bug500 (missed optimization)

Percentage Accurate: 68.7% → 99.7%

Time: 7.3s

Alternatives: 9

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: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ t_1 := x\_m + \sin x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\left(t\_0 \cdot t\_1\right) \cdot \frac{1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x\_m}^{3} + 0.008333333333333333 \cdot {x\_m}^{5}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (sin x_m) x_m)) (t_1 (+ x_m (sin x_m))))
   (*
    x_s
    (if (<= t_0 -1e-7)
      (* (* t_0 t_1) (/ 1.0 t_1))
      (+
       (* -0.16666666666666666 (pow x_m 3.0))
       (* 0.008333333333333333 (pow x_m 5.0)))))))

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 t_1 = x_m + sin(x_m);
	double tmp;
	if (t_0 <= -1e-7) {
		tmp = (t_0 * t_1) * (1.0 / t_1);
	} else {
		tmp = (-0.16666666666666666 * pow(x_m, 3.0)) + (0.008333333333333333 * pow(x_m, 5.0));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sin(x_m) - x_m
    t_1 = x_m + sin(x_m)
    if (t_0 <= (-1d-7)) then
        tmp = (t_0 * t_1) * (1.0d0 / t_1)
    else
        tmp = ((-0.16666666666666666d0) * (x_m ** 3.0d0)) + (0.008333333333333333d0 * (x_m ** 5.0d0))
    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 t_1 = x_m + Math.sin(x_m);
	double tmp;
	if (t_0 <= -1e-7) {
		tmp = (t_0 * t_1) * (1.0 / t_1);
	} else {
		tmp = (-0.16666666666666666 * Math.pow(x_m, 3.0)) + (0.008333333333333333 * Math.pow(x_m, 5.0));
	}
	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
	t_1 = x_m + math.sin(x_m)
	tmp = 0
	if t_0 <= -1e-7:
		tmp = (t_0 * t_1) * (1.0 / t_1)
	else:
		tmp = (-0.16666666666666666 * math.pow(x_m, 3.0)) + (0.008333333333333333 * math.pow(x_m, 5.0))
	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)
	t_1 = Float64(x_m + sin(x_m))
	tmp = 0.0
	if (t_0 <= -1e-7)
		tmp = Float64(Float64(t_0 * t_1) * Float64(1.0 / t_1));
	else
		tmp = Float64(Float64(-0.16666666666666666 * (x_m ^ 3.0)) + Float64(0.008333333333333333 * (x_m ^ 5.0)));
	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;
	t_1 = x_m + sin(x_m);
	tmp = 0.0;
	if (t_0 <= -1e-7)
		tmp = (t_0 * t_1) * (1.0 / t_1);
	else
		tmp = (-0.16666666666666666 * (x_m ^ 3.0)) + (0.008333333333333333 * (x_m ^ 5.0));
	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]}, Block[{t$95$1 = N[(x$95$m + N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -1e-7], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(-0.16666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.008333333333333333 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -9.9999999999999995e-8

   1. Initial program 90.3%

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

      1. flip-- 89.0%

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

      2. div-inv 89.0%

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

      3. pow2 89.0%

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

      4. pow2 89.0%

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

      5. +-commutative 89.0%

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

   4. Applied egg-rr 89.0%

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

      1. unpow2 89.0%

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

      2. unpow2 89.0%

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

      3. difference-of-squares 90.3%

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

      4. +-commutative 90.3%

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

   6. Applied egg-rr 90.3%

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

   if -9.9999999999999995e-8 < (-.f64 (sin.f64 x) x)

   1. Initial program 70.8%

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

   3. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + 0.008333333333333333 \cdot {x}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(\sin x - x\right) \cdot \left(x + \sin x\right)\right) \cdot \frac{1}{x + \sin x}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + 0.008333333333333333 \cdot {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(-0.16666666666666666 \cdot {x\_m}^{3} + \left(-0.0001984126984126984 \cdot {x\_m}^{7} + \left(2.7557319223985893 \cdot 10^{-6} \cdot {x\_m}^{9} + 0.008333333333333333 \cdot {x\_m}^{5}\right)\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (+
   (* -0.16666666666666666 (pow x_m 3.0))
   (+
    (* -0.0001984126984126984 (pow x_m 7.0))
    (+
     (* 2.7557319223985893e-6 (pow x_m 9.0))
     (* 0.008333333333333333 (pow x_m 5.0)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((-0.16666666666666666 * pow(x_m, 3.0)) + ((-0.0001984126984126984 * pow(x_m, 7.0)) + ((2.7557319223985893e-6 * pow(x_m, 9.0)) + (0.008333333333333333 * pow(x_m, 5.0)))));
}

