bug500 (missed optimization)

Percentage Accurate: 69.7% → 99.7%

Time: 11.9s

Alternatives: 11

Speedup: 6.5×

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: 69.7% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \cos \left(x\_m + x\_m\right)\\ t_1 := \mathsf{fma}\left(x\_m, x\_m \cdot -0.0001984126984126984, 0.008333333333333333\right)\\ t_2 := x\_m \cdot t\_1\\ t_3 := \mathsf{fma}\left(x\_m, x\_m + \sin x\_m, 0.5\right)\\ t_4 := 0.5 \cdot t\_0\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m - x\_m \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\sin x\_m, \frac{0.5 + -0.5 \cdot t\_0}{t\_3 - t\_4}, \frac{x\_m \cdot \left(x\_m \cdot x\_m\right)}{t\_4 - t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{\mathsf{fma}\left(x\_m, x\_m, 0\right) \cdot \mathsf{fma}\left(x\_m \cdot t\_2, t\_1 \cdot \left(t\_1 \cdot \left(x\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(x\_m, x\_m, 0\right)\right)\right)\right), -0.004629629629629629\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x\_m, x\_m, 0\right), t\_2 \cdot t\_2, 0.027777777777777776\right) - \mathsf{fma}\left(x\_m, x\_m, 0\right) \cdot \left(t\_1 \cdot -0.16666666666666666\right)}, 0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (cos (+ x_m x_m)))
        (t_1 (fma x_m (* x_m -0.0001984126984126984) 0.008333333333333333))
        (t_2 (* x_m t_1))
        (t_3 (fma x_m (+ x_m (sin x_m)) 0.5))
        (t_4 (* 0.5 t_0)))
   (*
    x_s
    (if (<= (- (sin x_m) x_m) -0.02)
      (fma
       (sin x_m)
       (/ (+ 0.5 (* -0.5 t_0)) (- t_3 t_4))
       (/ (* x_m (* x_m x_m)) (- t_4 t_3)))
      (fma
       x_m
       (/
        (*
         (fma x_m x_m 0.0)
         (fma
          (* x_m t_2)
          (* t_1 (* t_1 (* x_m (* x_m (fma x_m x_m 0.0)))))
          -0.004629629629629629))
        (-
         (fma (fma x_m x_m 0.0) (* t_2 t_2) 0.027777777777777776)
         (* (fma x_m x_m 0.0) (* t_1 -0.16666666666666666))))
       0.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = cos((x_m + x_m));
	double t_1 = fma(x_m, (x_m * -0.0001984126984126984), 0.008333333333333333);
	double t_2 = x_m * t_1;
	double t_3 = fma(x_m, (x_m + sin(x_m)), 0.5);
	double t_4 = 0.5 * t_0;
	double tmp;
	if ((sin(x_m) - x_m) <= -0.02) {
		tmp = fma(sin(x_m), ((0.5 + (-0.5 * t_0)) / (t_3 - t_4)), ((x_m * (x_m * x_m)) / (t_4 - t_3)));
	} else {
		tmp = fma(x_m, ((fma(x_m, x_m, 0.0) * fma((x_m * t_2), (t_1 * (t_1 * (x_m * (x_m * fma(x_m, x_m, 0.0))))), -0.004629629629629629)) / (fma(fma(x_m, x_m, 0.0), (t_2 * t_2), 0.027777777777777776) - (fma(x_m, x_m, 0.0) * (t_1 * -0.16666666666666666)))), 0.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = cos(Float64(x_m + x_m))
	t_1 = fma(x_m, Float64(x_m * -0.0001984126984126984), 0.008333333333333333)
	t_2 = Float64(x_m * t_1)
	t_3 = fma(x_m, Float64(x_m + sin(x_m)), 0.5)
	t_4 = Float64(0.5 * t_0)
	tmp = 0.0
	if (Float64(sin(x_m) - x_m) <= -0.02)
		tmp = fma(sin(x_m), Float64(Float64(0.5 + Float64(-0.5 * t_0)) / Float64(t_3 - t_4)), Float64(Float64(x_m * Float64(x_m * x_m)) / Float64(t_4 - t_3)));
	else
		tmp = fma(x_m, Float64(Float64(fma(x_m, x_m, 0.0) * fma(Float64(x_m * t_2), Float64(t_1 * Float64(t_1 * Float64(x_m * Float64(x_m * fma(x_m, x_m, 0.0))))), -0.004629629629629629)) / Float64(fma(fma(x_m, x_m, 0.0), Float64(t_2 * t_2), 0.027777777777777776) - Float64(fma(x_m, x_m, 0.0) * Float64(t_1 * -0.16666666666666666)))), 0.0);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * N[(x$95$m * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, Block[{t$95$2 = N[(x$95$m * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x$95$m * N[(x$95$m + N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * t$95$0), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision], -0.02], N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[(0.5 + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 - t$95$4), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(N[(x$95$m * x$95$m + 0.0), $MachinePrecision] * N[(N[(x$95$m * t$95$2), $MachinePrecision] * N[(t$95$1 * N[(t$95$1 * N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.004629629629629629), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x$95$m * x$95$m + 0.0), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision] + 0.027777777777777776), $MachinePrecision] - N[(N[(x$95$m * x$95$m + 0.0), $MachinePrecision] * N[(t$95$1 * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -0.0200000000000000004

