bug500 (missed optimization)

Percentage Accurate: 69.9% → 98.8%

Time: 15.9s

Alternatives: 8

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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ -0.16666666666666666 \cdot {x}^{3} + \left(-0.0001984126984126984 \cdot {x}^{7} + 0.008333333333333333 \cdot {x}^{5}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* -0.16666666666666666 (pow x 3.0))
  (+
   (* -0.0001984126984126984 (pow x 7.0))
   (* 0.008333333333333333 (pow x 5.0)))))

C

double code(double x) {
	return (-0.16666666666666666 * pow(x, 3.0)) + ((-0.0001984126984126984 * pow(x, 7.0)) + (0.008333333333333333 * pow(x, 5.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-0.16666666666666666d0) * (x ** 3.0d0)) + (((-0.0001984126984126984d0) * (x ** 7.0d0)) + (0.008333333333333333d0 * (x ** 5.0d0)))
end function

Java

public static double code(double x) {
	return (-0.16666666666666666 * Math.pow(x, 3.0)) + ((-0.0001984126984126984 * Math.pow(x, 7.0)) + (0.008333333333333333 * Math.pow(x, 5.0)));
}

Python

def code(x):
	return (-0.16666666666666666 * math.pow(x, 3.0)) + ((-0.0001984126984126984 * math.pow(x, 7.0)) + (0.008333333333333333 * math.pow(x, 5.0)))

Julia

function code(x)
	return Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(Float64(-0.0001984126984126984 * (x ^ 7.0)) + Float64(0.008333333333333333 * (x ^ 5.0))))
end

MATLAB

function tmp = code(x)
	tmp = (-0.16666666666666666 * (x ^ 3.0)) + ((-0.0001984126984126984 * (x ^ 7.0)) + (0.008333333333333333 * (x ^ 5.0)));
end

Wolfram

code[x_] := N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.0001984126984126984 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.008333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.16666666666666666 \cdot {x}^{3} + 0.008333333333333333 \cdot {x}^{5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (* -0.16666666666666666 (pow x 3.0)) (* 0.008333333333333333 (pow x 5.0))))

C

double code(double x) {
	return (-0.16666666666666666 * pow(x, 3.0)) + (0.008333333333333333 * pow(x, 5.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-0.16666666666666666d0) * (x ** 3.0d0)) + (0.008333333333333333d0 * (x ** 5.0d0))
end function

Java

public static double code(double x) {
	return (-0.16666666666666666 * Math.pow(x, 3.0)) + (0.008333333333333333 * Math.pow(x, 5.0));
}

Python

def code(x):
	return (-0.16666666666666666 * math.pow(x, 3.0)) + (0.008333333333333333 * math.pow(x, 5.0))

Julia

function code(x)
	return Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(0.008333333333333333 * (x ^ 5.0)))
end

MATLAB

function tmp = code(x)
	tmp = (-0.16666666666666666 * (x ^ 3.0)) + (0.008333333333333333 * (x ^ 5.0));
end

Wolfram

code[x_] := N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.008333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in x around 0 98.9%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + 0.008333333333333333 \cdot {x}^{5}} \]

4. Final simplification 98.9%

   \[\leadsto -0.16666666666666666 \cdot {x}^{3} + 0.008333333333333333 \cdot {x}^{5} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot -0.007857142857142858 - \left(\frac{0.3}{x} + \frac{6}{{x}^{3}}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (- (* x -0.007857142857142858) (+ (/ 0.3 x) (/ 6.0 (pow x 3.0))))))

C

double code(double x) {
	return 1.0 / ((x * -0.007857142857142858) - ((0.3 / x) + (6.0 / pow(x, 3.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((x * (-0.007857142857142858d0)) - ((0.3d0 / x) + (6.0d0 / (x ** 3.0d0))))
end function

Java

public static double code(double x) {
	return 1.0 / ((x * -0.007857142857142858) - ((0.3 / x) + (6.0 / Math.pow(x, 3.0))));
}