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 * (((-0.16666666666666666d0) * (x_m ** 3.0d0)) + (((-0.0001984126984126984d0) * (x_m ** 7.0d0)) + ((2.7557319223985893d-6 * (x_m ** 9.0d0)) + (0.008333333333333333d0 * (x_m ** 5.0d0)))))
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 * ((-0.16666666666666666 * Math.pow(x_m, 3.0)) + ((-0.0001984126984126984 * Math.pow(x_m, 7.0)) + ((2.7557319223985893e-6 * Math.pow(x_m, 9.0)) + (0.008333333333333333 * Math.pow(x_m, 5.0)))));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((-0.16666666666666666 * math.pow(x_m, 3.0)) + ((-0.0001984126984126984 * math.pow(x_m, 7.0)) + ((2.7557319223985893e-6 * math.pow(x_m, 9.0)) + (0.008333333333333333 * math.pow(x_m, 5.0)))))

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 ^ 3.0)) + Float64(Float64(-0.0001984126984126984 * (x_m ^ 7.0)) + Float64(Float64(2.7557319223985893e-6 * (x_m ^ 9.0)) + Float64(0.008333333333333333 * (x_m ^ 5.0))))))
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 ^ 3.0)) + ((-0.0001984126984126984 * (x_m ^ 7.0)) + ((2.7557319223985893e-6 * (x_m ^ 9.0)) + (0.008333333333333333 * (x_m ^ 5.0)))));
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[(N[(-0.16666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0001984126984126984 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.7557319223985893e-6 * N[Power[x$95$m, 9.0], $MachinePrecision]), $MachinePrecision] + N[(0.008333333333333333 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in x around 0 98.3%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(-0.0001984126984126984 \cdot {x}^{7} + \left(2.7557319223985893 \cdot 10^{-6} \cdot {x}^{9} + 0.008333333333333333 \cdot {x}^{5}\right)\right)} \]

4. Final simplification 98.3%

   \[\leadsto -0.16666666666666666 \cdot {x}^{3} + \left(-0.0001984126984126984 \cdot {x}^{7} + \left(2.7557319223985893 \cdot 10^{-6} \cdot {x}^{9} + 0.008333333333333333 \cdot {x}^{5}\right)\right) \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x\_m}^{3} + 0.008333333333333333 \cdot {x\_m}^{5}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (sin x_m) x_m)))
   (*
    x_s
    (if (<= t_0 -1e-7)
      t_0
      (+
       (* -0.16666666666666666 (pow x_m 3.0))
       (* 0.008333333333333333 (pow x_m 5.0)))))))

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-7) {
		tmp = t_0;
	} else {
		tmp = (-0.16666666666666666 * pow(x_m, 3.0)) + (0.008333333333333333 * pow(x_m, 5.0));
	}
	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-7)) then
        tmp = t_0
    else
        tmp = ((-0.16666666666666666d0) * (x_m ** 3.0d0)) + (0.008333333333333333d0 * (x_m ** 5.0d0))
    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-7) {
		tmp = t_0;
	} else {
		tmp = (-0.16666666666666666 * Math.pow(x_m, 3.0)) + (0.008333333333333333 * Math.pow(x_m, 5.0));
	}
	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-7:
		tmp = t_0
	else:
		tmp = (-0.16666666666666666 * math.pow(x_m, 3.0)) + (0.008333333333333333 * math.pow(x_m, 5.0))
	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-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.16666666666666666 * (x_m ^ 3.0)) + Float64(0.008333333333333333 * (x_m ^ 5.0)));
	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-7)
		tmp = t_0;
	else
		tmp = (-0.16666666666666666 * (x_m ^ 3.0)) + (0.008333333333333333 * (x_m ^ 5.0));
	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-7], t$95$0, N[(N[(-0.16666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.008333333333333333 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -9.9999999999999995e-8

