bug500 (missed optimization)

Percentage Accurate: 69.6% → 99.5%

Time: 8.3s

Alternatives: 6

Speedup: 14.7×

Specification

\[-1000 < x \land x < 1000\]

Math

\[\begin{array}{l} \\ \sin x - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (sin x) x))

C

double code(double x) {
	return sin(x) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sin(x) - x
end function

Java

public static double code(double x) {
	return Math.sin(x) - x;
}

Python

def code(x):
	return math.sin(x) - x

Julia

function code(x)
	return Float64(sin(x) - x)
end

MATLAB

function tmp = code(x)
	tmp = sin(x) - x;
end

Wolfram

code[x_] := N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]

Initial Program: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (sin x) x))

C

double code(double x) {
	return sin(x) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sin(x) - x
end function

Java

public static double code(double x) {
	return Math.sin(x) - x;
}

Python

def code(x):
	return math.sin(x) - x

Julia

function code(x)
	return Float64(sin(x) - x)
end

MATLAB

function tmp = code(x)
	tmp = sin(x) - x;
end

Wolfram

code[x_] := N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m - x\_m \leq -0.05:\\ \;\;\;\;\frac{\left(0.5 + -0.5 \cdot \cos \left(x\_m \cdot 2\right)\right) - x\_m \cdot x\_m}{x\_m + \sin x\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{x\_m \cdot \left(x\_m \cdot 0.027777777777777776\right)}{-0.16666666666666666 + \left(x\_m \cdot x\_m\right) \cdot -0.008333333333333333}\\ \end{array} \end{array} \]

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 (<= (- (sin x_m) x_m) -0.05)
    (/ (- (+ 0.5 (* -0.5 (cos (* x_m 2.0)))) (* x_m x_m)) (+ x_m (sin x_m)))
    (*
     x_m
     (/
      (* x_m (* x_m 0.027777777777777776))
      (+ -0.16666666666666666 (* (* x_m x_m) -0.008333333333333333)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if ((sin(x_m) - x_m) <= -0.05) {
		tmp = ((0.5 + (-0.5 * cos((x_m * 2.0)))) - (x_m * x_m)) / (x_m + sin(x_m));
	} else {
		tmp = x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if ((sin(x_m) - x_m) <= (-0.05d0)) then
        tmp = ((0.5d0 + ((-0.5d0) * cos((x_m * 2.0d0)))) - (x_m * x_m)) / (x_m + sin(x_m))
    else
        tmp = x_m * ((x_m * (x_m * 0.027777777777777776d0)) / ((-0.16666666666666666d0) + ((x_m * x_m) * (-0.008333333333333333d0))))
    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 ((Math.sin(x_m) - x_m) <= -0.05) {
		tmp = ((0.5 + (-0.5 * Math.cos((x_m * 2.0)))) - (x_m * x_m)) / (x_m + Math.sin(x_m));
	} else {
		tmp = x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333)));
	}
	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 (math.sin(x_m) - x_m) <= -0.05:
		tmp = ((0.5 + (-0.5 * math.cos((x_m * 2.0)))) - (x_m * x_m)) / (x_m + math.sin(x_m))
	else:
		tmp = x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333)))
	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 (Float64(sin(x_m) - x_m) <= -0.05)
		tmp = Float64(Float64(Float64(0.5 + Float64(-0.5 * cos(Float64(x_m * 2.0)))) - Float64(x_m * x_m)) / Float64(x_m + sin(x_m)));
	else
		tmp = Float64(x_m * Float64(Float64(x_m * Float64(x_m * 0.027777777777777776)) / Float64(-0.16666666666666666 + Float64(Float64(x_m * x_m) * -0.008333333333333333))));
	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 ((sin(x_m) - x_m) <= -0.05)
		tmp = ((0.5 + (-0.5 * cos((x_m * 2.0)))) - (x_m * x_m)) / (x_m + sin(x_m));
	else
		tmp = x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333)));
	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[N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision], -0.05], N[(N[(N[(0.5 + N[(-0.5 * N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$95$m + N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(x$95$m * N[(x$95$m * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(-0.16666666666666666 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -0.050000000000000003

   1. Initial program 100.0%

      \[\sin x - x \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sin x \cdot \sin x - x \cdot x\right), \color{blue}{\left(\sin x + x\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sin x \cdot \sin x\right), \left(x \cdot x\right)\right), \left(\color{blue}{\sin x} + x\right)\right) \]