Python

def code(x):
	return 1.0 / ((x * -0.007857142857142858) - ((0.3 / x) + (6.0 / math.pow(x, 3.0))))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(x * -0.007857142857142858) - Float64(Float64(0.3 / x) + Float64(6.0 / (x ^ 3.0)))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((x * -0.007857142857142858) - ((0.3 / x) + (6.0 / (x ^ 3.0))));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(x * -0.007857142857142858), $MachinePrecision] - N[(N[(0.3 / x), $MachinePrecision] + N[(6.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

   1. flip3-- 21.5%

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

   2. clear-num 21.5%

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

   3. +-commutative 21.5%

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

   4. distribute-rgt-out 21.5%

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

   5. +-commutative 21.5%

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

   6. fma-def 21.5%

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

   7. +-commutative 21.5%

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

   8. pow2 21.5%

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

4. Applied egg-rr 21.5%

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

5. Taylor expanded in x around 0 98.9%

   \[\leadsto \frac{1}{\color{blue}{-0.007857142857142858 \cdot x - \left(0.3 \cdot \frac{1}{x} + 6 \cdot \frac{1}{{x}^{3}}\right)}} \]
6. Step-by-step derivation

   1. *-commutative 98.9%

      \[\leadsto \frac{1}{\color{blue}{x \cdot -0.007857142857142858} - \left(0.3 \cdot \frac{1}{x} + 6 \cdot \frac{1}{{x}^{3}}\right)} \]

   2. associate-*r/ 98.9%

      \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \left(\color{blue}{\frac{0.3 \cdot 1}{x}} + 6 \cdot \frac{1}{{x}^{3}}\right)} \]

   3. metadata-eval 98.9%

      \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \left(\frac{\color{blue}{0.3}}{x} + 6 \cdot \frac{1}{{x}^{3}}\right)} \]

   4. associate-*r/ 98.9%

      \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \left(\frac{0.3}{x} + \color{blue}{\frac{6 \cdot 1}{{x}^{3}}}\right)} \]

   5. metadata-eval 98.9%

      \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \left(\frac{0.3}{x} + \frac{\color{blue}{6}}{{x}^{3}}\right)} \]

7. Simplified 98.9%

   \[\leadsto \frac{1}{\color{blue}{x \cdot -0.007857142857142858 - \left(\frac{0.3}{x} + \frac{6}{{x}^{3}}\right)}} \]

8. Final simplification 98.9%

   \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \left(\frac{0.3}{x} + \frac{6}{{x}^{3}}\right)} \]
9. Add Preprocessing

Alternative 4: 98.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot -0.007857142857142858 - \left(\frac{0.3}{x} + \frac{1}{x} \cdot \frac{\frac{6}{x}}{x}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (- (* x -0.007857142857142858) (+ (/ 0.3 x) (* (/ 1.0 x) (/ (/ 6.0 x) x))))))

C

double code(double x) {
	return 1.0 / ((x * -0.007857142857142858) - ((0.3 / x) + ((1.0 / x) * ((6.0 / x) / x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((x * (-0.007857142857142858d0)) - ((0.3d0 / x) + ((1.0d0 / x) * ((6.0d0 / x) / x))))
end function

Java

public static double code(double x) {
	return 1.0 / ((x * -0.007857142857142858) - ((0.3 / x) + ((1.0 / x) * ((6.0 / x) / x))));
}

Python

def code(x):
	return 1.0 / ((x * -0.007857142857142858) - ((0.3 / x) + ((1.0 / x) * ((6.0 / x) / x))))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(x * -0.007857142857142858) - Float64(Float64(0.3 / x) + Float64(Float64(1.0 / x) * Float64(Float64(6.0 / x) / x)))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((x * -0.007857142857142858) - ((0.3 / x) + ((1.0 / x) * ((6.0 / x) / x))));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(x * -0.007857142857142858), $MachinePrecision] - N[(N[(0.3 / x), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * N[(N[(6.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

   1. flip3-- 21.5%

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

   2. clear-num 21.5%

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

   3. +-commutative 21.5%

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

   4. distribute-rgt-out 21.5%

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

   5. +-commutative 21.5%

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

   6. fma-def 21.5%

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

   7. +-commutative 21.5%

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

   8. pow2 21.5%

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

4. Applied egg-rr 21.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(x, x + \sin x, {\sin x}^{2}\right)}{{\sin x}^{3} - {x}^{3}}}} \]
5. Step-by-step derivation

