invcot (example 3.9)

Percentage Accurate: 6.3% → 99.6%

Time: 15.7s

Alternatives: 4

Speedup: 35.7×

Specification

\[-0.026 < x \land x < 0.026\]

Math

\[\begin{array}{l} \\ \frac{1}{x} - \frac{1}{\tan x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))

C

double code(double x) {
	return (1.0 / x) - (1.0 / tan(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) - (1.0d0 / tan(x))
end function

Java

public static double code(double x) {
	return (1.0 / x) - (1.0 / Math.tan(x));
}

Python

def code(x):
	return (1.0 / x) - (1.0 / math.tan(x))

Julia

function code(x)
	return Float64(Float64(1.0 / x) - Float64(1.0 / tan(x)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / x) - (1.0 / tan(x));
end

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 6.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} - \frac{1}{\tan x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 x) (/ 1.0 (tan x))))

C

double code(double x) {
	return (1.0 / x) - (1.0 / tan(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) - (1.0d0 / tan(x))
end function

Java

public static double code(double x) {
	return (1.0 / x) - (1.0 / Math.tan(x));
}

Python

def code(x):
	return (1.0 / x) - (1.0 / math.tan(x))

Julia

function code(x)
	return Float64(Float64(1.0 / x) - Float64(1.0 / tan(x)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / x) - (1.0 / tan(x));
end

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{x \cdot 0.1111111111111111}{0.3333333333333333}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* 0.0021164021164021165 (pow x 5.0))
  (+
   (* 0.022222222222222223 (pow x 3.0))
   (/ (* x 0.1111111111111111) 0.3333333333333333))))

C

double code(double x) {
	return (0.0021164021164021165 * pow(x, 5.0)) + ((0.022222222222222223 * pow(x, 3.0)) + ((x * 0.1111111111111111) / 0.3333333333333333));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.0021164021164021165d0 * (x ** 5.0d0)) + ((0.022222222222222223d0 * (x ** 3.0d0)) + ((x * 0.1111111111111111d0) / 0.3333333333333333d0))
end function

Java

public static double code(double x) {
	return (0.0021164021164021165 * Math.pow(x, 5.0)) + ((0.022222222222222223 * Math.pow(x, 3.0)) + ((x * 0.1111111111111111) / 0.3333333333333333));
}

Python

def code(x):
	return (0.0021164021164021165 * math.pow(x, 5.0)) + ((0.022222222222222223 * math.pow(x, 3.0)) + ((x * 0.1111111111111111) / 0.3333333333333333))

Julia

function code(x)
	return Float64(Float64(0.0021164021164021165 * (x ^ 5.0)) + Float64(Float64(0.022222222222222223 * (x ^ 3.0)) + Float64(Float64(x * 0.1111111111111111) / 0.3333333333333333)))
end

MATLAB

function tmp = code(x)
	tmp = (0.0021164021164021165 * (x ^ 5.0)) + ((0.022222222222222223 * (x ^ 3.0)) + ((x * 0.1111111111111111) / 0.3333333333333333));
end

Wolfram

code[x_] := N[(N[(0.0021164021164021165 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.022222222222222223 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * 0.1111111111111111), $MachinePrecision] / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + 0.3333333333333333 \cdot x\right)} \]
4. Step-by-step derivation

   1. add0 99.5%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\left(0.3333333333333333 \cdot x + 0\right)}\right) \]

   2. fma-define 99.5%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 0\right)}\right) \]

   3. metadata-eval 99.5%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \mathsf{fma}\left(0.3333333333333333, x, \color{blue}{-0}\right)\right) \]

   4. fma-neg 99.5%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\left(0.3333333333333333 \cdot x - 0\right)}\right) \]

   5. *-commutative 99.5%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \left(\color{blue}{x \cdot 0.3333333333333333} - 0\right)\right) \]

   6. add-sqr-sqrt 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 0.3333333333333333 - 0\right)\right) \]

   7. associate-*l* 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \left(\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot 0.3333333333333333\right)} - 0\right)\right) \]

   8. fma-neg 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x} \cdot 0.3333333333333333, -0\right)}\right) \]

   9. metadata-eval 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \mathsf{fma}\left(\sqrt{x}, \sqrt{x} \cdot 0.3333333333333333, \color{blue}{0}\right)\right) \]

5. Applied egg-rr 51.1%

   \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x} \cdot 0.3333333333333333, 0\right)}\right) \]
6. Step-by-step derivation

   1. fma-undefine 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot 0.3333333333333333\right) + 0\right)}\right) \]

   2. +-rgt-identity 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot 0.3333333333333333\right)}\right) \]

   3. *-commutative 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \sqrt{x} \cdot \color{blue}{\left(0.3333333333333333 \cdot \sqrt{x}\right)}\right) \]

