bug500 (missed optimization)

Percentage Accurate: 69.1% → 99.6%

Time: 6.5s

Alternatives: 14

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: 69.1% 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: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := {\sin x\_m}^{2}\\ t_1 := \sin x\_m + x\_m\\ t_2 := t\_1 \cdot x\_m\\ t_3 := {t\_2}^{3} + {\sin x\_m}^{6}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m - x\_m \leq -0.001:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{{\sin x\_m}^{3}}{t\_3} \cdot \left(\left(t\_2 - t\_0\right) \cdot x\_m\right), \frac{{\sin x\_m}^{7}}{t\_3} - \frac{{x\_m}^{3}}{\mathsf{fma}\left(t\_1, x\_m, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.027777777777777776 \cdot {x\_m}^{3}}{\mathsf{fma}\left(0.008333333333333333, x\_m \cdot x\_m, 0.16666666666666666\right)}\\ \end{array} \end{array} \end{array} \]

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 (pow (sin x_m) 2.0))
        (t_1 (+ (sin x_m) x_m))
        (t_2 (* t_1 x_m))
        (t_3 (+ (pow t_2 3.0) (pow (sin x_m) 6.0))))
   (*
    x_s
    (if (<= (- (sin x_m) x_m) -0.001)
      (fma
       t_1
       (* (/ (pow (sin x_m) 3.0) t_3) (* (- t_2 t_0) x_m))
       (- (/ (pow (sin x_m) 7.0) t_3) (/ (pow x_m 3.0) (fma t_1 x_m t_0))))
      (/
       (* -0.027777777777777776 (pow x_m 3.0))
       (fma 0.008333333333333333 (* 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) {
	double t_0 = pow(sin(x_m), 2.0);
	double t_1 = sin(x_m) + x_m;
	double t_2 = t_1 * x_m;
	double t_3 = pow(t_2, 3.0) + pow(sin(x_m), 6.0);
	double tmp;
	if ((sin(x_m) - x_m) <= -0.001) {
		tmp = fma(t_1, ((pow(sin(x_m), 3.0) / t_3) * ((t_2 - t_0) * x_m)), ((pow(sin(x_m), 7.0) / t_3) - (pow(x_m, 3.0) / fma(t_1, x_m, t_0))));
	} else {
		tmp = (-0.027777777777777776 * pow(x_m, 3.0)) / fma(0.008333333333333333, (x_m * x_m), 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 = sin(x_m) ^ 2.0
	t_1 = Float64(sin(x_m) + x_m)
	t_2 = Float64(t_1 * x_m)
	t_3 = Float64((t_2 ^ 3.0) + (sin(x_m) ^ 6.0))
	tmp = 0.0
	if (Float64(sin(x_m) - x_m) <= -0.001)
		tmp = fma(t_1, Float64(Float64((sin(x_m) ^ 3.0) / t_3) * Float64(Float64(t_2 - t_0) * x_m)), Float64(Float64((sin(x_m) ^ 7.0) / t_3) - Float64((x_m ^ 3.0) / fma(t_1, x_m, t_0))));
	else
		tmp = Float64(Float64(-0.027777777777777776 * (x_m ^ 3.0)) / fma(0.008333333333333333, Float64(x_m * x_m), 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[Power[N[Sin[x$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x$95$m], $MachinePrecision] + x$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * x$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$2, 3.0], $MachinePrecision] + N[Power[N[Sin[x$95$m], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision], -0.001], N[(t$95$1 * N[(N[(N[Power[N[Sin[x$95$m], $MachinePrecision], 3.0], $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[(t$95$2 - t$95$0), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Sin[x$95$m], $MachinePrecision], 7.0], $MachinePrecision] / t$95$3), $MachinePrecision] - N[(N[Power[x$95$m, 3.0], $MachinePrecision] / N[(t$95$1 * x$95$m + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.027777777777777776 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(0.008333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -1e-3

   1. Initial program 98.8%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. flip3-+ N/A

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

      6. associate-/r/ N/A

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x + x, \left(\left(\left(\sin x + x\right) \cdot x - {\sin x}^{2}\right) \cdot x\right) \cdot \frac{{\sin x}^{3}}{{\sin x}^{6} + {\left(\left(\sin x + x\right) \cdot x\right)}^{3}}, \frac{{\sin x}^{7}}{{\sin x}^{6} + {\left(\left(\sin x + x\right) \cdot x\right)}^{3}} - \frac{{x}^{3}}{\mathsf{fma}\left(\sin x + x, x, {\sin x}^{2}\right)}\right)} \]

   if -1e-3 < (-.f64 (sin.f64 x) x)