   1. Initial program 90.3%

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

   if -9.9999999999999995e-8 < (-.f64 (sin.f64 x) x)

   1. Initial program 70.8%

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

   3. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + 0.008333333333333333 \cdot {x}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\sin x - x\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + 0.008333333333333333 \cdot {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(-0.16666666666666666 \cdot {x\_m}^{3} + \left(-0.0001984126984126984 \cdot {x\_m}^{7} + 0.008333333333333333 \cdot {x\_m}^{5}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (+
   (* -0.16666666666666666 (pow x_m 3.0))
   (+
    (* -0.0001984126984126984 (pow x_m 7.0))
    (* 0.008333333333333333 (pow x_m 5.0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((-0.16666666666666666 * pow(x_m, 3.0)) + ((-0.0001984126984126984 * pow(x_m, 7.0)) + (0.008333333333333333 * pow(x_m, 5.0))));
}

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 * (((-0.16666666666666666d0) * (x_m ** 3.0d0)) + (((-0.0001984126984126984d0) * (x_m ** 7.0d0)) + (0.008333333333333333d0 * (x_m ** 5.0d0))))
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 * ((-0.16666666666666666 * Math.pow(x_m, 3.0)) + ((-0.0001984126984126984 * Math.pow(x_m, 7.0)) + (0.008333333333333333 * Math.pow(x_m, 5.0))));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((-0.16666666666666666 * math.pow(x_m, 3.0)) + ((-0.0001984126984126984 * math.pow(x_m, 7.0)) + (0.008333333333333333 * math.pow(x_m, 5.0))))

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 ^ 3.0)) + Float64(Float64(-0.0001984126984126984 * (x_m ^ 7.0)) + Float64(0.008333333333333333 * (x_m ^ 5.0)))))
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 ^ 3.0)) + ((-0.0001984126984126984 * (x_m ^ 7.0)) + (0.008333333333333333 * (x_m ^ 5.0))));
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[(N[(-0.16666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0001984126984126984 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.008333333333333333 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in x around 0 98.1%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(-0.0001984126984126984 \cdot {x}^{7} + 0.008333333333333333 \cdot {x}^{5}\right)} \]

4. Final simplification 98.1%

   \[\leadsto -0.16666666666666666 \cdot {x}^{3} + \left(-0.0001984126984126984 \cdot {x}^{7} + 0.008333333333333333 \cdot {x}^{5}\right) \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(-0.0001984126984126984 \cdot {x\_m}^{7} + \left(-0.16666666666666666 \cdot {x\_m}^{3} + 0.008333333333333333 \cdot {x\_m}^{5}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (+
   (* -0.0001984126984126984 (pow x_m 7.0))
   (+
    (* -0.16666666666666666 (pow x_m 3.0))
    (* 0.008333333333333333 (pow x_m 5.0))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((-0.0001984126984126984 * pow(x_m, 7.0)) + ((-0.16666666666666666 * pow(x_m, 3.0)) + (0.008333333333333333 * pow(x_m, 5.0))));
}

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 * (((-0.0001984126984126984d0) * (x_m ** 7.0d0)) + (((-0.16666666666666666d0) * (x_m ** 3.0d0)) + (0.008333333333333333d0 * (x_m ** 5.0d0))))
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 * ((-0.0001984126984126984 * Math.pow(x_m, 7.0)) + ((-0.16666666666666666 * Math.pow(x_m, 3.0)) + (0.008333333333333333 * Math.pow(x_m, 5.0))));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((-0.0001984126984126984 * math.pow(x_m, 7.0)) + ((-0.16666666666666666 * math.pow(x_m, 3.0)) + (0.008333333333333333 * math.pow(x_m, 5.0))))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(-0.0001984126984126984 * (x_m ^ 7.0)) + Float64(Float64(-0.16666666666666666 * (x_m ^ 3.0)) + Float64(0.008333333333333333 * (x_m ^ 5.0)))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((-0.0001984126984126984 * (x_m ^ 7.0)) + ((-0.16666666666666666 * (x_m ^ 3.0)) + (0.008333333333333333 * (x_m ^ 5.0))));
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[(N[(-0.0001984126984126984 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.008333333333333333 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

