bug500 (missed optimization)

Percentage Accurate: 69.9% → 99.6%

Time: 7.2s

Alternatives: 7

Speedup: 14.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.9% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.0001:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{3} \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.008333333333333333 - 0.16666666666666666\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (sin x_m) x_m)))
   (*
    x_s
    (if (<= t_0 -0.0001)
      t_0
      (*
       (pow x_m 3.0)
       (- (* (* x_m x_m) 0.008333333333333333) 0.16666666666666666))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = t_0;
	} else {
		tmp = pow(x_m, 3.0) * (((x_m * x_m) * 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 <= (-0.0001d0)) then
        tmp = t_0
    else
        tmp = (x_m ** 3.0d0) * (((x_m * x_m) * 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 <= -0.0001) {
		tmp = t_0;
	} else {
		tmp = Math.pow(x_m, 3.0) * (((x_m * x_m) * 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 <= -0.0001:
		tmp = t_0
	else:
		tmp = math.pow(x_m, 3.0) * (((x_m * x_m) * 0.008333333333333333) - 0.16666666666666666)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(sin(x_m) - x_m)
	tmp = 0.0
	if (t_0 <= -0.0001)
		tmp = t_0;
	else
		tmp = Float64((x_m ^ 3.0) * Float64(Float64(Float64(x_m * x_m) * 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 <= -0.0001)
		tmp = t_0;
	else
		tmp = (x_m ^ 3.0) * (((x_m * x_m) * 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, -0.0001], t$95$0, N[(N[Power[x$95$m, 3.0], $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -1.00000000000000005e-4

   1. Initial program 100.0%

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

   if -1.00000000000000005e-4 < (-.f64 (sin.f64 x) x)

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(0.008333333333333333 \cdot {x}^{2} - 0.16666666666666666\right)} \]
   4. Step-by-step derivation

      1. unpow2 99.0%

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

   5. Applied egg-rr 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.0001:\\ \;\;\;\;\sin x - x\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.0001:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \end{array} \]

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 -0.0001) 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 <= -0.0001) {
		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 <= (-0.0001d0)) 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 <= -0.0001) {
		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 <= -0.0001:
		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 <= -0.0001)
		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 <= -0.0001)
		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, -0.0001], 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) < -1.00000000000000005e-4

   1. Initial program 100.0%

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

   if -1.00000000000000005e-4 < (-.f64 (sin.f64 x) x)

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.0001:\\ \;\;\;\;\sin x - x\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.0001:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \end{array} \]

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 -0.0001) t_0 (* x_m (* (* x_m x_m) -0.16666666666666666))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = sin(x_m) - x_m;
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = t_0;
	} else {
		tmp = x_m * ((x_m * x_m) * -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 <= (-0.0001d0)) then
        tmp = t_0
    else
        tmp = x_m * ((x_m * x_m) * (-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 <= -0.0001) {
		tmp = t_0;
	} else {
		tmp = x_m * ((x_m * x_m) * -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 <= -0.0001:
		tmp = t_0
	else:
		tmp = x_m * ((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 <= -0.0001)
		tmp = t_0;
	else
		tmp = Float64(x_m * Float64(Float64(x_m * x_m) * -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 <= -0.0001)
		tmp = t_0;
	else
		tmp = x_m * ((x_m * x_m) * -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, -0.0001], t$95$0, N[(x$95$m * N[(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) < -1.00000000000000005e-4

   1. Initial program 100.0%

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

   if -1.00000000000000005e-4 < (-.f64 (sin.f64 x) x)

   1. Initial program 70.1%

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

   3. Taylor expanded in x around inf 70.1%

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

   4. Taylor expanded in x around 0 98.8%

      \[\leadsto x \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)} \]

   6. Simplified 98.8%

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)} \]
   7. Step-by-step derivation

