bug500 (missed optimization)

Percentage Accurate: 69.4% → 99.7%

Time: 7.9s

Alternatives: 8

Speedup: 1.0×

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: 69.4% 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, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m - x\_m \leq -0.2:\\ \;\;\;\;\frac{\sqrt{{\sin x\_m}^{4}} - {x\_m}^{2}}{x\_m + \sin x\_m}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{3} \cdot \left({x\_m}^{2} \cdot \left(0.008333333333333333 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot 2.7557319223985893 \cdot 10^{-6} - 0.0001984126984126984\right)\right) - 0.16666666666666666\right)\\ \end{array} \end{array} \]

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) -0.2)
    (/ (- (sqrt (pow (sin x_m) 4.0)) (pow x_m 2.0)) (+ x_m (sin x_m)))
    (*
     (pow x_m 3.0)
     (-
      (*
       (pow x_m 2.0)
       (+
        0.008333333333333333
        (*
         (pow x_m 2.0)
         (- (* (pow x_m 2.0) 2.7557319223985893e-6) 0.0001984126984126984))))
      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) <= -0.2) {
		tmp = (sqrt(pow(sin(x_m), 4.0)) - pow(x_m, 2.0)) / (x_m + sin(x_m));
	} else {
		tmp = pow(x_m, 3.0) * ((pow(x_m, 2.0) * (0.008333333333333333 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 0.16666666666666666);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if ((sin(x_m) - x_m) <= (-0.2d0)) then
        tmp = (sqrt((sin(x_m) ** 4.0d0)) - (x_m ** 2.0d0)) / (x_m + sin(x_m))
    else
        tmp = (x_m ** 3.0d0) * (((x_m ** 2.0d0) * (0.008333333333333333d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 2.7557319223985893d-6) - 0.0001984126984126984d0)))) - 0.16666666666666666d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if ((Math.sin(x_m) - x_m) <= -0.2) {
		tmp = (Math.sqrt(Math.pow(Math.sin(x_m), 4.0)) - Math.pow(x_m, 2.0)) / (x_m + Math.sin(x_m));
	} else {
		tmp = Math.pow(x_m, 3.0) * ((Math.pow(x_m, 2.0) * (0.008333333333333333 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 0.16666666666666666);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if (math.sin(x_m) - x_m) <= -0.2:
		tmp = (math.sqrt(math.pow(math.sin(x_m), 4.0)) - math.pow(x_m, 2.0)) / (x_m + math.sin(x_m))
	else:
		tmp = math.pow(x_m, 3.0) * ((math.pow(x_m, 2.0) * (0.008333333333333333 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 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) <= -0.2)
		tmp = Float64(Float64(sqrt((sin(x_m) ^ 4.0)) - (x_m ^ 2.0)) / Float64(x_m + sin(x_m)));
	else
		tmp = Float64((x_m ^ 3.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.008333333333333333 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 0.16666666666666666));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if ((sin(x_m) - x_m) <= -0.2)
		tmp = (sqrt((sin(x_m) ^ 4.0)) - (x_m ^ 2.0)) / (x_m + sin(x_m));
	else
		tmp = (x_m ^ 3.0) * (((x_m ^ 2.0) * (0.008333333333333333 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 0.16666666666666666);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision], -0.2], N[(N[(N[Sqrt[N[Power[N[Sin[x$95$m], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] - N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(x$95$m + N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 3.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.008333333333333333 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 2.7557319223985893e-6), $MachinePrecision] - 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -0.20000000000000001

   1. Initial program 100.0%

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

      1. flip-- 100.0%

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

      2. div-inv 99.6%

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

      3. pow2 99.6%

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

      4. pow2 99.6%

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

      5. +-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

   6. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. pow-prod-up 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \frac{\sqrt{{\sin x}^{\color{blue}{4}}} - {x}^{2}}{x + \sin x} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\sqrt{{\sin x}^{4}}} - {x}^{2}}{x + \sin x} \]

   if -0.20000000000000001 < (-.f64 (sin.f64 x) x)