      4. sqr-sin-a N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(x \cdot x\right)\right), \left(\sin \color{blue}{x} + x\right)\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot x\right)\right), \left(x \cdot x\right)\right), \left(\sin \color{blue}{x} + x\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot x\right)\right)\right), \left(x \cdot x\right)\right), \left(\sin \color{blue}{x} + x\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos \left(2 \cdot x\right)\right)\right), \left(x \cdot x\right)\right), \left(\sin x + x\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \cos \left(2 \cdot x\right)\right)\right), \left(x \cdot x\right)\right), \left(\sin x + x\right)\right) \]

      9. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \cos \left(x + x\right)\right)\right), \left(x \cdot x\right)\right), \left(\sin x + x\right)\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\left(x + x\right)\right)\right)\right), \left(x \cdot x\right)\right), \left(\sin x + x\right)\right) \]

      11. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot x\right)\right)\right)\right), \left(x \cdot x\right)\right), \left(\sin x + x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\left(x \cdot 2\right)\right)\right)\right), \left(x \cdot x\right)\right), \left(\sin x + x\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right)\right)\right), \left(x \cdot x\right)\right), \left(\sin x + x\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \left(\sin x + x\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \left(x + \color{blue}{\sin x}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\sin x}\right)\right) \]

      17. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(x, \mathsf{sin.f64}\left(x\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   if -0.050000000000000003 < (-.f64 (sin.f64 x) x)

   1. Initial program 73.5%

      \[\sin x - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{{x}^{2}} - \frac{1}{6}\right) \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 97.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(x \cdot x\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{6} + \frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{-1}{6} \cdot \frac{-1}{6} - \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right)}{\frac{-1}{6} - \frac{1}{120} \cdot \left(x \cdot x\right)} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)}{\color{blue}{\frac{-1}{6} - \frac{1}{120} \cdot \left(x \cdot x\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\frac{-1}{6} - \frac{1}{120} \cdot \left(x \cdot x\right)\right)}\right)\right) \]

   7. Applied egg-rr 97.3%

      \[\leadsto x \cdot \color{blue}{\frac{\left(0.027777777777777776 - 6.944444444444444 \cdot 10^{-5} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}{-0.16666666666666666 + -0.008333333333333333 \cdot \left(x \cdot x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{36} \cdot {x}^{2}\right)}, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left({x}^{2} \cdot \frac{1}{36}\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \frac{1}{36}\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{36}\right)\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{36}\right)\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 97.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{36}\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   10. Simplified 97.7%

       \[\leadsto x \cdot \frac{\color{blue}{x \cdot \left(x \cdot 0.027777777777777776\right)}}{-0.16666666666666666 + -0.008333333333333333 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.05:\\ \;\;\;\;\frac{\left(0.5 + -0.5 \cdot \cos \left(x \cdot 2\right)\right) - x \cdot x}{x + \sin x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x \cdot \left(x \cdot 0.027777777777777776\right)}{-0.16666666666666666 + \left(x \cdot x\right) \cdot -0.008333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{x\_m \cdot \left(x\_m \cdot 0.027777777777777776\right)}{-0.16666666666666666 + \left(x\_m \cdot x\_m\right) \cdot -0.008333333333333333}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (sin x_m) x_m)))
   (*
    x_s
    (if (<= t_0 -0.05)
      t_0
      (*
       x_m
       (/
        (* x_m (* x_m 0.027777777777777776))
        (+ -0.16666666666666666 (* (* x_m x_m) -0.008333333333333333))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0;
	} else {
		tmp = x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333)));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x_m) - x_m
    if (t_0 <= (-0.05d0)) then
        tmp = t_0
    else
        tmp = x_m * ((x_m * (x_m * 0.027777777777777776d0)) / ((-0.16666666666666666d0) + ((x_m * x_m) * (-0.008333333333333333d0))))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0;
	} else {
		tmp = x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333)));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin(x_m) - x_m
	tmp = 0
	if t_0 <= -0.05:
		tmp = t_0
	else:
		tmp = x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333)))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(sin(x_m) - x_m)
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = t_0;
	else
		tmp = Float64(x_m * Float64(Float64(x_m * Float64(x_m * 0.027777777777777776)) / Float64(-0.16666666666666666 + Float64(Float64(x_m * x_m) * -0.008333333333333333))));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin(x_m) - x_m;
	tmp = 0.0;
	if (t_0 <= -0.05)
		tmp = t_0;
	else
		tmp = x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333)));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -0.05], t$95$0, N[(x$95$m * N[(N[(x$95$m * N[(x$95$m * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(-0.16666666666666666 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -0.050000000000000003

   1. Initial program 100.0%

      \[\sin x - x \]
   2. Add Preprocessing

   if -0.050000000000000003 < (-.f64 (sin.f64 x) x)