   1. Initial program 66.0%

      \[\sin x - x \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 98.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot {x}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

      \[\leadsto \frac{\mathsf{fma}\left(6.944444444444444 \cdot 10^{-5}, {x}^{4}, -0.027777777777777776\right) \cdot {x}^{3}}{\color{blue}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{-1}{36} \cdot {x}^{3}}{\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{1}{6}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.001:\\ \;\;\;\;\mathsf{fma}\left(\sin x + x, \frac{{\sin x}^{3}}{{\left(\left(\sin x + x\right) \cdot x\right)}^{3} + {\sin x}^{6}} \cdot \left(\left(\left(\sin x + x\right) \cdot x - {\sin x}^{2}\right) \cdot x\right), \frac{{\sin x}^{7}}{{\left(\left(\sin x + x\right) \cdot x\right)}^{3} + {\sin x}^{6}} - \frac{{x}^{3}}{\mathsf{fma}\left(\sin x + x, x, {\sin x}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.027777777777777776 \cdot {x}^{3}}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m - \sin x\_m\\ t_1 := {\sin x\_m}^{2}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m - x\_m \leq -0.001:\\ \;\;\;\;\frac{t\_1 - x\_m \cdot x\_m}{\frac{x\_m \cdot x\_m}{t\_0} - \frac{t\_1}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.027777777777777776 \cdot {x\_m}^{3}}{\mathsf{fma}\left(0.008333333333333333, x\_m \cdot x\_m, 0.16666666666666666\right)}\\ \end{array} \end{array} \end{array} \]

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 (- x_m (sin x_m))) (t_1 (pow (sin x_m) 2.0)))
   (*
    x_s
    (if (<= (- (sin x_m) x_m) -0.001)
      (/ (- t_1 (* x_m x_m)) (- (/ (* x_m x_m) t_0) (/ t_1 t_0)))
      (/
       (* -0.027777777777777776 (pow x_m 3.0))
       (fma 0.008333333333333333 (* 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) {
	double t_0 = x_m - sin(x_m);
	double t_1 = pow(sin(x_m), 2.0);
	double tmp;
	if ((sin(x_m) - x_m) <= -0.001) {
		tmp = (t_1 - (x_m * x_m)) / (((x_m * x_m) / t_0) - (t_1 / t_0));
	} else {
		tmp = (-0.027777777777777776 * pow(x_m, 3.0)) / fma(0.008333333333333333, (x_m * x_m), 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(x_m - sin(x_m))
	t_1 = sin(x_m) ^ 2.0
	tmp = 0.0
	if (Float64(sin(x_m) - x_m) <= -0.001)
		tmp = Float64(Float64(t_1 - Float64(x_m * x_m)) / Float64(Float64(Float64(x_m * x_m) / t_0) - Float64(t_1 / t_0)));
	else
		tmp = Float64(Float64(-0.027777777777777776 * (x_m ^ 3.0)) / fma(0.008333333333333333, Float64(x_m * x_m), 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[(x$95$m - N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision], -0.001], N[(N[(t$95$1 - N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.027777777777777776 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(0.008333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -1e-3

   1. Initial program 98.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. lower-/.f64 N/A

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

      6. sqr-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-neg.f64 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. sqr-neg N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. sub-neg N/A

          \[\leadsto \frac{x \cdot x - {\sin x}^{2}}{\frac{x \cdot x - {\sin x}^{2}}{\color{blue}{\sin x - x}}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{x \cdot x - {\sin x}^{2}}{\frac{x \cdot x - {\sin x}^{2}}{\color{blue}{\sin x} - x}} \]

      13. sub-div N/A

          \[\leadsto \frac{x \cdot x - {\sin x}^{2}}{\color{blue}{\frac{x \cdot x}{\sin x - x} - \frac{{\sin x}^{2}}{\sin x - x}}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{x \cdot x - {\sin x}^{2}}{\color{blue}{\frac{x \cdot x}{\sin x - x} - \frac{{\sin x}^{2}}{\sin x - x}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot x - {\sin x}^{2}}{\color{blue}{\frac{x \cdot x}{\sin x - x}} - \frac{{\sin x}^{2}}{\sin x - x}} \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{x \cdot x - {\sin x}^{2}}{\frac{x \cdot x}{\color{blue}{\sin x} - x} - \frac{{\sin x}^{2}}{\sin x - x}} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x \cdot x - {\sin x}^{2}}{\frac{x \cdot x}{\color{blue}{\sin x - x}} - \frac{{\sin x}^{2}}{\sin x - x}} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot x - {\sin x}^{2}}{\frac{x \cdot x}{\sin x - x} - \color{blue}{\frac{{\sin x}^{2}}{\sin x - x}}} \]