   4. *-commutative 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\left(0.3333333333333333 \cdot \sqrt{x}\right) \cdot \sqrt{x}}\right) \]

   5. associate-*l* 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{0.3333333333333333 \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \]

7. Simplified 51.1%

   \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{0.3333333333333333 \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \]
8. Step-by-step derivation

   1. metadata-eval 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\sqrt{0.1111111111111111}} \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right) \]

   2. sqrt-unprod 29.3%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \sqrt{0.1111111111111111} \cdot \color{blue}{\sqrt{x \cdot x}}\right) \]

   3. unpow2 29.3%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \sqrt{0.1111111111111111} \cdot \sqrt{\color{blue}{{x}^{2}}}\right) \]

   4. sqrt-prod 29.3%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\sqrt{0.1111111111111111 \cdot {x}^{2}}}\right) \]

   5. *-commutative 29.3%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \sqrt{\color{blue}{{x}^{2} \cdot 0.1111111111111111}}\right) \]

   6. add0 29.3%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\left(\sqrt{{x}^{2} \cdot 0.1111111111111111} + 0\right)}\right) \]

   7. flip-+ 25.9%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\frac{\sqrt{{x}^{2} \cdot 0.1111111111111111} \cdot \sqrt{{x}^{2} \cdot 0.1111111111111111} - 0 \cdot 0}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}}\right) \]

   8. metadata-eval 25.9%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\sqrt{{x}^{2} \cdot 0.1111111111111111} \cdot \sqrt{{x}^{2} \cdot 0.1111111111111111} - \color{blue}{0}}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}\right) \]

   9. fma-neg 25.9%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\color{blue}{\mathsf{fma}\left(\sqrt{{x}^{2} \cdot 0.1111111111111111}, \sqrt{{x}^{2} \cdot 0.1111111111111111}, -0\right)}}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}\right) \]

   10. metadata-eval 25.9%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\mathsf{fma}\left(\sqrt{{x}^{2} \cdot 0.1111111111111111}, \sqrt{{x}^{2} \cdot 0.1111111111111111}, \color{blue}{0}\right)}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}\right) \]

   11. fma-define 25.9%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\color{blue}{\sqrt{{x}^{2} \cdot 0.1111111111111111} \cdot \sqrt{{x}^{2} \cdot 0.1111111111111111} + 0}}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}\right) \]

   12. add0 25.9%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\color{blue}{\sqrt{{x}^{2} \cdot 0.1111111111111111} \cdot \sqrt{{x}^{2} \cdot 0.1111111111111111}}}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}\right) \]

   13. pow2 25.9%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\color{blue}{{\left(\sqrt{{x}^{2} \cdot 0.1111111111111111}\right)}^{2}}}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}\right) \]

   14. sqrt-prod 25.8%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{{\color{blue}{\left(\sqrt{{x}^{2}} \cdot \sqrt{0.1111111111111111}\right)}}^{2}}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}\right) \]

   15. unpow2 25.8%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{{\left(\sqrt{\color{blue}{x \cdot x}} \cdot \sqrt{0.1111111111111111}\right)}^{2}}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}\right) \]

   16. sqrt-unprod 25.1%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{{\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{0.1111111111111111}\right)}^{2}}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}\right) \]

   17. add-sqr-sqrt 25.8%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{{\left(\color{blue}{x} \cdot \sqrt{0.1111111111111111}\right)}^{2}}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}\right) \]

   18. metadata-eval 25.8%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{{\left(x \cdot \color{blue}{0.3333333333333333}\right)}^{2}}{\sqrt{{x}^{2} \cdot 0.1111111111111111} - 0}\right) \]

9. Applied egg-rr 53.7%

   \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\frac{{\left(x \cdot 0.3333333333333333\right)}^{2}}{x \cdot 0.3333333333333333}}\right) \]
10. Step-by-step derivation

    1. associate-/r* 53.5%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\frac{\frac{{\left(x \cdot 0.3333333333333333\right)}^{2}}{x}}{0.3333333333333333}}\right) \]

    2. unpow2 53.5%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\frac{\color{blue}{\left(x \cdot 0.3333333333333333\right) \cdot \left(x \cdot 0.3333333333333333\right)}}{x}}{0.3333333333333333}\right) \]

    3. swap-sqr 53.7%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}}{x}}{0.3333333333333333}\right) \]

    4. unpow2 53.7%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\frac{\color{blue}{{x}^{2}} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}{x}}{0.3333333333333333}\right) \]

    5. metadata-eval 53.7%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\frac{{x}^{2} \cdot \color{blue}{0.1111111111111111}}{x}}{0.3333333333333333}\right) \]