   1. Initial program 97.9%

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

      1. flip3-- N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   if -0.0200000000000000004 < (-.f64 (sin.f64 x) x)

   1. Initial program 70.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]

      4. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + 0} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right), 0\right)} \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)\right), 0\right)} \]

   7. Applied egg-rr 98.8%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right)\right)\right), -0.004629629629629629\right) \cdot \mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), 0.027777777777777776\right) - \mathsf{fma}\left(x, x, 0\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot -0.16666666666666666\right)}}, 0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \frac{0.5 + -0.5 \cdot \cos \left(x + x\right)}{\mathsf{fma}\left(x, x + \sin x, 0.5\right) - 0.5 \cdot \cos \left(x + x\right)}, \frac{x \cdot \left(x \cdot x\right)}{0.5 \cdot \cos \left(x + x\right) - \mathsf{fma}\left(x, x + \sin x, 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right)\right)\right), -0.004629629629629629\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), 0.027777777777777776\right) - \mathsf{fma}\left(x, x, 0\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot -0.16666666666666666\right)}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m, x\_m \cdot -0.0001984126984126984, 0.008333333333333333\right)\\ t_1 := x\_m \cdot t\_0\\ t_2 := \cos \left(x\_m + x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m - x\_m \leq -0.02:\\ \;\;\;\;\frac{0.5 - \mathsf{fma}\left(0.5, t\_2, x\_m \cdot x\_m\right)}{\mathsf{fma}\left(x\_m, x\_m \cdot x\_m, {\sin x\_m}^{3}\right)} \cdot \mathsf{fma}\left(1 - t\_2, 0.5, x\_m \cdot \left(x\_m - \sin x\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{\mathsf{fma}\left(x\_m, x\_m, 0\right) \cdot \mathsf{fma}\left(x\_m \cdot t\_1, t\_0 \cdot \left(t\_0 \cdot \left(x\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(x\_m, x\_m, 0\right)\right)\right)\right), -0.004629629629629629\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x\_m, x\_m, 0\right), t\_1 \cdot t\_1, 0.027777777777777776\right) - \mathsf{fma}\left(x\_m, x\_m, 0\right) \cdot \left(t\_0 \cdot -0.16666666666666666\right)}, 0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (fma x_m (* x_m -0.0001984126984126984) 0.008333333333333333))
        (t_1 (* x_m t_0))
        (t_2 (cos (+ x_m x_m))))
   (*
    x_s
    (if (<= (- (sin x_m) x_m) -0.02)
      (*
       (/
        (- 0.5 (fma 0.5 t_2 (* x_m x_m)))
