bug500 (missed optimization)

Percentage Accurate: 69.9% → 99.7%

Time: 6.9s

Alternatives: 8

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: 69.9% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin x\_m + x\_m, x\_m, {\sin x\_m}^{2}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m - x\_m \leq -0.001:\\ \;\;\;\;\frac{1}{\frac{{t\_0}^{2}}{{\sin x\_m}^{3} \cdot t\_0 - {x\_m}^{3} \cdot t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 2.7557319223985893 \cdot 10^{-6}, -0.0001984126984126984\right), x\_m \cdot x\_m, 0.008333333333333333\right), x\_m \cdot x\_m, -0.16666666666666666\right) \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (+ (sin x_m) x_m) x_m (pow (sin x_m) 2.0))))
   (*
    x_s
    (if (<= (- (sin x_m) x_m) -0.001)
      (/
       1.0
       (/ (pow t_0 2.0) (- (* (pow (sin x_m) 3.0) t_0) (* (pow x_m 3.0) t_0))))
      (*
       (*
        (fma
         (fma
          (fma (* x_m x_m) 2.7557319223985893e-6 -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) {
	double t_0 = fma((sin(x_m) + x_m), x_m, pow(sin(x_m), 2.0));
	double tmp;
	if ((sin(x_m) - x_m) <= -0.001) {
		tmp = 1.0 / (pow(t_0, 2.0) / ((pow(sin(x_m), 3.0) * t_0) - (pow(x_m, 3.0) * t_0)));
	} else {
		tmp = (fma(fma(fma((x_m * x_m), 2.7557319223985893e-6, -0.0001984126984126984), (x_m * x_m), 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 = fma(Float64(sin(x_m) + x_m), x_m, (sin(x_m) ^ 2.0))
	tmp = 0.0
	if (Float64(sin(x_m) - x_m) <= -0.001)
		tmp = Float64(1.0 / Float64((t_0 ^ 2.0) / Float64(Float64((sin(x_m) ^ 3.0) * t_0) - Float64((x_m ^ 3.0) * t_0))));
	else
		tmp = Float64(Float64(fma(fma(fma(Float64(x_m * x_m), 2.7557319223985893e-6, -0.0001984126984126984), Float64(x_m * x_m), 0.008333333333333333), Float64(x_m * x_m), -0.16666666666666666) * x_m) * Float64(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[(N[Sin[x$95$m], $MachinePrecision] + x$95$m), $MachinePrecision] * x$95$m + N[Power[N[Sin[x$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision], -0.001], N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[(N[(N[Power[N[Sin[x$95$m], $MachinePrecision], 3.0], $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[Power[x$95$m, 3.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 2.7557319223985893e-6 + -0.0001984126984126984), $MachinePrecision] * 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] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.0%

      \[\sin x - x \]
   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. frac-sub N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 98.3%

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

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

   1. Initial program 67.5%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.7557319223985893 \cdot 10^{-6}, x \cdot x, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot {x}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.5%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 2.7557319223985893 \cdot 10^{-6}, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.001:\\ \;\;\;\;\frac{1}{\frac{{\left(\mathsf{fma}\left(\sin x + x, x, {\sin x}^{2}\right)\right)}^{2}}{{\sin x}^{3} \cdot \mathsf{fma}\left(\sin x + x, x, {\sin x}^{2}\right) - {x}^{3} \cdot \mathsf{fma}\left(\sin x + x, x, {\sin x}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 2.7557319223985893 \cdot 10^{-6}, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.001:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 2.7557319223985893 \cdot 10^{-6}, -0.0001984126984126984\right), x\_m \cdot x\_m, 0.008333333333333333\right), x\_m \cdot x\_m, -0.16666666666666666\right) \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\\ \end{array} \end{array} \end{array} \]

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
      (*
       (*
        (fma
         (fma
          (fma (* x_m x_m) 2.7557319223985893e-6 -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) {
	double t_0 = sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -0.001) {
		tmp = t_0;
	} else {
		tmp = (fma(fma(fma((x_m * x_m), 2.7557319223985893e-6, -0.0001984126984126984), (x_m * x_m), 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(fma(fma(Float64(x_m * x_m), 2.7557319223985893e-6, -0.0001984126984126984), Float64(x_m * x_m), 0.008333333333333333), Float64(x_m * x_m), -0.16666666666666666) * x_m) * Float64(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[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 2.7557319223985893e-6 + -0.0001984126984126984), $MachinePrecision] * 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] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.0%

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

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

   1. Initial program 67.5%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.7557319223985893 \cdot 10^{-6}, x \cdot x, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot {x}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.5%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 2.7557319223985893 \cdot 10^{-6}, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 2.7557319223985893 \cdot 10^{-6}, -0.0001984126984126984\right), x\_m \cdot x\_m, 0.008333333333333333\right), x\_m \cdot x\_m, -0.16666666666666666\right) \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\right) \end{array} \]

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
     (fma
      (fma (* x_m x_m) 2.7557319223985893e-6 -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(fma((x_m * x_m), 2.7557319223985893e-6, -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(fma(fma(fma(Float64(x_m * x_m), 2.7557319223985893e-6, -0.0001984126984126984), Float64(x_m * x_m), 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[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 2.7557319223985893e-6 + -0.0001984126984126984), $MachinePrecision] * 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] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.9%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.7557319223985893 \cdot 10^{-6}, x \cdot x, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot {x}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 97.5%

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

Alternative 4: 98.7% accurate, 2.7× speedup

Math

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

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

Derivation

1. Initial program 67.9%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.7557319223985893 \cdot 10^{-6}, x \cdot x, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot {x}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 97.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.4%

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

Alternative 5: 98.4% 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 67.9%

   \[\sin x - x \]
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 96.7

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

5. Applied rewrites 96.7%

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

7. Applied rewrites 96.7%

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

Alternative 6: 98.0% accurate, 6.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* (* -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 * x_m) * Float64(x_m * x_m)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((-0.16666666666666666 * x_m) * (x_m * x_m));
end

Wolfram

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

Derivation

1. Initial program 67.9%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.7557319223985893 \cdot 10^{-6}, x \cdot x, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot {x}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 97.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 96.1%

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

Alternative 7: 67.4% 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 67.9%

   \[\sin x - x \]
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. neg-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - x\right)} + \sin x \]

   5. associate-+l- N/A

      \[\leadsto \color{blue}{0 - \left(x - \sin x\right)} \]

   6. flip3-- N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower--.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower--.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower--.f64 5.1

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

4. Applied rewrites 5.1%

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

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 67.8

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

7. Applied rewrites 67.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 63.0%

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

Alternative 8: 6.5% 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 67.9%

   \[\sin x - x \]
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 6.8

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

5. Applied rewrites 6.8%

   \[\leadsto \color{blue}{-x} \]
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 2024244 
(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))