   1. expm1-log1p-u 20.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x + \sin x, {\sin x}^{2}\right)}{{\sin x}^{3} - {x}^{3}}\right)\right)}} \]

   2. expm1-udef 20.6%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, x + \sin x, {\sin x}^{2}\right)}{{\sin x}^{3} - {x}^{3}}\right)} - 1}} \]

6. Applied egg-rr 64.3%

   \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sin x - x}\right)} - 1}} \]
7. Step-by-step derivation

   1. expm1-def 64.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sin x - x}\right)\right)}} \]

   2. expm1-log1p 65.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin x - x}}} \]

8. Simplified 65.2%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin x - x}}} \]

9. Taylor expanded in x around 0 98.9%

   \[\leadsto \frac{1}{\color{blue}{-0.007857142857142858 \cdot x - \left(0.3 \cdot \frac{1}{x} + 6 \cdot \frac{1}{{x}^{3}}\right)}} \]
10. Step-by-step derivation

    1. *-commutative 98.9%

       \[\leadsto \frac{1}{\color{blue}{x \cdot -0.007857142857142858} - \left(0.3 \cdot \frac{1}{x} + 6 \cdot \frac{1}{{x}^{3}}\right)} \]

    2. associate-*r/ 98.9%

       \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \left(\color{blue}{\frac{0.3 \cdot 1}{x}} + 6 \cdot \frac{1}{{x}^{3}}\right)} \]

    3. metadata-eval 98.9%

       \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \left(\frac{\color{blue}{0.3}}{x} + 6 \cdot \frac{1}{{x}^{3}}\right)} \]

    4. +-commutative 98.9%

       \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \color{blue}{\left(6 \cdot \frac{1}{{x}^{3}} + \frac{0.3}{x}\right)}} \]

    5. associate-*r/ 98.9%

       \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \left(\color{blue}{\frac{6 \cdot 1}{{x}^{3}}} + \frac{0.3}{x}\right)} \]

    6. metadata-eval 98.9%

       \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \left(\frac{\color{blue}{6}}{{x}^{3}} + \frac{0.3}{x}\right)} \]

11. Simplified 98.9%

    \[\leadsto \frac{1}{\color{blue}{x \cdot -0.007857142857142858 - \left(\frac{6}{{x}^{3}} + \frac{0.3}{x}\right)}} \]
12. Step-by-step derivation

    1. pow3 98.7%

       \[\leadsto \frac{1}{\frac{-0.3}{x} - \frac{6}{\color{blue}{\left(x \cdot x\right) \cdot x}}} \]

    2. associate-/l/ 98.7%

       \[\leadsto \frac{1}{\frac{-0.3}{x} - \color{blue}{\frac{\frac{6}{x}}{x \cdot x}}} \]

    3. *-un-lft-identity 98.7%

       \[\leadsto \frac{1}{\frac{-0.3}{x} - \frac{\color{blue}{1 \cdot \frac{6}{x}}}{x \cdot x}} \]

    4. times-frac 98.6%

       \[\leadsto \frac{1}{\frac{-0.3}{x} - \color{blue}{\frac{1}{x} \cdot \frac{\frac{6}{x}}{x}}} \]

13. Applied egg-rr 98.8%

    \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \left(\color{blue}{\frac{1}{x} \cdot \frac{\frac{6}{x}}{x}} + \frac{0.3}{x}\right)} \]

14. Final simplification 98.8%

    \[\leadsto \frac{1}{x \cdot -0.007857142857142858 - \left(\frac{0.3}{x} + \frac{1}{x} \cdot \frac{\frac{6}{x}}{x}\right)} \]
15. Add Preprocessing

Alternative 5: 98.0% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-0.3}{x} + \frac{\frac{6}{x}}{x} \cdot \frac{-1}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ (/ -0.3 x) (* (/ (/ 6.0 x) x) (/ -1.0 x)))))

C

double code(double x) {
	return 1.0 / ((-0.3 / x) + (((6.0 / x) / x) * (-1.0 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((-0.3d0) / x) + (((6.0d0 / x) / x) * ((-1.0d0) / x)))
end function

Java

public static double code(double x) {
	return 1.0 / ((-0.3 / x) + (((6.0 / x) / x) * (-1.0 / x)));
}

