bug500 (missed optimization)

Percentage Accurate: 69.8% → 98.9%

Time: 10.8s

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: 69.8% 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

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

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) t_0))
     (+
      (* 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) / t_0)) + ((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) / t_0)) + ((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(0.0002314814814814815 * Float64((x ^ 5.0) / t_0)) + 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[(0.0002314814814814815 * N[(N[Power[x, 5.0], $MachinePrecision] / t$95$0), $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 69.5%

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

   1. add-cbrt-cube 69.5%

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

   2. pow1/3 67.8%

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

   3. pow3 67.8%

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

3. Applied egg-rr 67.8%

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

4. Taylor expanded in x around 0 98.8%

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

   1. cancel-sign-sub-inv 98.8%

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

   2. metadata-eval 98.8%

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

   3. pow-flip 98.8%

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

   4. metadata-eval 98.8%

      \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.0002314814814814815 \cdot \frac{{x}^{5}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot \left(8.083480305702528 \cdot 10^{-7} + -1.2403628765940151 \cdot 10^{-11} \cdot {\left(\sqrt[3]{\color{blue}{0.027777777777777776 \cdot -0.16666666666666666}}\right)}^{\left(-6\right)}\right)}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]

   5. metadata-eval 98.8%

      \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.0002314814814814815 \cdot \frac{{x}^{5}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot \left(8.083480305702528 \cdot 10^{-7} + -1.2403628765940151 \cdot 10^{-11} \cdot {\left(\sqrt[3]{\color{blue}{\left(-0.16666666666666666 \cdot -0.16666666666666666\right)} \cdot -0.16666666666666666}\right)}^{\left(-6\right)}\right)}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]

   6. add-cbrt-cube 98.8%

      \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.0002314814814814815 \cdot \frac{{x}^{5}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot \left(8.083480305702528 \cdot 10^{-7} + -1.2403628765940151 \cdot 10^{-11} \cdot {\color{blue}{-0.16666666666666666}}^{\left(-6\right)}\right)}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]

   7. metadata-eval 98.8%

      \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.0002314814814814815 \cdot \frac{{x}^{5}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot \left(8.083480305702528 \cdot 10^{-7} + -1.2403628765940151 \cdot 10^{-11} \cdot {-0.16666666666666666}^{\color{blue}{-6}}\right)}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]

   8. metadata-eval 98.8%

      \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.0002314814814814815 \cdot \frac{{x}^{5}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot \left(8.083480305702528 \cdot 10^{-7} + -1.2403628765940151 \cdot 10^{-11} \cdot \color{blue}{46656}\right)}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]

   9. metadata-eval 98.8%

      \[\leadsto -5.5114638447971785 \cdot 10^{-6} \cdot \frac{{x}^{7}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.0002314814814814815 \cdot \frac{{x}^{5}}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + \left(0.3333333333333333 \cdot \frac{{x}^{9} \cdot \left(8.083480305702528 \cdot 10^{-7} + \color{blue}{-5.787037037037037 \cdot 10^{-7}}\right)}{{\left(\sqrt[3]{-0.004629629629629629}\right)}^{2}} + {x}^{3} \cdot \sqrt[3]{-0.004629629629629629}\right)\right) \]

   10. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 98.8% 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 69.5%

   \[\sin x - x \]

2. Taylor expanded in x around 0 98.7%

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

3. Final simplification 98.7%

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

Alternative 3: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left({x}^{3} \cdot -0.0001349206349206349 + x \cdot -0.007857142857142858\right) + \left(0.3 \cdot \frac{-1}{x} + 6 \cdot \frac{-1}{{x}^{3}}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (+
   (+ (* (pow x 3.0) -0.0001349206349206349) (* x -0.007857142857142858))
   (+ (* 0.3 (/ -1.0 x)) (* 6.0 (/ -1.0 (pow x 3.0)))))))

C

double code(double x) {
	return 1.0 / (((pow(x, 3.0) * -0.0001349206349206349) + (x * -0.007857142857142858)) + ((0.3 * (-1.0 / x)) + (6.0 * (-1.0 / pow(x, 3.0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((((x ** 3.0d0) * (-0.0001349206349206349d0)) + (x * (-0.007857142857142858d0))) + ((0.3d0 * ((-1.0d0) / x)) + (6.0d0 * ((-1.0d0) / (x ** 3.0d0)))))
end function

Java

public static double code(double x) {
	return 1.0 / (((Math.pow(x, 3.0) * -0.0001349206349206349) + (x * -0.007857142857142858)) + ((0.3 * (-1.0 / x)) + (6.0 * (-1.0 / Math.pow(x, 3.0)))));
}