   1. Initial program 67.4%

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

   3. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.2:\\ \;\;\;\;\frac{\sqrt{{\sin x}^{4}} - {x}^{2}}{x + \sin x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left({x}^{2} \cdot \left(0.008333333333333333 + {x}^{2} \cdot \left({x}^{2} \cdot 2.7557319223985893 \cdot 10^{-6} - 0.0001984126984126984\right)\right) - 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{3} \cdot \left({x\_m}^{2} \cdot \left(0.008333333333333333 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot 2.7557319223985893 \cdot 10^{-6} - 0.0001984126984126984\right)\right) - 0.16666666666666666\right)\\ \end{array} \end{array} \end{array} \]

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 -0.2)
      t_0
      (*
       (pow x_m 3.0)
       (-
        (*
         (pow x_m 2.0)
         (+
          0.008333333333333333
          (*
           (pow x_m 2.0)
           (- (* (pow x_m 2.0) 2.7557319223985893e-6) 0.0001984126984126984))))
        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 <= -0.2) {
		tmp = t_0;
	} else {
		tmp = pow(x_m, 3.0) * ((pow(x_m, 2.0) * (0.008333333333333333 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 0.16666666666666666);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x_m) - x_m
    if (t_0 <= (-0.2d0)) then
        tmp = t_0
    else
        tmp = (x_m ** 3.0d0) * (((x_m ** 2.0d0) * (0.008333333333333333d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 2.7557319223985893d-6) - 0.0001984126984126984d0)))) - 0.16666666666666666d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -0.2) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x_m, 3.0) * ((Math.pow(x_m, 2.0) * (0.008333333333333333 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 0.16666666666666666);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin(x_m) - x_m
	tmp = 0
	if t_0 <= -0.2:
		tmp = t_0
	else:
		tmp = math.pow(x_m, 3.0) * ((math.pow(x_m, 2.0) * (0.008333333333333333 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 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 <= -0.2)
		tmp = t_0;
	else
		tmp = Float64((x_m ^ 3.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.008333333333333333 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 0.16666666666666666));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin(x_m) - x_m;
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = t_0;
	else
		tmp = (x_m ^ 3.0) * (((x_m ^ 2.0) * (0.008333333333333333 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 0.16666666666666666);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -0.2], t$95$0, N[(N[Power[x$95$m, 3.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.008333333333333333 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 2.7557319223985893e-6), $MachinePrecision] - 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -0.20000000000000001

   1. Initial program 100.0%

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

   if -0.20000000000000001 < (-.f64 (sin.f64 x) x)

   1. Initial program 67.4%

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

   3. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.2:\\ \;\;\;\;\sin x - x\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left({x}^{2} \cdot \left(0.008333333333333333 + {x}^{2} \cdot \left({x}^{2} \cdot 2.7557319223985893 \cdot 10^{-6} - 0.0001984126984126984\right)\right) - 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m - x\_m \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{\sin x\_m}^{2} - {x\_m}^{2}}{x\_m + \sin x\_m}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{3} \cdot \left({x\_m}^{2} \cdot \left(0.008333333333333333 + {x\_m}^{2} \cdot -0.0001984126984126984\right) - 0.16666666666666666\right)\\ \end{array} \end{array} \]