        (fma x_m (* x_m x_m) (pow (sin x_m) 3.0)))
       (fma (- 1.0 t_2) 0.5 (* x_m (- x_m (sin x_m)))))
      (fma
       x_m
       (/
        (*
         (fma x_m x_m 0.0)
         (fma
          (* x_m t_1)
          (* t_0 (* t_0 (* x_m (* x_m (fma x_m x_m 0.0)))))
          -0.004629629629629629))
        (-
         (fma (fma x_m x_m 0.0) (* t_1 t_1) 0.027777777777777776)
         (* (fma x_m x_m 0.0) (* t_0 -0.16666666666666666))))
       0.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = fma(x_m, (x_m * -0.0001984126984126984), 0.008333333333333333);
	double t_1 = x_m * t_0;
	double t_2 = cos((x_m + x_m));
	double tmp;
	if ((sin(x_m) - x_m) <= -0.02) {
		tmp = ((0.5 - fma(0.5, t_2, (x_m * x_m))) / fma(x_m, (x_m * x_m), pow(sin(x_m), 3.0))) * fma((1.0 - t_2), 0.5, (x_m * (x_m - sin(x_m))));
	} else {
		tmp = fma(x_m, ((fma(x_m, x_m, 0.0) * fma((x_m * t_1), (t_0 * (t_0 * (x_m * (x_m * fma(x_m, x_m, 0.0))))), -0.004629629629629629)) / (fma(fma(x_m, x_m, 0.0), (t_1 * t_1), 0.027777777777777776) - (fma(x_m, x_m, 0.0) * (t_0 * -0.16666666666666666)))), 0.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = fma(x_m, Float64(x_m * -0.0001984126984126984), 0.008333333333333333)
	t_1 = Float64(x_m * t_0)
	t_2 = cos(Float64(x_m + x_m))
	tmp = 0.0
	if (Float64(sin(x_m) - x_m) <= -0.02)
		tmp = Float64(Float64(Float64(0.5 - fma(0.5, t_2, Float64(x_m * x_m))) / fma(x_m, Float64(x_m * x_m), (sin(x_m) ^ 3.0))) * fma(Float64(1.0 - t_2), 0.5, Float64(x_m * Float64(x_m - sin(x_m)))));
	else
		tmp = fma(x_m, Float64(Float64(fma(x_m, x_m, 0.0) * fma(Float64(x_m * t_1), Float64(t_0 * Float64(t_0 * Float64(x_m * Float64(x_m * fma(x_m, x_m, 0.0))))), -0.004629629629629629)) / Float64(fma(fma(x_m, x_m, 0.0), Float64(t_1 * t_1), 0.027777777777777776) - Float64(fma(x_m, x_m, 0.0) * Float64(t_0 * -0.16666666666666666)))), 0.0);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision], -0.02], N[(N[(N[(0.5 - N[(0.5 * t$95$2 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision] + N[Power[N[Sin[x$95$m], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - t$95$2), $MachinePrecision] * 0.5 + N[(x$95$m * N[(x$95$m - N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(N[(x$95$m * x$95$m + 0.0), $MachinePrecision] * N[(N[(x$95$m * t$95$1), $MachinePrecision] * N[(t$95$0 * N[(t$95$0 * N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.004629629629629629), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x$95$m * x$95$m + 0.0), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision] + 0.027777777777777776), $MachinePrecision] - N[(N[(x$95$m * x$95$m + 0.0), $MachinePrecision] * N[(t$95$0 * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -0.0200000000000000004