      19. lift-sin.f64 N/A

          \[\leadsto \frac{x \cdot x - {\sin x}^{2}}{\frac{x \cdot x}{\sin x - x} - \frac{{\sin x}^{2}}{\color{blue}{\sin x} - x}} \]

      20. lower--.f64 98.8

          \[\leadsto \frac{x \cdot x - {\sin x}^{2}}{\frac{x \cdot x}{\sin x - x} - \frac{{\sin x}^{2}}{\color{blue}{\sin x - x}}} \]

   6. Applied rewrites 98.8%

      \[\leadsto \frac{x \cdot x - {\sin x}^{2}}{\color{blue}{\frac{x \cdot x}{\sin x - x} - \frac{{\sin x}^{2}}{\sin x - x}}} \]

   if -1e-3 < (-.f64 (sin.f64 x) x)

   1. Initial program 66.0%

      \[\sin x - x \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 98.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot {x}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

      \[\leadsto \frac{\mathsf{fma}\left(6.944444444444444 \cdot 10^{-5}, {x}^{4}, -0.027777777777777776\right) \cdot {x}^{3}}{\color{blue}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{-1}{36} \cdot {x}^{3}}{\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{1}{6}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.001:\\ \;\;\;\;\frac{{\sin x}^{2} - x \cdot x}{\frac{x \cdot x}{x - \sin x} - \frac{{\sin x}^{2}}{x - \sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.027777777777777776 \cdot {x}^{3}}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m - x\_m \leq -0.001:\\ \;\;\;\;\frac{\left(\frac{\sin x\_m}{x\_m} - 1\right) \cdot {x\_m}^{3}}{x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.027777777777777776 \cdot {x\_m}^{3}}{\mathsf{fma}\left(0.008333333333333333, x\_m \cdot x\_m, 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= (- (sin x_m) x_m) -0.001)
    (/ (* (- (/ (sin x_m) x_m) 1.0) (pow x_m 3.0)) (* x_m x_m))
    (/
     (* -0.027777777777777776 (pow x_m 3.0))
     (fma 0.008333333333333333 (* 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) {
	double tmp;
	if ((sin(x_m) - x_m) <= -0.001) {
		tmp = (((sin(x_m) / x_m) - 1.0) * pow(x_m, 3.0)) / (x_m * x_m);
	} else {
		tmp = (-0.027777777777777776 * pow(x_m, 3.0)) / fma(0.008333333333333333, (x_m * x_m), 0.16666666666666666);
	}
	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) <= -0.001)
		tmp = Float64(Float64(Float64(Float64(sin(x_m) / x_m) - 1.0) * (x_m ^ 3.0)) / Float64(x_m * x_m));
	else
		tmp = Float64(Float64(-0.027777777777777776 * (x_m ^ 3.0)) / fma(0.008333333333333333, Float64(x_m * x_m), 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_] := N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision], -0.001], N[(N[(N[(N[(N[Sin[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-0.027777777777777776 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(0.008333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -1e-3

   1. Initial program 98.8%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. pow2 N/A

          \[\leadsto \frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, \color{blue}{{\sin x}^{2}}\right)} \]

      14. lower-pow.f64 99.2

          \[\leadsto \frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, \color{blue}{{\sin x}^{2}}\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, {\sin x}^{2}\right)}} \]

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 98.4

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 100.0%

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

   if -1e-3 < (-.f64 (sin.f64 x) x)

   1. Initial program 66.0%

      \[\sin x - x \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 98.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot {x}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

      \[\leadsto \frac{\mathsf{fma}\left(6.944444444444444 \cdot 10^{-5}, {x}^{4}, -0.027777777777777776\right) \cdot {x}^{3}}{\color{blue}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{-1}{36} \cdot {x}^{3}}{\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{1}{6}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.001:\\ \;\;\;\;\frac{\left(\frac{\sin x}{x} - 1\right) \cdot {x}^{3}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.027777777777777776 \cdot {x}^{3}}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.001:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.027777777777777776 \cdot {x\_m}^{3}}{\mathsf{fma}\left(0.008333333333333333, x\_m \cdot x\_m, 0.16666666666666666\right)}\\ \end{array} \end{array} \end{array} \]