   1. Initial program 73.5%

      \[\sin x - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{{x}^{2}} - \frac{1}{6}\right) \]

      3. associate-*r* N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 97.3%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(x \cdot x\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{6} + \frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{-1}{6} \cdot \frac{-1}{6} - \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right)}{\frac{-1}{6} - \frac{1}{120} \cdot \left(x \cdot x\right)} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)}{\color{blue}{\frac{-1}{6} - \frac{1}{120} \cdot \left(x \cdot x\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\frac{-1}{6} - \frac{1}{120} \cdot \left(x \cdot x\right)\right)}\right)\right) \]

   7. Applied egg-rr 97.3%

      \[\leadsto x \cdot \color{blue}{\frac{\left(0.027777777777777776 - 6.944444444444444 \cdot 10^{-5} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}{-0.16666666666666666 + -0.008333333333333333 \cdot \left(x \cdot x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{36} \cdot {x}^{2}\right)}, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left({x}^{2} \cdot \frac{1}{36}\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \frac{1}{36}\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{36}\right)\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{36}\right)\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 97.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{36}\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   10. Simplified 97.7%

       \[\leadsto x \cdot \frac{\color{blue}{x \cdot \left(x \cdot 0.027777777777777776\right)}}{-0.16666666666666666 + -0.008333333333333333 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.05:\\ \;\;\;\;\sin x - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x \cdot \left(x \cdot 0.027777777777777776\right)}{-0.16666666666666666 + \left(x \cdot x\right) \cdot -0.008333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 6.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \frac{x\_m \cdot \left(x\_m \cdot 0.027777777777777776\right)}{-0.16666666666666666 + \left(x\_m \cdot x\_m\right) \cdot -0.008333333333333333}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   x_m
   (/
    (* x_m (* x_m 0.027777777777777776))
    (+ -0.16666666666666666 (* (* x_m x_m) -0.008333333333333333))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333))));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (x_m * ((x_m * (x_m * 0.027777777777777776d0)) / ((-0.16666666666666666d0) + ((x_m * x_m) * (-0.008333333333333333d0)))))
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 * (x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m * Float64(Float64(x_m * Float64(x_m * 0.027777777777777776)) / Float64(-0.16666666666666666 + Float64(Float64(x_m * x_m) * -0.008333333333333333)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (x_m * ((x_m * (x_m * 0.027777777777777776)) / (-0.16666666666666666 + ((x_m * x_m) * -0.008333333333333333))));
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[(x$95$m * N[(N[(x$95$m * N[(x$95$m * 0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(-0.16666666666666666 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.6%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right) \]

   2. unpow2 N/A

      \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{{x}^{2}} - \frac{1}{6}\right) \]

   3. associate-*r* N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 96.9%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

5. Simplified 96.9%

   \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(x \cdot x\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{6} + \frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{-1}{6} \cdot \frac{-1}{6} - \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right)}{\frac{-1}{6} - \frac{1}{120} \cdot \left(x \cdot x\right)} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)}{\color{blue}{\frac{-1}{6} - \frac{1}{120} \cdot \left(x \cdot x\right)}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\frac{-1}{6} - \frac{1}{120} \cdot \left(x \cdot x\right)\right)}\right)\right) \]

7. Applied egg-rr 96.9%

   \[\leadsto x \cdot \color{blue}{\frac{\left(0.027777777777777776 - 6.944444444444444 \cdot 10^{-5} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \left(x \cdot x\right)}{-0.16666666666666666 + -0.008333333333333333 \cdot \left(x \cdot x\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{36} \cdot {x}^{2}\right)}, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left({x}^{2} \cdot \frac{1}{36}\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \frac{1}{36}\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{36}\right)\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{36}\right)\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 97.4%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{36}\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right) \]

10. Simplified 97.4%

    \[\leadsto x \cdot \frac{\color{blue}{x \cdot \left(x \cdot 0.027777777777777776\right)}}{-0.16666666666666666 + -0.008333333333333333 \cdot \left(x \cdot x\right)} \]

11. Final simplification 97.4%

    \[\leadsto x \cdot \frac{x \cdot \left(x \cdot 0.027777777777777776\right)}{-0.16666666666666666 + \left(x \cdot x\right) \cdot -0.008333333333333333} \]
12. Add Preprocessing

Alternative 4: 98.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot -0.16666666666666666\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* (* x_m x_m) (* x_m -0.16666666666666666))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m * x_m) * (x_m * -0.16666666666666666));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((x_m * x_m) * (x_m * (-0.16666666666666666d0)))
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 * ((x_m * x_m) * (x_m * -0.16666666666666666));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m * x_m) * (x_m * -0.16666666666666666))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * x_m) * Float64(x_m * -0.16666666666666666)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m * x_m) * (x_m * -0.16666666666666666));
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[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.6%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{3}\right)}\right) \]

