bug500 (missed optimization)

Percentage Accurate: 69.2% → 98.9%

Time: 8.8s

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: 69.2% 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: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}\\ -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{t_0} + \left(\frac{0.0002314814814814815 \cdot {x}^{5}}{\sqrt[3]{2.143347050754458 \cdot 10^{-5}}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot 2.296443268665491 \cdot 10^{-7}}{t_0} + \sqrt[3]{-0.004629629629629629} \cdot {x}^{3}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (cbrt -0.004629629629629629) 2.0)))
   (+
    (* -5.5114638447971785e-6 (/ (pow x 7.0) t_0))
    (+
     (/ (* 0.0002314814814814815 (pow x 5.0)) (cbrt 2.143347050754458e-5))
     (+
      (* 0.3333333333333333 (/ (* (pow x 9.0) 2.296443268665491e-7) t_0))
      (* (cbrt -0.004629629629629629) (pow x 3.0)))))))

C

double code(double x) {
	double t_0 = pow(cbrt(-0.004629629629629629), 2.0);
	return (-5.5114638447971785e-6 * (pow(x, 7.0) / t_0)) + (((0.0002314814814814815 * pow(x, 5.0)) / cbrt(2.143347050754458e-5)) + ((0.3333333333333333 * ((pow(x, 9.0) * 2.296443268665491e-7) / t_0)) + (cbrt(-0.004629629629629629) * pow(x, 3.0))));
}

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.cbrt(-0.004629629629629629), 2.0);
	return (-5.5114638447971785e-6 * (Math.pow(x, 7.0) / t_0)) + (((0.0002314814814814815 * Math.pow(x, 5.0)) / Math.cbrt(2.143347050754458e-5)) + ((0.3333333333333333 * ((Math.pow(x, 9.0) * 2.296443268665491e-7) / t_0)) + (Math.cbrt(-0.004629629629629629) * Math.pow(x, 3.0))));
}

Julia

function code(x)
	t_0 = cbrt(-0.004629629629629629) ^ 2.0
	return Float64(Float64(-5.5114638447971785e-6 * Float64((x ^ 7.0) / t_0)) + Float64(Float64(Float64(0.0002314814814814815 * (x ^ 5.0)) / cbrt(2.143347050754458e-5)) + Float64(Float64(0.3333333333333333 * Float64(Float64((x ^ 9.0) * 2.296443268665491e-7) / t_0)) + Float64(cbrt(-0.004629629629629629) * (x ^ 3.0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[-0.004629629629629629, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(-5.5114638447971785e-6 * N[(N[Power[x, 7.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.0002314814814814815 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] / N[Power[2.143347050754458e-5, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(N[(N[Power[x, 9.0], $MachinePrecision] * 2.296443268665491e-7), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[-0.004629629629629629, 1/3], $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 67.6%

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

   1. add-cbrt-cube 67.6%

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

   2. flip-- 67.6%

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

   3. associate-*r/ 67.6%

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

   4. cbrt-div 67.6%

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

   5. pow2 67.6%

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

   6. pow2 67.6%

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

   7. pow2 67.6%

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

   8. +-commutative 67.6%

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

3. Applied egg-rr 67.6%

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

4. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{-5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.0002314814814814815 \cdot \frac{{x}^{5}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot \left(8.083480305702528 \cdot 10^{-7} - 1.2403628765940151 \cdot 10^{-11} \cdot \frac{1}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{6}}\right)}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right)} \]

6. Applied egg-rr 99.4%

   \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.0002314814814814815 \cdot \frac{{x}^{5}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot \color{blue}{2.296443268665491 \cdot 10^{-7}}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]
7. Step-by-step derivation

   1. associate-*r/ 99.4%

      \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(\color{blue}{\frac{0.0002314814814814815 \cdot {x}^{5}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot 2.296443268665491 \cdot 10^{-7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]

   2. unpow2 99.4%

      \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(\frac{0.0002314814814814815 \cdot {x}^{5}}{\color{blue}{\sqrt[3]{-0.004629629629629629} \cdot \sqrt[3]{-0.004629629629629629}}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot 2.296443268665491 \cdot 10^{-7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]

   3. cbrt-unprod 99.4%

      \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(\frac{0.0002314814814814815 \cdot {x}^{5}}{\color{blue}{\sqrt[3]{-0.004629629629629629 \cdot -0.004629629629629629}}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot 2.296443268665491 \cdot 10^{-7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]

   4. metadata-eval 99.4%

      \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(\frac{0.0002314814814814815 \cdot {x}^{5}}{\sqrt[3]{\color{blue}{2.143347050754458 \cdot 10^{-5}}}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot 2.296443268665491 \cdot 10^{-7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]