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) -2e-5)
    (/ (- (pow (sin x_m) 2.0) (pow x_m 2.0)) (+ x_m (sin x_m)))
    (*
     (pow x_m 3.0)
     (-
      (*
       (pow x_m 2.0)
       (+ 0.008333333333333333 (* (pow x_m 2.0) -0.0001984126984126984)))
      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) <= -2e-5) {
		tmp = (pow(sin(x_m), 2.0) - pow(x_m, 2.0)) / (x_m + sin(x_m));
	} else {
		tmp = pow(x_m, 3.0) * ((pow(x_m, 2.0) * (0.008333333333333333 + (pow(x_m, 2.0) * -0.0001984126984126984))) - 0.16666666666666666);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if ((sin(x_m) - x_m) <= (-2d-5)) then
        tmp = ((sin(x_m) ** 2.0d0) - (x_m ** 2.0d0)) / (x_m + sin(x_m))
    else
        tmp = (x_m ** 3.0d0) * (((x_m ** 2.0d0) * (0.008333333333333333d0 + ((x_m ** 2.0d0) * (-0.0001984126984126984d0)))) - 0.16666666666666666d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if ((Math.sin(x_m) - x_m) <= -2e-5) {
		tmp = (Math.pow(Math.sin(x_m), 2.0) - Math.pow(x_m, 2.0)) / (x_m + Math.sin(x_m));
	} else {
		tmp = Math.pow(x_m, 3.0) * ((Math.pow(x_m, 2.0) * (0.008333333333333333 + (Math.pow(x_m, 2.0) * -0.0001984126984126984))) - 0.16666666666666666);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if (math.sin(x_m) - x_m) <= -2e-5:
		tmp = (math.pow(math.sin(x_m), 2.0) - math.pow(x_m, 2.0)) / (x_m + math.sin(x_m))
	else:
		tmp = math.pow(x_m, 3.0) * ((math.pow(x_m, 2.0) * (0.008333333333333333 + (math.pow(x_m, 2.0) * -0.0001984126984126984))) - 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) <= -2e-5)
		tmp = Float64(Float64((sin(x_m) ^ 2.0) - (x_m ^ 2.0)) / Float64(x_m + sin(x_m)));
	else
		tmp = Float64((x_m ^ 3.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.008333333333333333 + Float64((x_m ^ 2.0) * -0.0001984126984126984))) - 0.16666666666666666));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if ((sin(x_m) - x_m) <= -2e-5)
		tmp = ((sin(x_m) ^ 2.0) - (x_m ^ 2.0)) / (x_m + sin(x_m));
	else
		tmp = (x_m ^ 3.0) * (((x_m ^ 2.0) * (0.008333333333333333 + ((x_m ^ 2.0) * -0.0001984126984126984))) - 0.16666666666666666);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision], -2e-5], N[(N[(N[Power[N[Sin[x$95$m], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(x$95$m + N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 3.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.008333333333333333 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -2.00000000000000016e-5

   1. Initial program 100.0%

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

      1. flip-- 100.0%

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

      2. div-inv 99.6%

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

      3. pow2 99.6%

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

      4. pow2 99.6%

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

      5. +-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

   6. Simplified 100.0%

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

   if -2.00000000000000016e-5 < (-.f64 (sin.f64 x) x)

   1. Initial program 67.4%

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

   3. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot {x}^{2}\right) - 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

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

Alternative 4: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{3} \cdot \left({x\_m}^{2} \cdot \left(0.008333333333333333 + {x\_m}^{2} \cdot -0.0001984126984126984\right) - 0.16666666666666666\right)\\ \end{array} \end{array} \end{array} \]

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 -2e-5)
      t_0
      (*
       (pow x_m 3.0)
       (-
        (*
         (pow x_m 2.0)
         (+ 0.008333333333333333 (* (pow x_m 2.0) -0.0001984126984126984)))
        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 <= -2e-5) {
		tmp = t_0;
	} else {
		tmp = pow(x_m, 3.0) * ((pow(x_m, 2.0) * (0.008333333333333333 + (pow(x_m, 2.0) * -0.0001984126984126984))) - 0.16666666666666666);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x_m) - x_m
    if (t_0 <= (-2d-5)) then
        tmp = t_0
    else
        tmp = (x_m ** 3.0d0) * (((x_m ** 2.0d0) * (0.008333333333333333d0 + ((x_m ** 2.0d0) * (-0.0001984126984126984d0)))) - 0.16666666666666666d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -2e-5) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x_m, 3.0) * ((Math.pow(x_m, 2.0) * (0.008333333333333333 + (Math.pow(x_m, 2.0) * -0.0001984126984126984))) - 0.16666666666666666);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin(x_m) - x_m
	tmp = 0
	if t_0 <= -2e-5:
		tmp = t_0
	else:
		tmp = math.pow(x_m, 3.0) * ((math.pow(x_m, 2.0) * (0.008333333333333333 + (math.pow(x_m, 2.0) * -0.0001984126984126984))) - 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 <= -2e-5)
		tmp = t_0;
	else
		tmp = Float64((x_m ^ 3.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.008333333333333333 + Float64((x_m ^ 2.0) * -0.0001984126984126984))) - 0.16666666666666666));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin(x_m) - x_m;
	tmp = 0.0;
	if (t_0 <= -2e-5)
		tmp = t_0;
	else
		tmp = (x_m ^ 3.0) * (((x_m ^ 2.0) * (0.008333333333333333 + ((x_m ^ 2.0) * -0.0001984126984126984))) - 0.16666666666666666);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e-5], t$95$0, N[(N[Power[x$95$m, 3.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.008333333333333333 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -2.00000000000000016e-5

