.0 / ((1.5 + (-0.125 * pow(x_m, 2.0))) / 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 * (1.0d0 / ((1.5d0 + ((-0.125d0) * (x_m ** 2.0d0))) / 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 * (1.0 / ((1.5 + (-0.125 * Math.pow(x_m, 2.0))) / x_m));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (1.0 / ((1.5 + (-0.125 * math.pow(x_m, 2.0))) / x_m))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(1.0 / Float64(Float64(1.5 + Float64(-0.125 * (x_m ^ 2.0))) / 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 * (1.0 / ((1.5 + (-0.125 * (x_m ^ 2.0))) / 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[(1.0 / N[(N[(1.5 + N[(-0.125 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. sqr-sin-a N/A

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

   7. sub-flip N/A

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

   8. distribute-rgt-in N/A

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

   9. lower-+.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

3. Applied rewrites 52.3%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f64 52.3

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}} \]

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval 52.4

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

      \[\leadsto \frac{1}{\frac{\frac{3}{2} + \frac{-1}{8} \cdot {x}^{2}}{x}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\frac{3}{2} + \frac{-1}{8} \cdot {x}^{2}}{x}} \]

   4. lower-pow.f64 51.7

      \[\leadsto \frac{1}{\frac{1.5 + -0.125 \cdot {x}^{2}}{x}} \]

8. Applied rewrites 51.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5 + -0.125 \cdot {x}^{2}}{x}}} \]
9. Add Preprocessing

Alternative 11: 51.3% accurate, 10.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{1}{\frac{1}{\frac{x\_m}{1.5}}} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ 1.0 (/ 1.0 (/ x_m 1.5)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (1.0 / (1.0 / (x_m / 1.5)));
}

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 * (1.0d0 / (1.0d0 / (x_m / 1.5d0)))
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 * (1.0 / (1.0 / (x_m / 1.5)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (1.0 / (1.0 / (x_m / 1.5)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(1.0 / Float64(1.0 / Float64(x_m / 1.5))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (1.0 / (1.0 / (x_m / 1.5)));
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[(1.0 / N[(1.0 / N[(x$95$m / 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. sqr-sin-a N/A

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

   7. sub-flip N/A

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

   8. distribute-rgt-in N/A

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

   9. lower-+.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

3. Applied rewrites 52.3%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f64 52.3

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}} \]

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval 52.4

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 51.2

      \[\leadsto \frac{1}{\frac{1.5}{\color{blue}{x}}} \]

8. Applied rewrites 51.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{\frac{\frac{3}{2}}{\color{blue}{x}}} \]

   2. div-flip N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

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

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{\frac{3}{2}}}}} \]

   4. lower-unsound-/.f64 51.3

      \[\leadsto \frac{1}{\frac{1}{\frac{x}{\color{blue}{1.5}}}} \]

10. Applied rewrites 51.3%

    \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{1.5}}}} \]
11. Add Preprocessing

Alternative 12: 51.2% accurate, 14.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{1}{\frac{1.5}{x\_m}} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (/ 1.0 (/ 1.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (1.0 / (1.5 / 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 * (1.0d0 / (1.5d0 / 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 * (1.0 / (1.5 / x_m));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (1.0 / (1.5 / x_m))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(1.0 / Float64(1.5 / 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 * (1.0 / (1.5 / 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[(1.0 / N[(1.5 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. sqr-sin-a N/A

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

   7. sub-flip N/A

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

   8. distribute-rgt-in N/A

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

   9. lower-+.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

3. Applied rewrites 52.3%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f64 52.3

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin x}{1.3333333333333333 + \left(-0.5 \cdot \cos x\right) \cdot 2.6666666666666665}}} \]

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval 52.4

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 51.2

      \[\leadsto \frac{1}{\frac{1.5}{\color{blue}{x}}} \]

8. Applied rewrites 51.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{1.5}{x}}} \]
9. Add Preprocessing

Alternative 13: 51.1% accurate, 29.0× 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 77.2%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 51.1

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

4. Applied rewrites 51.1%

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

Reproduce

herbie shell --seed 2025159 
(FPCore (x)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
  :precision binary64
  (/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))