   1. Initial program 97.9%

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

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{0.5 - \mathsf{fma}\left(0.5, \cos \left(x + x\right), x \cdot x\right)}{\mathsf{fma}\left(x, x \cdot x, {\sin x}^{3}\right)} \cdot \mathsf{fma}\left(1 - \cos \left(x + x\right), 0.5, x \cdot \left(x - \sin x\right)\right)} \]

   if -0.0200000000000000004 < (-.f64 (sin.f64 x) x)

   1. Initial program 70.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]

      4. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + 0} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right), 0\right)} \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)\right), 0\right)} \]

   7. Applied egg-rr 98.8%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right)\right)\right), -0.004629629629629629\right) \cdot \mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), 0.027777777777777776\right) - \mathsf{fma}\left(x, x, 0\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot -0.16666666666666666\right)}}, 0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.02:\\ \;\;\;\;\frac{0.5 - \mathsf{fma}\left(0.5, \cos \left(x + x\right), x \cdot x\right)}{\mathsf{fma}\left(x, x \cdot x, {\sin x}^{3}\right)} \cdot \mathsf{fma}\left(1 - \cos \left(x + x\right), 0.5, x \cdot \left(x - \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right)\right)\right), -0.004629629629629629\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), 0.027777777777777776\right) - \mathsf{fma}\left(x, x, 0\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot -0.16666666666666666\right)}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m, x\_m \cdot -0.0001984126984126984, 0.008333333333333333\right)\\ t_1 := x\_m \cdot t\_0\\ t_2 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.02:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{\mathsf{fma}\left(x\_m, x\_m, 0\right) \cdot \mathsf{fma}\left(x\_m \cdot t\_1, t\_0 \cdot \left(t\_0 \cdot \left(x\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(x\_m, x\_m, 0\right)\right)\right)\right), -0.004629629629629629\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x\_m, x\_m, 0\right), t\_1 \cdot t\_1, 0.027777777777777776\right) - \mathsf{fma}\left(x\_m, x\_m, 0\right) \cdot \left(t\_0 \cdot -0.16666666666666666\right)}, 0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (fma x_m (* x_m -0.0001984126984126984) 0.008333333333333333))
        (t_1 (* x_m t_0))
        (t_2 (- (sin x_m) x_m)))
   (*
    x_s
    (if (<= t_2 -0.02)
      t_2
      (fma
       x_m
       (/
        (*
         (fma x_m x_m 0.0)
         (fma
          (* x_m t_1)
          (* t_0 (* t_0 (* x_m (* x_m (fma x_m x_m 0.0)))))
          -0.004629629629629629))
        (-
         (fma (fma x_m x_m 0.0) (* t_1 t_1) 0.027777777777777776)
         (* (fma x_m x_m 0.0) (* t_0 -0.16666666666666666))))
       0.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = fma(x_m, (x_m * -0.0001984126984126984), 0.008333333333333333);
	double t_1 = x_m * t_0;
	double t_2 = sin(x_m) - x_m;
	double tmp;
	if (t_2 <= -0.02) {
		tmp = t_2;
	} else {
		tmp = fma(x_m, ((fma(x_m, x_m, 0.0) * fma((x_m * t_1), (t_0 * (t_0 * (x_m * (x_m * fma(x_m, x_m, 0.0))))), -0.004629629629629629)) / (fma(fma(x_m, x_m, 0.0), (t_1 * t_1), 0.027777777777777776) - (fma(x_m, x_m, 0.0) * (t_0 * -0.16666666666666666)))), 0.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = fma(x_m, Float64(x_m * -0.0001984126984126984), 0.008333333333333333)
	t_1 = Float64(x_m * t_0)
	t_2 = Float64(sin(x_m) - x_m)
	tmp = 0.0
	if (t_2 <= -0.02)
		tmp = t_2;
	else
		tmp = fma(x_m, Float64(Float64(fma(x_m, x_m, 0.0) * fma(Float64(x_m * t_1), Float64(t_0 * Float64(t_0 * Float64(x_m * Float64(x_m * fma(x_m, x_m, 0.0))))), -0.004629629629629629)) / Float64(fma(fma(x_m, x_m, 0.0), Float64(t_1 * t_1), 0.027777777777777776) - Float64(fma(x_m, x_m, 0.0) * Float64(t_0 * -0.16666666666666666)))), 0.0);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$2, -0.02], t$95$2, N[(x$95$m * N[(N[(N[(x$95$m * x$95$m + 0.0), $MachinePrecision] * N[(N[(x$95$m * t$95$1), $MachinePrecision] * N[(t$95$0 * N[(t$95$0 * N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.004629629629629629), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x$95$m * x$95$m + 0.0), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision] + 0.027777777777777776), $MachinePrecision] - N[(N[(x$95$m * x$95$m + 0.0), $MachinePrecision] * N[(t$95$0 * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -0.0200000000000000004

   1. Initial program 97.9%

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

   if -0.0200000000000000004 < (-.f64 (sin.f64 x) x)

   1. Initial program 70.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]

      4. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + 0} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right), 0\right)} \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)\right), 0\right)} \]

   7. Applied egg-rr 98.8%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right)\right)\right), -0.004629629629629629\right) \cdot \mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), 0.027777777777777776\right) - \mathsf{fma}\left(x, x, 0\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot -0.16666666666666666\right)}}, 0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.02:\\ \;\;\;\;\sin x - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right)\right)\right), -0.004629629629629629\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), 0.027777777777777776\right) - \mathsf{fma}\left(x, x, 0\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot -0.16666666666666666\right)}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \mathsf{fma}\left(x\_m, x\_m \cdot 0.008333333333333333, -0.16666666666666666\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (sin x_m) x_m)))
   (*
    x_s
    (if (<= t_0 -0.02)
      t_0
      (*
       (* x_m (* x_m x_m))
       (fma x_m (* x_m 0.008333333333333333) -0.16666666666666666))))))