Python

def code(x):
	return 1.0 / (((math.pow(x, 3.0) * -0.0001349206349206349) + (x * -0.007857142857142858)) + ((0.3 * (-1.0 / x)) + (6.0 * (-1.0 / math.pow(x, 3.0)))))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64((x ^ 3.0) * -0.0001349206349206349) + Float64(x * -0.007857142857142858)) + Float64(Float64(0.3 * Float64(-1.0 / x)) + Float64(6.0 * Float64(-1.0 / (x ^ 3.0))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((((x ^ 3.0) * -0.0001349206349206349) + (x * -0.007857142857142858)) + ((0.3 * (-1.0 / x)) + (6.0 * (-1.0 / (x ^ 3.0)))));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.0001349206349206349), $MachinePrecision] + N[(x * -0.007857142857142858), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(-1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

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

   1. add-cbrt-cube 69.5%

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

   2. pow1/3 67.8%

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

   3. pow3 67.8%

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

3. Applied egg-rr 67.8%

   \[\leadsto \color{blue}{{\left({\left(\sin x - x\right)}^{3}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 69.5%

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

   2. rem-cbrt-cube 69.5%

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

   3. flip3-- 23.0%

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

   4. unpow2 23.0%

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

   5. distribute-rgt-in 23.0%

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

   6. clear-num 23.0%

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

   7. clear-num 23.0%

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

   8. unpow2 23.0%

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

   9. distribute-rgt-in 23.0%

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

   10. flip3-- 69.5%

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

5. Applied egg-rr 69.5%

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

6. Taylor expanded in x around 0 98.4%

   \[\leadsto \frac{1}{\color{blue}{\left(-0.007857142857142858 \cdot x + -0.0001349206349206349 \cdot {x}^{3}\right) - \left(0.3 \cdot \frac{1}{x} + 6 \cdot \frac{1}{{x}^{3}}\right)}} \]

7. Final simplification 98.4%

   \[\leadsto \frac{1}{\left({x}^{3} \cdot -0.0001349206349206349 + x \cdot -0.007857142857142858\right) + \left(0.3 \cdot \frac{-1}{x} + 6 \cdot \frac{-1}{{x}^{3}}\right)} \]

Alternative 4: 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 69.5%

   \[\sin x - x \]

2. Taylor expanded in x around 0 98.4%

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

3. Final simplification 98.4%

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

Alternative 5: 98.3% 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 69.5%

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

   1. add-cbrt-cube 69.5%

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

   2. pow1/3 67.8%

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

   3. pow3 67.8%

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

3. Applied egg-rr 67.8%

   \[\leadsto \color{blue}{{\left({\left(\sin x - x\right)}^{3}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 69.5%

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

   2. rem-cbrt-cube 69.5%

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

   3. flip3-- 23.0%

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

   4. unpow2 23.0%

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

   5. distribute-rgt-in 23.0%

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

   6. clear-num 23.0%

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

   7. clear-num 23.0%

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

   8. unpow2 23.0%

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

   9. distribute-rgt-in 23.0%

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

   10. flip3-- 69.5%

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

5. Applied egg-rr 69.5%

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

6. Taylor expanded in x around 0 98.2%

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

   1. *-commutative 98.2%

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

   2. associate-*r/ 98.2%

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

   3. metadata-eval 98.2%

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

   4. associate-*r/ 98.3%

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

   5. metadata-eval 98.3%

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

8. Simplified 98.3%

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

9. Final simplification 98.3%

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

Alternative 6: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-0.3}{x} + \frac{-6}{{x}^{3}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. add-cbrt-cube 69.5%

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

   2. pow1/3 67.8%

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

   3. pow3 67.8%

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

3. Applied egg-rr 67.8%

   \[\leadsto \color{blue}{{\left({\left(\sin x - x\right)}^{3}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 69.5%

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

   2. rem-cbrt-cube 69.5%

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

   3. flip3-- 23.0%

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

   4. unpow2 23.0%

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

   5. distribute-rgt-in 23.0%

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

   6. clear-num 23.0%

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

   7. clear-num 23.0%

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

   8. unpow2 23.0%

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

   9. distribute-rgt-in 23.0%

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

   10. flip3-- 69.5%

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

5. Applied egg-rr 69.5%

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

6. Taylor expanded in x around 0 98.0%

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

   1. distribute-neg-in 98.0%

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

   2. associate-*r/ 98.0%

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

   3. metadata-eval 98.0%

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

   4. distribute-neg-frac 98.0%

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

   5. metadata-eval 98.0%

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

   6. associate-*r/ 98.0%

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

   7. metadata-eval 98.0%

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

   8. distribute-neg-frac 98.0%

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

   9. metadata-eval 98.0%

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

8. Simplified 98.0%

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

9. Final simplification 98.0%

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

Alternative 7: 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 69.5%

   \[\sin x - x \]

2. Taylor expanded in x around 0 98.0%

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

3. Final simplification 98.0%

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

Alternative 8: 69.8% 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 69.5%

   \[\sin x - x \]

2. Final simplification 69.5%

   \[\leadsto \sin x - x \]

Alternative 9: 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 69.5%

   \[\sin x - x \]

2. Taylor expanded in x around inf 6.3%

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

   1. neg-mul-1 6.3%

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

4. Simplified 6.3%

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

5. Final simplification 6.3%

   \[\leadsto -x \]

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 2023293 
(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))