      1. unpow2 99.0%

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

   8. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -0.0001:\\ \;\;\;\;\sin x - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left({x\_m}^{3} \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(0.008333333333333333 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 2.7557319223985893 \cdot 10^{-6} - 0.0001984126984126984\right)\right) - 0.16666666666666666\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   (pow x_m 3.0)
   (-
    (*
     (* x_m x_m)
     (+
      0.008333333333333333
      (*
       (* x_m x_m)
       (- (* (* x_m x_m) 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) {
	return x_s * (pow(x_m, 3.0) * (((x_m * x_m) * (0.008333333333333333 + ((x_m * x_m) * (((x_m * x_m) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 0.16666666666666666));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((x_m ** 3.0d0) * (((x_m * x_m) * (0.008333333333333333d0 + ((x_m * x_m) * (((x_m * x_m) * 2.7557319223985893d-6) - 0.0001984126984126984d0)))) - 0.16666666666666666d0))
end function

Java

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

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (math.pow(x_m, 3.0) * (((x_m * x_m) * (0.008333333333333333 + ((x_m * x_m) * (((x_m * x_m) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 0.16666666666666666))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64((x_m ^ 3.0) * Float64(Float64(Float64(x_m * x_m) * Float64(0.008333333333333333 + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 0.16666666666666666)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m ^ 3.0) * (((x_m * x_m) * (0.008333333333333333 + ((x_m * x_m) * (((x_m * x_m) * 2.7557319223985893e-6) - 0.0001984126984126984)))) - 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[Power[x$95$m, 3.0], $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 2.7557319223985893e-6), $MachinePrecision] - 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around 0 98.3%

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(0.008333333333333333 + {x}^{2} \cdot \left(2.7557319223985893 \cdot 10^{-6} \cdot {x}^{2} - 0.0001984126984126984\right)\right) - 0.16666666666666666\right)} \]
4. Step-by-step derivation

   1. unpow2 98.0%

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

5. Applied egg-rr 98.3%

   \[\leadsto {x}^{3} \cdot \left({x}^{2} \cdot \left(0.008333333333333333 + {x}^{2} \cdot \left(2.7557319223985893 \cdot 10^{-6} \cdot \color{blue}{\left(x \cdot x\right)} - 0.0001984126984126984\right)\right) - 0.16666666666666666\right) \]
6. Step-by-step derivation

   1. unpow2 98.0%

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

7. Applied egg-rr 98.3%

   \[\leadsto {x}^{3} \cdot \left({x}^{2} \cdot \left(0.008333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot \left(2.7557319223985893 \cdot 10^{-6} \cdot \left(x \cdot x\right) - 0.0001984126984126984\right)\right) - 0.16666666666666666\right) \]
8. Step-by-step derivation

   1. unpow2 98.0%

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

9. Applied egg-rr 98.3%

   \[\leadsto {x}^{3} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(2.7557319223985893 \cdot 10^{-6} \cdot \left(x \cdot x\right) - 0.0001984126984126984\right)\right) - 0.16666666666666666\right) \]

10. Final simplification 98.3%

    \[\leadsto {x}^{3} \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 2.7557319223985893 \cdot 10^{-6} - 0.0001984126984126984\right)\right) - 0.16666666666666666\right) \]
11. Add Preprocessing

Alternative 5: 98.0% accurate, 14.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (x_m * ((x_m * x_m) * -0.16666666666666666));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around inf 70.4%

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

4. Taylor expanded in x around 0 97.8%

   \[\leadsto x \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right)} \]
5. Step-by-step derivation

   1. *-commutative 97.8%

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)} \]

6. Simplified 97.8%

   \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666\right)} \]
7. Step-by-step derivation

   1. unpow2 98.0%

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

8. Applied egg-rr 97.8%

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

9. Final simplification 97.8%

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

Alternative 6: 67.5% accurate, 34.3× speedup

Math

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

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 0.0)))

C

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

Fortran

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

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (x_m * 0.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

3. Taylor expanded in x around inf 70.4%

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

4. Taylor expanded in x around 0 67.9%

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

5. Final simplification 67.9%

   \[\leadsto x \cdot 0 \]
6. Add Preprocessing

Alternative 7: 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 70.5%

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

3. Taylor expanded in x around inf 6.4%

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

   1. neg-mul-1 6.4%

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

5. Simplified 6.4%

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

6. Final simplification 6.4%

   \[\leadsto -x \]
7. 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 2024080 
(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))