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 <= -0.02) {
		tmp = t_0;
	} else {
		tmp = (x_m * (x_m * x_m)) * fma(x_m, (x_m * 0.008333333333333333), -0.16666666666666666);
	}
	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 <= -0.02)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_m * Float64(x_m * x_m)) * fma(x_m, Float64(x_m * 0.008333333333333333), -0.16666666666666666));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -0.02], t$95$0, N[(N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -0.0200000000000000004

   1. Initial program 97.9%

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

   if -0.0200000000000000004 < (-.f64 (sin.f64 x) x)

   1. Initial program 70.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

      2. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]

      4. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + 0} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right), 0\right)} \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)\right), 0\right)} \]

   6. Taylor expanded in x around 0

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

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

         \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \]

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 98.6

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right) \]

   8. Simplified 98.6%

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right) \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m, x\_m \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)\right) \end{array} \]

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))
   (fma
    (* x_m x_m)
    (fma x_m (* x_m -0.0001984126984126984) 0.008333333333333333)
    -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)) * fma((x_m * x_m), fma(x_m, (x_m * -0.0001984126984126984), 0.008333333333333333), -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 * Float64(x_m * x_m)) * fma(Float64(x_m * x_m), fma(x_m, Float64(x_m * -0.0001984126984126984), 0.008333333333333333), -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 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

   2. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]

   4. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + 0} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right), 0\right)} \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)\right), 0\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
7. Step-by-step derivation

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

      \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]

   2. cube-mult N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

   3. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

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

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

   5. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

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

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

   7. sub-neg N/A

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{6}}\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}, \frac{-1}{6}\right)} \]

   10. unpow2 N/A

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}, \frac{-1}{6}\right) \]

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

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}, \frac{-1}{6}\right) \]

   12. +-commutative N/A

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{5040} \cdot {x}^{2} + \frac{1}{120}}, \frac{-1}{6}\right) \]

   13. unpow2 N/A

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{5040} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{120}, \frac{-1}{6}\right) \]

   14. associate-*r* N/A

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{-1}{5040} \cdot x\right) \cdot x} + \frac{1}{120}, \frac{-1}{6}\right) \]

   15. *-commutative N/A

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{-1}{5040} \cdot x\right)} + \frac{1}{120}, \frac{-1}{6}\right) \]

   16. accelerator-lowering-fma.f64 N/A

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{-1}{5040} \cdot x, \frac{1}{120}\right)}, \frac{-1}{6}\right) \]

   17. *-commutative N/A

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{5040}}, \frac{1}{120}\right), \frac{-1}{6}\right) \]

   18. *-lowering-*.f64 98.1

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot -0.0001984126984126984}, 0.008333333333333333\right), -0.16666666666666666\right) \]

8. Simplified 98.1%

   \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)} \]
9. Add Preprocessing

Alternative 6: 98.5% accurate, 3.9× speedup

Math

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

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))
   (fma x_m (* x_m 0.008333333333333333) -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)) * fma(x_m, (x_m * 0.008333333333333333), -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 * Float64(x_m * x_m)) * fma(x_m, Float64(x_m * 0.008333333333333333), -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 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

   2. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]

   4. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + 0} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right), 0\right)} \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)\right), 0\right)} \]

6. Taylor expanded in x around 0

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

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

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)} \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \]

   11. metadata-eval N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. *-lowering-*.f64 97.8

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right) \]

8. Simplified 97.8%

   \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right)} \]
9. Add Preprocessing

Alternative 7: 98.5% accurate, 3.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   x_m
   (*
    x_m
    (* x_m (fma x_m (* x_m 0.008333333333333333) -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 * fma(x_m, (x_m * 0.008333333333333333), -0.16666666666666666))));
}

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(x_m * Float64(x_m * fma(x_m, Float64(x_m * 0.008333333333333333), -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[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right) \]

   2. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right) \cdot {x}^{2}\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(\left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right) \cdot {x}^{2}\right)} \]