Python

def code(x):
	return 1.0 / ((-0.3 / x) + (((6.0 / x) / x) * (-1.0 / x)))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(-0.3 / x) + Float64(Float64(Float64(6.0 / x) / x) * Float64(-1.0 / x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((-0.3 / x) + (((6.0 / x) / x) * (-1.0 / x)));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(-0.3 / x), $MachinePrecision] + N[(N[(N[(6.0 / x), $MachinePrecision] / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

   1. flip3-- 21.5%

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

   2. clear-num 21.5%

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

   3. +-commutative 21.5%

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

   4. distribute-rgt-out 21.5%

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

   5. +-commutative 21.5%

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

   6. fma-def 21.5%

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

   7. +-commutative 21.5%

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

   8. pow2 21.5%

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

4. Applied egg-rr 21.5%

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

5. Taylor expanded in x around 0 98.7%

   \[\leadsto \frac{1}{\color{blue}{-\left(0.3 \cdot \frac{1}{x} + 6 \cdot \frac{1}{{x}^{3}}\right)}} \]
6. Step-by-step derivation

   1. distribute-neg-in 98.7%

      \[\leadsto \frac{1}{\color{blue}{\left(-0.3 \cdot \frac{1}{x}\right) + \left(-6 \cdot \frac{1}{{x}^{3}}\right)}} \]

   2. unsub-neg 98.7%

      \[\leadsto \frac{1}{\color{blue}{\left(-0.3 \cdot \frac{1}{x}\right) - 6 \cdot \frac{1}{{x}^{3}}}} \]

   3. associate-*r/ 98.7%

      \[\leadsto \frac{1}{\left(-\color{blue}{\frac{0.3 \cdot 1}{x}}\right) - 6 \cdot \frac{1}{{x}^{3}}} \]

   4. metadata-eval 98.7%

      \[\leadsto \frac{1}{\left(-\frac{\color{blue}{0.3}}{x}\right) - 6 \cdot \frac{1}{{x}^{3}}} \]

   5. distribute-neg-frac 98.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.3}{x}} - 6 \cdot \frac{1}{{x}^{3}}} \]

   6. metadata-eval 98.7%

      \[\leadsto \frac{1}{\frac{\color{blue}{-0.3}}{x} - 6 \cdot \frac{1}{{x}^{3}}} \]

   7. associate-*r/ 98.7%

      \[\leadsto \frac{1}{\frac{-0.3}{x} - \color{blue}{\frac{6 \cdot 1}{{x}^{3}}}} \]

   8. metadata-eval 98.7%

      \[\leadsto \frac{1}{\frac{-0.3}{x} - \frac{\color{blue}{6}}{{x}^{3}}} \]

7. Simplified 98.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{-0.3}{x} - \frac{6}{{x}^{3}}}} \]
8. Step-by-step derivation

   1. pow3 98.7%

      \[\leadsto \frac{1}{\frac{-0.3}{x} - \frac{6}{\color{blue}{\left(x \cdot x\right) \cdot x}}} \]

   2. associate-/l/ 98.7%

      \[\leadsto \frac{1}{\frac{-0.3}{x} - \color{blue}{\frac{\frac{6}{x}}{x \cdot x}}} \]

   3. *-un-lft-identity 98.7%

      \[\leadsto \frac{1}{\frac{-0.3}{x} - \frac{\color{blue}{1 \cdot \frac{6}{x}}}{x \cdot x}} \]

   4. times-frac 98.6%

      \[\leadsto \frac{1}{\frac{-0.3}{x} - \color{blue}{\frac{1}{x} \cdot \frac{\frac{6}{x}}{x}}} \]

9. Applied egg-rr 98.6%

   \[\leadsto \frac{1}{\frac{-0.3}{x} - \color{blue}{\frac{1}{x} \cdot \frac{\frac{6}{x}}{x}}} \]

10. Final simplification 98.6%

    \[\leadsto \frac{1}{\frac{-0.3}{x} + \frac{\frac{6}{x}}{x} \cdot \frac{-1}{x}} \]
11. Add Preprocessing

Alternative 6: 98.0% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* -0.16666666666666666 (* x (* x x))))

C

double code(double x) {
	return -0.16666666666666666 * (x * (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-0.16666666666666666d0) * (x * (x * x))
end function