   2. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   6. *-lowering-*.f64 96.9%

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

5. Simplified 96.9%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-1}{6} \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot x\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{-1}{6} \cdot x\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{-1}{6}} \cdot x\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{\frac{-1}{6}}\right)\right) \]

   6. *-lowering-*.f64 96.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right) \]

7. Applied egg-rr 96.9%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)} \]
8. Add Preprocessing

Alternative 5: 98.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(-0.16666666666666666 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* -0.16666666666666666 (* x_m (* x_m x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (-0.16666666666666666 * (x_m * (x_m * x_m)));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((-0.16666666666666666d0) * (x_m * (x_m * 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 * (-0.16666666666666666 * (x_m * (x_m * x_m)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (-0.16666666666666666 * (x_m * (x_m * x_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(-0.16666666666666666 * Float64(x_m * Float64(x_m * 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 * (-0.16666666666666666 * (x_m * (x_m * 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[(-0.16666666666666666 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.6%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{3}\right)}\right) \]

   2. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   6. *-lowering-*.f64 96.9%

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

5. Simplified 96.9%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
6. Add Preprocessing

Alternative 6: 67.2% accurate, 103.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 0 \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s 0.0))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 0.0;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * 0.0d0
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 * 0.0;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 0.0

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 0.0)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 0.0;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 73.6%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, x\right) \]
4. Step-by-step derivation

5. Simplified 70.0%

   \[\leadsto \color{blue}{x} - x \]
6. Step-by-step derivation

   1. +-inverses 70.0%

      \[\leadsto 0 \]

7. Applied egg-rr 70.0%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| < 0.07:\\ \;\;\;\;-\left(\left(\frac{{x}^{3}}{6} - \frac{{x}^{5}}{120}\right) + \frac{{x}^{7}}{5040}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (< (fabs x) 0.07)
   (- (+ (- (/ (pow x 3.0) 6.0) (/ (pow x 5.0) 120.0)) (/ (pow x 7.0) 5040.0)))
   (- (sin x) x)))

C

double code(double x) {
	double tmp;
	if (fabs(x) < 0.07) {
		tmp = -(((pow(x, 3.0) / 6.0) - (pow(x, 5.0) / 120.0)) + (pow(x, 7.0) / 5040.0));
	} else {
		tmp = sin(x) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) < 0.07d0) then
        tmp = -((((x ** 3.0d0) / 6.0d0) - ((x ** 5.0d0) / 120.0d0)) + ((x ** 7.0d0) / 5040.0d0))
    else
        tmp = sin(x) - x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) < 0.07) {
		tmp = -(((Math.pow(x, 3.0) / 6.0) - (Math.pow(x, 5.0) / 120.0)) + (Math.pow(x, 7.0) / 5040.0));
	} else {
		tmp = Math.sin(x) - x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) < 0.07:
		tmp = -(((math.pow(x, 3.0) / 6.0) - (math.pow(x, 5.0) / 120.0)) + (math.pow(x, 7.0) / 5040.0))
	else:
		tmp = math.sin(x) - x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) < 0.07)
		tmp = Float64(-Float64(Float64(Float64((x ^ 3.0) / 6.0) - Float64((x ^ 5.0) / 120.0)) + Float64((x ^ 7.0) / 5040.0)));
	else
		tmp = Float64(sin(x) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) < 0.07)
		tmp = -((((x ^ 3.0) / 6.0) - ((x ^ 5.0) / 120.0)) + ((x ^ 7.0) / 5040.0));
	else
		tmp = sin(x) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Less[N[Abs[x], $MachinePrecision], 0.07], (-N[(N[(N[(N[Power[x, 3.0], $MachinePrecision] / 6.0), $MachinePrecision] - N[(N[Power[x, 5.0], $MachinePrecision] / 120.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 7.0], $MachinePrecision] / 5040.0), $MachinePrecision]), $MachinePrecision]), N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]]

Reproduce

herbie shell --seed 2024138 
(FPCore (x)
  :name "bug500 (missed optimization)"
  :precision binary64
  :pre (and (< -1000.0 x) (< x 1000.0))

  :alt
  (! :herbie-platform default (if (< (fabs x) 7/100) (- (+ (- (/ (pow x 3) 6) (/ (pow x 5) 120)) (/ (pow x 7) 5040))) (- (sin x) x)))

  (- (sin x) x))