   6. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)} \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right) \]

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

      \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) + \color{blue}{\frac{-1}{6}}\right)\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right), \frac{-1}{6}\right)}\right) \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 2.7557319223985893 \cdot 10^{-6}, -0.0001984126984126984\right), 0.008333333333333333\right), -0.16666666666666666\right)\right)} \]

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right) \]

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-lowering-*.f64 97.8

       \[\leadsto x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right)\right)\right) \]

8. Simplified 97.8%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right)\right)\right)} \]
9. Add Preprocessing

Alternative 8: 98.1% accurate, 6.5× speedup

Math

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

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 * Float64(x_m * 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 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

   2. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]

   4. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + 0} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right), 0\right)} \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)\right), 0\right)} \]

7. Applied egg-rr 98.1%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right)\right)\right), -0.004629629629629629\right) \cdot \mathsf{fma}\left(x, x, 0\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right)\right), 0.027777777777777776\right) - \mathsf{fma}\left(x, x, 0\right) \cdot \left(\mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right) \cdot -0.16666666666666666\right)}}, 0\right) \]

8. Taylor expanded in x around 0

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

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

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 97.6

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

10. Simplified 97.6%

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

11. Final simplification 97.6%

    \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot -0.16666666666666666 \]
12. Add Preprocessing

Alternative 9: 67.4% accurate, 26.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (- 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 * (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 * (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 * (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 * (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(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 * (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[(x$95$m - x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 68.0%

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

Alternative 10: 6.4% accurate, 26.0× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (0.0 - 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 * (0.0d0 - 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 * (0.0 - x_m);
}

Python

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(0.0 - 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 * (0.0 - 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[(0.0 - x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - x} \]

   3. --lowering--.f64 6.5

      \[\leadsto \color{blue}{0 - x} \]

5. Simplified 6.5%

   \[\leadsto \color{blue}{0 - x} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. neg-lowering-neg.f64 6.5

      \[\leadsto \color{blue}{-x} \]

7. Applied egg-rr 6.5%

   \[\leadsto \color{blue}{-x} \]

8. Final simplification 6.5%

   \[\leadsto 0 - x \]
9. Add Preprocessing

Alternative 11: 5.0% accurate, 104.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - x} \]

   3. --lowering--.f64 6.5

      \[\leadsto \color{blue}{0 - x} \]

5. Simplified 6.5%

   \[\leadsto \color{blue}{0 - x} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. neg-lowering-neg.f64 6.5

      \[\leadsto \color{blue}{-x} \]

7. Applied egg-rr 6.5%

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

   1. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - x} \]

   2. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   4. sub0-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({x}^{3}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   5. cube-neg N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   6. neg-sub0 N/A

      \[\leadsto \frac{{\color{blue}{\left(0 - x\right)}}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   7. sqr-pow N/A

      \[\leadsto \frac{\color{blue}{{\left(0 - x\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - x\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   8. unpow-prod-down N/A

      \[\leadsto \frac{\color{blue}{{\left(\left(0 - x\right) \cdot \left(0 - x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   9. neg-sub0 N/A

      \[\leadsto \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(0 - x\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   10. neg-sub0 N/A

       \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   11. sqr-neg N/A

       \[\leadsto \frac{{\color{blue}{\left(x \cdot x\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   12. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   13. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{x}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{{x}^{3}}{\color{blue}{0} + \left(x \cdot x + 0 \cdot x\right)} \]

   15. +-lft-identity N/A

       \[\leadsto \frac{{x}^{3}}{\color{blue}{x \cdot x + 0 \cdot x}} \]

   16. mul0-lft N/A

       \[\leadsto \frac{{x}^{3}}{x \cdot x + \color{blue}{0}} \]

   17. +-rgt-identity N/A

       \[\leadsto \frac{{x}^{3}}{\color{blue}{x \cdot x}} \]

   18. pow2 N/A

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

   19. pow-div N/A

       \[\leadsto \color{blue}{{x}^{\left(3 - 2\right)}} \]

   20. metadata-eval N/A

       \[\leadsto {x}^{\color{blue}{1}} \]

   21. unpow1 5.0

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

9. Applied egg-rr 5.0%

   \[\leadsto \color{blue}{x} \]
10. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.3× 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 2024195 
(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))