   1. Initial program 100.0%

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

   if -2.00000000000000016e-5 < (-.f64 (sin.f64 x) x)

   1. Initial program 67.4%

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

   3. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot {x}^{2}\right) - 0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

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

Alternative 5: 99.7% accurate, 0.3× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (sin x_m) x_m)))
   (*
    x_s
    (if (<= t_0 -1e-6)
      t_0
      (*
       (pow x_m 3.0)
       (- (* (pow x_m 2.0) 0.008333333333333333) 0.16666666666666666))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -1e-6) {
		tmp = t_0;
	} else {
		tmp = pow(x_m, 3.0) * ((pow(x_m, 2.0) * 0.008333333333333333) - 0.16666666666666666);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x_m) - x_m
    if (t_0 <= (-1d-6)) then
        tmp = t_0
    else
        tmp = (x_m ** 3.0d0) * (((x_m ** 2.0d0) * 0.008333333333333333d0) - 0.16666666666666666d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -1e-6) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x_m, 3.0) * ((Math.pow(x_m, 2.0) * 0.008333333333333333) - 0.16666666666666666);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin(x_m) - x_m
	tmp = 0
	if t_0 <= -1e-6:
		tmp = t_0
	else:
		tmp = math.pow(x_m, 3.0) * ((math.pow(x_m, 2.0) * 0.008333333333333333) - 0.16666666666666666)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(sin(x_m) - x_m)
	tmp = 0.0
	if (t_0 <= -1e-6)
		tmp = t_0;
	else
		tmp = Float64((x_m ^ 3.0) * Float64(Float64((x_m ^ 2.0) * 0.008333333333333333) - 0.16666666666666666));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin(x_m) - x_m;
	tmp = 0.0;
	if (t_0 <= -1e-6)
		tmp = t_0;
	else
		tmp = (x_m ^ 3.0) * (((x_m ^ 2.0) * 0.008333333333333333) - 0.16666666666666666);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -1e-6], t$95$0, N[(N[Power[x$95$m, 3.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -9.99999999999999955e-7

   1. Initial program 96.1%

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

   if -9.99999999999999955e-7 < (-.f64 (sin.f64 x) x)

   1. Initial program 67.3%

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

   3. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.9%

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

Alternative 6: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \end{array} \]

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 -2e-9) t_0 (* (pow x_m 3.0) -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 <= -2e-9) {
		tmp = t_0;
	} else {
		tmp = pow(x_m, 3.0) * -0.16666666666666666;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x_m) - x_m
    if (t_0 <= (-2d-9)) then
        tmp = t_0
    else
        tmp = (x_m ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -2e-9) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x_m, 3.0) * -0.16666666666666666;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.sin(x_m) - x_m
	tmp = 0
	if t_0 <= -2e-9:
		tmp = t_0
	else:
		tmp = math.pow(x_m, 3.0) * -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 <= -2e-9)
		tmp = t_0;
	else
		tmp = Float64((x_m ^ 3.0) * -0.16666666666666666);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = sin(x_m) - x_m;
	tmp = 0.0;
	if (t_0 <= -2e-9)
		tmp = t_0;
	else
		tmp = (x_m ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(N[Sin[x$95$m], $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e-9], t$95$0, N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -2.00000000000000012e-9

   1. Initial program 96.1%

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

   if -2.00000000000000012e-9 < (-.f64 (sin.f64 x) x)

   1. Initial program 67.3%

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

   3. Taylor expanded in x around 0 98.4%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\sin x - x\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 67.9%

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

Alternative 8: 6.5% accurate, 51.5× 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 6.5%

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

   1. neg-mul-1 6.5%

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

5. Simplified 6.5%

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

Developer target: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024108 
(FPCore (x)
  :name "bug500 (missed optimization)"
  :precision binary64
  :pre (and (< -1000.0 x) (< x 1000.0))

  :alt
  (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))

  (- (sin x) x))