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 -0.001)
      t_0
      (/
       (* -0.027777777777777776 (pow x_m 3.0))
       (fma 0.008333333333333333 (* 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) {
	double t_0 = sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -0.001) {
		tmp = t_0;
	} else {
		tmp = (-0.027777777777777776 * pow(x_m, 3.0)) / fma(0.008333333333333333, (x_m * x_m), 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 <= -0.001)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.027777777777777776 * (x_m ^ 3.0)) / fma(0.008333333333333333, Float64(x_m * x_m), 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, -0.001], t$95$0, N[(N[(-0.027777777777777776 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(0.008333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -1e-3

   1. Initial program 98.8%

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

   if -1e-3 < (-.f64 (sin.f64 x) x)

   1. Initial program 66.0%

      \[\sin x - x \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 98.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot {x}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

      \[\leadsto \frac{\mathsf{fma}\left(6.944444444444444 \cdot 10^{-5}, {x}^{4}, -0.027777777777777776\right) \cdot {x}^{3}}{\color{blue}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{-1}{36} \cdot {x}^{3}}{\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{1}{6}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.3%

       \[\leadsto \frac{-0.027777777777777776 \cdot {x}^{3}}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.001:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.008333333333333333, x\_m \cdot x\_m, -0.16666666666666666\right) \cdot \left(x\_m \cdot x\_m\right)\right) \cdot x\_m\\ \end{array} \end{array} \end{array} \]

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 -0.001)
      t_0
      (*
       (*
        (fma 0.008333333333333333 (* x_m x_m) -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) {
	double t_0 = sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -0.001) {
		tmp = t_0;
	} else {
		tmp = (fma(0.008333333333333333, (x_m * x_m), -0.16666666666666666) * (x_m * x_m)) * x_m;
	}
	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 <= -0.001)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(0.008333333333333333, Float64(x_m * x_m), -0.16666666666666666) * Float64(x_m * x_m)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -0.001], t$95$0, N[(N[(N[(0.008333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -1e-3

   1. Initial program 98.8%

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

   if -1e-3 < (-.f64 (sin.f64 x) x)

   1. Initial program 66.0%

      \[\sin x - x \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 98.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot {x}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

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

Alternative 6: 98.9% accurate, 2.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   (*
    (*
     (fma
      (fma -0.0001984126984126984 (* x_m x_m) 0.008333333333333333)
      (* x_m x_m)
      -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 * (((fma(fma(-0.0001984126984126984, (x_m * x_m), 0.008333333333333333), (x_m * x_m), -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(Float64(Float64(fma(fma(-0.0001984126984126984, Float64(x_m * x_m), 0.008333333333333333), Float64(x_m * x_m), -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[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. pow2 N/A

       \[\leadsto \frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, \color{blue}{{\sin x}^{2}}\right)} \]

   14. lower-pow.f64 24.5

       \[\leadsto \frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, \color{blue}{{\sin x}^{2}}\right)} \]

4. Applied rewrites 24.5%

   \[\leadsto \color{blue}{\frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, {\sin x}^{2}\right)}} \]

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 66.3

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

7. Applied rewrites 66.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.7%

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

Alternative 7: 98.7% accurate, 2.7× speedup

Math

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

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) -0.027777777777777776) x_m)
   (fma 0.008333333333333333 (* 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) * -0.027777777777777776) * x_m) / fma(0.008333333333333333, (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(Float64(Float64(x_m * x_m) * -0.027777777777777776) * x_m) / fma(0.008333333333333333, Float64(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[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.027777777777777776), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(0.008333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\sin x - x \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 97.4

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

5. Applied rewrites 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot {x}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 97.5%

   \[\leadsto \frac{\mathsf{fma}\left(6.944444444444444 \cdot 10^{-5}, {x}^{4}, -0.027777777777777776\right) \cdot {x}^{3}}{\color{blue}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-1}{36} \cdot {x}^{3}}{\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{1}{6}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 97.7%

    \[\leadsto \frac{-0.027777777777777776 \cdot {x}^{3}}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)} \]
11. Step-by-step derivation

12. Applied rewrites 97.6%

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

13. Final simplification 97.6%

    \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)} \]
14. Add Preprocessing

Alternative 8: 98.7% accurate, 2.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (* (* -0.027777777777777776 x_m) (* x_m x_m))
   (fma 0.008333333333333333 (* 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 * (((-0.027777777777777776 * x_m) * (x_m * x_m)) / fma(0.008333333333333333, (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(Float64(-0.027777777777777776 * x_m) * Float64(x_m * x_m)) / fma(0.008333333333333333, Float64(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[(N[(-0.027777777777777776 * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(0.008333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\sin x - x \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 97.4