    6. associate-*l/ 53.9%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\color{blue}{\frac{{x}^{2}}{x} \cdot 0.1111111111111111}}{0.3333333333333333}\right) \]

    7. unpow2 53.9%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\frac{\color{blue}{x \cdot x}}{x} \cdot 0.1111111111111111}{0.3333333333333333}\right) \]

    8. associate-*r/ 99.7%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\color{blue}{\left(x \cdot \frac{x}{x}\right)} \cdot 0.1111111111111111}{0.3333333333333333}\right) \]

    9. *-inverses 99.7%

       \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\left(x \cdot \color{blue}{1}\right) \cdot 0.1111111111111111}{0.3333333333333333}\right) \]

    10. *-rgt-identity 99.7%

        \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{\color{blue}{x} \cdot 0.1111111111111111}{0.3333333333333333}\right) \]

11. Simplified 99.7%

    \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\frac{x \cdot 0.1111111111111111}{0.3333333333333333}}\right) \]

12. Final simplification 99.7%

    \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \frac{x \cdot 0.1111111111111111}{0.3333333333333333}\right) \]
13. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + x \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* 0.0021164021164021165 (pow x 5.0))
  (+ (* 0.022222222222222223 (pow x 3.0)) (* x 0.3333333333333333))))

C

double code(double x) {
	return (0.0021164021164021165 * pow(x, 5.0)) + ((0.022222222222222223 * pow(x, 3.0)) + (x * 0.3333333333333333));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.0021164021164021165d0 * (x ** 5.0d0)) + ((0.022222222222222223d0 * (x ** 3.0d0)) + (x * 0.3333333333333333d0))
end function

Java

public static double code(double x) {
	return (0.0021164021164021165 * Math.pow(x, 5.0)) + ((0.022222222222222223 * Math.pow(x, 3.0)) + (x * 0.3333333333333333));
}

Python

def code(x):
	return (0.0021164021164021165 * math.pow(x, 5.0)) + ((0.022222222222222223 * math.pow(x, 3.0)) + (x * 0.3333333333333333))

Julia

function code(x)
	return Float64(Float64(0.0021164021164021165 * (x ^ 5.0)) + Float64(Float64(0.022222222222222223 * (x ^ 3.0)) + Float64(x * 0.3333333333333333)))
end

MATLAB

function tmp = code(x)
	tmp = (0.0021164021164021165 * (x ^ 5.0)) + ((0.022222222222222223 * (x ^ 3.0)) + (x * 0.3333333333333333));
end

Wolfram

code[x_] := N[(N[(0.0021164021164021165 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.022222222222222223 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + 0.3333333333333333 \cdot x\right)} \]

4. Final simplification 99.5%

   \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + x \cdot 0.3333333333333333\right) \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.3333333333333333))

C

double code(double x) {
	return x * 0.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return x * 0.3333333333333333;
}

Python

def code(x):
	return x * 0.3333333333333333

Julia

function code(x)
	return Float64(x * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.3333333333333333;
end

Wolfram

code[x_] := N[(x * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 6.4%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot x} \]

4. Final simplification 99.0%

   \[\leadsto x \cdot 0.3333333333333333 \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x 3.0))

C

double code(double x) {
	return x / 3.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / 3.0d0
end function

Java

public static double code(double x) {
	return x / 3.0;
}

Python

def code(x):
	return x / 3.0

Julia

function code(x)
	return Float64(x / 3.0)
end

MATLAB

function tmp = code(x)
	tmp = x / 3.0;
end

Wolfram

code[x_] := N[(x / 3.0), $MachinePrecision]

Derivation

1. Initial program 6.4%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot x} \]
4. Step-by-step derivation

   1. add0 99.5%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\left(0.3333333333333333 \cdot x + 0\right)}\right) \]

   2. fma-define 99.5%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 0\right)}\right) \]

   3. metadata-eval 99.5%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \mathsf{fma}\left(0.3333333333333333, x, \color{blue}{-0}\right)\right) \]

   4. fma-neg 99.5%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\left(0.3333333333333333 \cdot x - 0\right)}\right) \]

   5. *-commutative 99.5%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \left(\color{blue}{x \cdot 0.3333333333333333} - 0\right)\right) \]

   6. add-sqr-sqrt 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 0.3333333333333333 - 0\right)\right) \]

   7. associate-*l* 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \left(\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot 0.3333333333333333\right)} - 0\right)\right) \]

   8. fma-neg 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x} \cdot 0.3333333333333333, -0\right)}\right) \]

   9. metadata-eval 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \mathsf{fma}\left(\sqrt{x}, \sqrt{x} \cdot 0.3333333333333333, \color{blue}{0}\right)\right) \]