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

   1. add-cube-cbrt 71.4%

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

   2. pow3 71.4%

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

4. Applied egg-rr 71.4%

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

5. Taylor expanded in x around 0 97.5%

   \[\leadsto {\left(\sqrt[3]{\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(-0.0001984126984126984 \cdot {x}^{7} + 0.008333333333333333 \cdot {x}^{5}\right)}}\right)}^{3} \]
6. Step-by-step derivation

   1. rem-cube-cbrt 98.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(-0.0001984126984126984 \cdot {x}^{7} + 0.008333333333333333 \cdot {x}^{5}\right)} \]

   2. +-commutative 98.1%

      \[\leadsto -0.16666666666666666 \cdot {x}^{3} + \color{blue}{\left(0.008333333333333333 \cdot {x}^{5} + -0.0001984126984126984 \cdot {x}^{7}\right)} \]

   3. associate-+r+ 98.1%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + 0.008333333333333333 \cdot {x}^{5}\right) + -0.0001984126984126984 \cdot {x}^{7}} \]

7. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + 0.008333333333333333 \cdot {x}^{5}\right) + -0.0001984126984126984 \cdot {x}^{7}} \]

8. Final simplification 98.1%

   \[\leadsto -0.0001984126984126984 \cdot {x}^{7} + \left(-0.16666666666666666 \cdot {x}^{3} + 0.008333333333333333 \cdot {x}^{5}\right) \]
9. Add Preprocessing

Alternative 6: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \sin x\_m - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x\_m}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (sin x_m) x_m)))
   (* x_s (if (<= t_0 -2e-12) t_0 (* -0.16666666666666666 (pow x_m 3.0))))))

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-12) {
		tmp = t_0;
	} else {
		tmp = -0.16666666666666666 * pow(x_m, 3.0);
	}
	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-12)) then
        tmp = t_0
    else
        tmp = (-0.16666666666666666d0) * (x_m ** 3.0d0)
    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-12) {
		tmp = t_0;
	} else {
		tmp = -0.16666666666666666 * Math.pow(x_m, 3.0);
	}
	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-12:
		tmp = t_0
	else:
		tmp = -0.16666666666666666 * math.pow(x_m, 3.0)
	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-12)
		tmp = t_0;
	else
		tmp = Float64(-0.16666666666666666 * (x_m ^ 3.0));
	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-12)
		tmp = t_0;
	else
		tmp = -0.16666666666666666 * (x_m ^ 3.0);
	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-12], t$95$0, N[(-0.16666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 x) x) < -1.99999999999999996e-12

   1. Initial program 86.3%

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

   if -1.99999999999999996e-12 < (-.f64 (sin.f64 x) x)

   1. Initial program 70.9%

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

   3. Taylor expanded in x around 0 99.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin x - x \leq -2 \cdot 10^{-12}:\\ \;\;\;\;\sin x - x\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% 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 1 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 71.5%

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

3. Final simplification 71.5%

   \[\leadsto \sin x - x \]
4. Add Preprocessing

Alternative 8: 6.4% 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 1 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 71.5%

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

3. Taylor expanded in x around inf 6.8%

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

   1. neg-mul-1 6.8%

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

5. Simplified 6.8%

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

6. Final simplification 6.8%

   \[\leadsto -x \]
7. Add Preprocessing

Alternative 9: 66.5% accurate, 103.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s 0.0))

C

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

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 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 * 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 * 0.0), $MachinePrecision]

Derivation

1. Initial program 71.5%

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

   1. add-log-exp 71.2%

      \[\leadsto \color{blue}{\log \left(e^{\sin x - x}\right)} \]

4. Applied egg-rr 71.2%

   \[\leadsto \color{blue}{\log \left(e^{\sin x - x}\right)} \]

5. Taylor expanded in x around 0 67.5%

   \[\leadsto \log \color{blue}{1} \]

6. Final simplification 67.5%

   \[\leadsto 0 \]
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 2024039 
(FPCore (x)
  :name "bug500 (missed optimization)"
  :precision binary64
  :pre (and (< -1000.0 x) (< x 1000.0))

  :herbie-target
  (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))