Java

public static double code(double x) {
	return -0.16666666666666666 * (x * (x * x));
}

Python

def code(x):
	return -0.16666666666666666 * (x * (x * x))

Julia

function code(x)
	return Float64(-0.16666666666666666 * Float64(x * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = -0.16666666666666666 * (x * (x * x));
end

Wolfram

code[x_] := N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in x around 0 98.1%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3}} \]
4. Step-by-step derivation

   1. add-cube-cbrt 96.9%

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

   2. unpow-prod-down 96.9%

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

   3. pow2 96.9%

      \[\leadsto -0.16666666666666666 \cdot \left({\color{blue}{\left({\left(\sqrt[3]{x}\right)}^{2}\right)}}^{3} \cdot {\left(\sqrt[3]{x}\right)}^{3}\right) \]

   4. pow3 96.9%

      \[\leadsto -0.16666666666666666 \cdot \left({\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{3} \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}\right) \]

   5. add-cube-cbrt 97.2%

      \[\leadsto -0.16666666666666666 \cdot \left({\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{3} \cdot \color{blue}{x}\right) \]

5. Applied egg-rr 97.2%

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

   1. pow-pow 97.2%

      \[\leadsto -0.16666666666666666 \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(2 \cdot 3\right)}} \cdot x\right) \]

   2. pow1/3 49.6%

      \[\leadsto -0.16666666666666666 \cdot \left({\color{blue}{\left({x}^{0.3333333333333333}\right)}}^{\left(2 \cdot 3\right)} \cdot x\right) \]

   3. metadata-eval 49.6%

      \[\leadsto -0.16666666666666666 \cdot \left({\left({x}^{0.3333333333333333}\right)}^{\color{blue}{6}} \cdot x\right) \]

   4. pow-pow 98.1%

      \[\leadsto -0.16666666666666666 \cdot \left(\color{blue}{{x}^{\left(0.3333333333333333 \cdot 6\right)}} \cdot x\right) \]

   5. metadata-eval 98.1%

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

   6. unpow2 98.1%

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

7. Applied egg-rr 98.1%

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

8. Final simplification 98.1%

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

Alternative 7: 6.5% accurate, 51.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in x around inf 6.3%

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

   1. neg-mul-1 6.3%

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

5. Simplified 6.3%

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

6. Final simplification 6.3%

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

Alternative 8: 5.1% accurate, 103.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 65.2%

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

   1. flip3-- 21.5%

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

   2. clear-num 21.5%

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

   3. +-commutative 21.5%

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

   4. distribute-rgt-out 21.5%

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

   5. +-commutative 21.5%

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

   6. fma-def 21.5%

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

   7. +-commutative 21.5%

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

   8. pow2 21.5%

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

4. Applied egg-rr 21.5%

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

5. Taylor expanded in x around inf 6.3%

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

   1. add-sqr-sqrt 3.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{-1}{x}}} \cdot \sqrt{\frac{1}{\frac{-1}{x}}}} \]

   2. sqrt-unprod 46.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{-1}{x}} \cdot \frac{1}{\frac{-1}{x}}}} \]

   3. associate-/r/ 46.3%

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

   4. metadata-eval 46.3%

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

   5. associate-/r/ 46.3%

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

   6. metadata-eval 46.3%

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

   7. swap-sqr 46.3%

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

   8. metadata-eval 46.3%

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

   9. unpow2 46.3%

      \[\leadsto \sqrt{1 \cdot \color{blue}{{x}^{2}}} \]

   10. *-un-lft-identity 46.3%

       \[\leadsto \sqrt{\color{blue}{{x}^{2}}} \]

   11. sqrt-pow1 4.9%

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

   12. metadata-eval 4.9%

       \[\leadsto {x}^{\color{blue}{1}} \]

   13. expm1-log1p-u 4.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{1}\right)\right)} \]

   14. pow1 4.9%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x}\right)\right) \]

   15. expm1-udef 62.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1} \]

7. Applied egg-rr 62.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 4.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]

   2. expm1-log1p 4.9%

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

9. Simplified 4.9%

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

10. Final simplification 4.9%

    \[\leadsto x \]
11. 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 2024014 
(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))