5. Applied egg-rr 50.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x} \cdot 0.3333333333333333, 0\right)} \]
6. Step-by-step derivation

   1. fma-undefine 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot 0.3333333333333333\right) + 0\right)}\right) \]

   2. +-rgt-identity 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot 0.3333333333333333\right)}\right) \]

   3. *-commutative 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \sqrt{x} \cdot \color{blue}{\left(0.3333333333333333 \cdot \sqrt{x}\right)}\right) \]

   4. *-commutative 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{\left(0.3333333333333333 \cdot \sqrt{x}\right) \cdot \sqrt{x}}\right) \]

   5. associate-*l* 51.1%

      \[\leadsto 0.0021164021164021165 \cdot {x}^{5} + \left(0.022222222222222223 \cdot {x}^{3} + \color{blue}{0.3333333333333333 \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \]

7. Simplified 51.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)} \]
8. Step-by-step derivation

   1. metadata-eval 51.0%

      \[\leadsto \color{blue}{\sqrt{0.1111111111111111}} \cdot \left(\sqrt{x} \cdot \sqrt{x}\right) \]

   2. sqrt-unprod 29.1%

      \[\leadsto \sqrt{0.1111111111111111} \cdot \color{blue}{\sqrt{x \cdot x}} \]

   3. unpow2 29.1%

      \[\leadsto \sqrt{0.1111111111111111} \cdot \sqrt{\color{blue}{{x}^{2}}} \]

   4. sqrt-prod 29.1%

      \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot {x}^{2}}} \]

   5. *-commutative 29.1%

      \[\leadsto \sqrt{\color{blue}{{x}^{2} \cdot 0.1111111111111111}} \]

   6. add-cube-cbrt 28.6%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{{x}^{2} \cdot 0.1111111111111111}} \cdot \sqrt[3]{\sqrt{{x}^{2} \cdot 0.1111111111111111}}\right) \cdot \sqrt[3]{\sqrt{{x}^{2} \cdot 0.1111111111111111}}} \]

   7. pow3 28.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{{x}^{2} \cdot 0.1111111111111111}}\right)}^{3}} \]

   8. sqrt-prod 28.6%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{0.1111111111111111}}}\right)}^{3} \]

   9. unpow2 28.6%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\color{blue}{x \cdot x}} \cdot \sqrt{0.1111111111111111}}\right)}^{3} \]

   10. sqrt-unprod 50.2%

       \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{0.1111111111111111}}\right)}^{3} \]

   11. add-sqr-sqrt 97.2%

       \[\leadsto {\left(\sqrt[3]{\color{blue}{x} \cdot \sqrt{0.1111111111111111}}\right)}^{3} \]

   12. metadata-eval 97.2%

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

9. Applied egg-rr 97.2%

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

    1. rem-cube-cbrt 99.0%

       \[\leadsto \color{blue}{x \cdot 0.3333333333333333} \]

    2. metadata-eval 99.0%

       \[\leadsto x \cdot \color{blue}{\frac{1}{3}} \]

    3. div-inv 99.5%

       \[\leadsto \color{blue}{\frac{x}{3}} \]

11. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{\frac{x}{3}} \]

12. Final simplification 99.5%

    \[\leadsto \frac{x}{3} \]
13. Add Preprocessing

Developer target: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| < 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (< (fabs x) 0.026)
   (* (/ x 3.0) (+ 1.0 (/ (* x x) 15.0)))
   (- (/ 1.0 x) (/ 1.0 (tan x)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) < 0.026) {
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0));
	} else {
		tmp = (1.0 / x) - (1.0 / tan(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) < 0.026d0) then
        tmp = (x / 3.0d0) * (1.0d0 + ((x * x) / 15.0d0))
    else
        tmp = (1.0d0 / x) - (1.0d0 / tan(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) < 0.026) {
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0));
	} else {
		tmp = (1.0 / x) - (1.0 / Math.tan(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) < 0.026:
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0))
	else:
		tmp = (1.0 / x) - (1.0 / math.tan(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) < 0.026)
		tmp = Float64(Float64(x / 3.0) * Float64(1.0 + Float64(Float64(x * x) / 15.0)));
	else
		tmp = Float64(Float64(1.0 / x) - Float64(1.0 / tan(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) < 0.026)
		tmp = (x / 3.0) * (1.0 + ((x * x) / 15.0));
	else
		tmp = (1.0 / x) - (1.0 / tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Less[N[Abs[x], $MachinePrecision], 0.026], N[(N[(x / 3.0), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] / 15.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024034 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3.0) (+ 1.0 (/ (* x x) 15.0))) (- (/ 1.0 x) (/ 1.0 (tan x))))

  (- (/ 1.0 x) (/ 1.0 (tan x))))