8. Applied egg-rr 99.4%

   \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(\color{blue}{\frac{0.0002314814814814815 \cdot {x}^{5}}{\sqrt[3]{2.143347050754458 \cdot 10^{-5}}}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot 2.296443268665491 \cdot 10^{-7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]

9. Final simplification 99.4%

   \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(\frac{0.0002314814814814815 \cdot {x}^{5}}{\sqrt[3]{2.143347050754458 \cdot 10^{-5}}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot 2.296443268665491 \cdot 10^{-7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \sqrt[3]{-0.004629629629629629} \cdot {x}^{3}\right)\right) \]

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {x}^{3} \cdot -0.16666666666666666 + \left({x}^{7} \cdot -0.0001984126984126984 + \left({x}^{9} \cdot 2.7557319223985893 \cdot 10^{-6} + {x}^{5} \cdot 0.008333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* (pow x 3.0) -0.16666666666666666)
  (+
   (* (pow x 7.0) -0.0001984126984126984)
   (+
    (* (pow x 9.0) 2.7557319223985893e-6)
    (* (pow x 5.0) 0.008333333333333333)))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x ** 3.0d0) * (-0.16666666666666666d0)) + (((x ** 7.0d0) * (-0.0001984126984126984d0)) + (((x ** 9.0d0) * 2.7557319223985893d-6) + ((x ** 5.0d0) * 0.008333333333333333d0)))
end function

Java

public static double code(double x) {
	return (Math.pow(x, 3.0) * -0.16666666666666666) + ((Math.pow(x, 7.0) * -0.0001984126984126984) + ((Math.pow(x, 9.0) * 2.7557319223985893e-6) + (Math.pow(x, 5.0) * 0.008333333333333333)));
}

Python

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

Julia

function code(x)
	return Float64(Float64((x ^ 3.0) * -0.16666666666666666) + Float64(Float64((x ^ 7.0) * -0.0001984126984126984) + Float64(Float64((x ^ 9.0) * 2.7557319223985893e-6) + Float64((x ^ 5.0) * 0.008333333333333333))))
end

MATLAB

function tmp = code(x)
	tmp = ((x ^ 3.0) * -0.16666666666666666) + (((x ^ 7.0) * -0.0001984126984126984) + (((x ^ 9.0) * 2.7557319223985893e-6) + ((x ^ 5.0) * 0.008333333333333333)));
end

Wolfram

code[x_] := N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(N[(N[Power[x, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + N[(N[(N[Power[x, 9.0], $MachinePrecision] * 2.7557319223985893e-6), $MachinePrecision] + N[(N[Power[x, 5.0], $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.6%

   \[\sin x - x \]

2. Taylor expanded in x around 0 99.4%

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

3. Final simplification 99.4%

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

Alternative 3: 98.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

   \[\sin x - x \]

2. Taylor expanded in x around 0 99.3%

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

3. Final simplification 99.3%

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

Alternative 4: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. add-cbrt-cube 67.6%

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

   2. flip-- 67.6%

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

   3. associate-*r/ 67.6%

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

   4. cbrt-div 67.6%

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

   5. pow2 67.6%

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

   6. pow2 67.6%

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

   7. pow2 67.6%

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

   8. +-commutative 67.6%

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

3. Applied egg-rr 67.6%

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

   1. clear-num 67.6%

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

   2. inv-pow 67.6%

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

5. Applied egg-rr 67.6%

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

6. Taylor expanded in x around 0 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-*r/ 99.0%

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

   3. metadata-eval 99.0%

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

   4. associate-*r/ 99.0%

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

   5. metadata-eval 99.0%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

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

Alternative 5: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

   \[\sin x - x \]

2. Taylor expanded in x around 0 98.9%

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

3. Final simplification 98.9%

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

Alternative 6: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

   \[\sin x - x \]

2. Taylor expanded in x around 0 98.3%

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

3. Final simplification 98.3%

   \[\leadsto {x}^{3} \cdot -0.16666666666666666 \]

Alternative 7: 69.2% 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]

Derivation

1. Initial program 67.6%

   \[\sin x - x \]

2. Final simplification 67.6%

   \[\leadsto \sin x - x \]

Alternative 8: 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 67.6%

   \[\sin x - x \]

2. Taylor expanded in x around inf 6.4%

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

   1. neg-mul-1 6.4%

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

4. Simplified 6.4%

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

5. Final simplification 6.4%

   \[\leadsto -x \]

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 2023318 
(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))