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

5. Applied rewrites 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot {x}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 97.5%

   \[\leadsto \frac{\mathsf{fma}\left(6.944444444444444 \cdot 10^{-5}, {x}^{4}, -0.027777777777777776\right) \cdot {x}^{3}}{\color{blue}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-1}{36} \cdot {x}^{3}}{\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{1}{6}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 97.7%

    \[\leadsto \frac{-0.027777777777777776 \cdot {x}^{3}}{\mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right)} \]
11. Step-by-step derivation

12. Applied rewrites 97.6%

    \[\leadsto \frac{\left(-0.027777777777777776 \cdot x\right) \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(\color{blue}{0.008333333333333333}, x \cdot x, 0.16666666666666666\right)} \]
13. Add Preprocessing

Alternative 9: 98.7% accurate, 3.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   (* (fma 0.008333333333333333 (* x_m x_m) -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 * ((fma(0.008333333333333333, (x_m * x_m), -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(Float64(fma(0.008333333333333333, Float64(x_m * x_m), -0.16666666666666666) * Float64(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[(N[(N[(0.008333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\sin x - x \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 97.4

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

5. Applied rewrites 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot {x}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 97.4%

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

Alternative 10: 98.7% accurate, 3.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   (* (fma 0.008333333333333333 (* x_m x_m) -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 * ((fma(0.008333333333333333, (x_m * x_m), -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(Float64(fma(0.008333333333333333, Float64(x_m * x_m), -0.16666666666666666) * x_m) * Float64(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[(N[(N[(0.008333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\sin x - x \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 97.4

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

5. Applied rewrites 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot {x}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 97.4%

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

Alternative 11: 98.2% accurate, 6.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* (* -0.16666666666666666 (* 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(Float64(-0.16666666666666666 * Float64(x_m * 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[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. pow2 N/A

       \[\leadsto \frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, \color{blue}{{\sin x}^{2}}\right)} \]

   14. lower-pow.f64 24.5

       \[\leadsto \frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, \color{blue}{{\sin x}^{2}}\right)} \]

4. Applied rewrites 24.5%

   \[\leadsto \color{blue}{\frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, {\sin x}^{2}\right)}} \]

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 66.3

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

7. Applied rewrites 66.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.1%

    \[\leadsto \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x \]

11. Final simplification 97.1%

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

Alternative 12: 98.2% accurate, 6.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* (* (* -0.16666666666666666 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(Float64(Float64(-0.16666666666666666 * x_m) * 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[(N[(N[(-0.16666666666666666 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. pow2 N/A

       \[\leadsto \frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, \color{blue}{{\sin x}^{2}}\right)} \]

   14. lower-pow.f64 24.5

       \[\leadsto \frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, \color{blue}{{\sin x}^{2}}\right)} \]

4. Applied rewrites 24.5%

   \[\leadsto \color{blue}{\frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, {\sin x}^{2}\right)}} \]

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 66.3

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

7. Applied rewrites 66.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.7%

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

11. Taylor expanded in x around 0

    \[\leadsto \left(\left(\frac{-1}{6} \cdot x\right) \cdot x\right) \cdot x \]
12. Step-by-step derivation

13. Applied rewrites 97.1%

    \[\leadsto \left(\left(x \cdot -0.16666666666666666\right) \cdot x\right) \cdot x \]

14. Final simplification 97.1%

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

Alternative 13: 66.9% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 66.3%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. pow2 N/A

       \[\leadsto \frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, \color{blue}{{\sin x}^{2}}\right)} \]

   14. lower-pow.f64 24.5

       \[\leadsto \frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, \color{blue}{{\sin x}^{2}}\right)} \]

4. Applied rewrites 24.5%

   \[\leadsto \color{blue}{\frac{{\sin x}^{3} - {x}^{3}}{\mathsf{fma}\left(\sin x + x, x, {\sin x}^{2}\right)}} \]

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 66.3

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

7. Applied rewrites 66.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \left(1 - 1\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 62.6%

    \[\leadsto \left(1 - 1\right) \cdot x \]
11. Add Preprocessing

Alternative 14: 6.4% accurate, 34.7× speedup

Math

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

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)))

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 * Float64(-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 66.3%

   \[\sin x - x \]
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. lower-neg.f64 6.6

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

5. Applied rewrites 6.6%

   \[\leadsto \color{blue}{-x} \]
6. 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 2024277 
(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))