invcot (example 3.9)

Percentage Accurate: 6.4% → 99.5%

Time: 14.9s

Alternatives: 7

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: 6.4% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(1.0973936899862826 \cdot 10^{-5} \cdot {x}^{6} + 0.037037037037037035\right)}{0.1111111111111111 + \left({x}^{4} \cdot 0.0004938271604938272 - 0.007407407407407408 \cdot {x}^{2}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* x (+ (* 1.0973936899862826e-5 (pow x 6.0)) 0.037037037037037035))
  (+
   0.1111111111111111
   (-
    (* (pow x 4.0) 0.0004938271604938272)
    (* 0.007407407407407408 (pow x 2.0))))))

C

double code(double x) {
	return (x * ((1.0973936899862826e-5 * pow(x, 6.0)) + 0.037037037037037035)) / (0.1111111111111111 + ((pow(x, 4.0) * 0.0004938271604938272) - (0.007407407407407408 * pow(x, 2.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * ((1.0973936899862826d-5 * (x ** 6.0d0)) + 0.037037037037037035d0)) / (0.1111111111111111d0 + (((x ** 4.0d0) * 0.0004938271604938272d0) - (0.007407407407407408d0 * (x ** 2.0d0))))
end function

Java

public static double code(double x) {
	return (x * ((1.0973936899862826e-5 * Math.pow(x, 6.0)) + 0.037037037037037035)) / (0.1111111111111111 + ((Math.pow(x, 4.0) * 0.0004938271604938272) - (0.007407407407407408 * Math.pow(x, 2.0))));
}

Python

def code(x):
	return (x * ((1.0973936899862826e-5 * math.pow(x, 6.0)) + 0.037037037037037035)) / (0.1111111111111111 + ((math.pow(x, 4.0) * 0.0004938271604938272) - (0.007407407407407408 * math.pow(x, 2.0))))

Julia

function code(x)
	return Float64(Float64(x * Float64(Float64(1.0973936899862826e-5 * (x ^ 6.0)) + 0.037037037037037035)) / Float64(0.1111111111111111 + Float64(Float64((x ^ 4.0) * 0.0004938271604938272) - Float64(0.007407407407407408 * (x ^ 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (x * ((1.0973936899862826e-5 * (x ^ 6.0)) + 0.037037037037037035)) / (0.1111111111111111 + (((x ^ 4.0) * 0.0004938271604938272) - (0.007407407407407408 * (x ^ 2.0))));
end

Wolfram

code[x_] := N[(N[(x * N[(N[(1.0973936899862826e-5 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + 0.037037037037037035), $MachinePrecision]), $MachinePrecision] / N[(0.1111111111111111 + N[(N[(N[Power[x, 4.0], $MachinePrecision] * 0.0004938271604938272), $MachinePrecision] - N[(0.007407407407407408 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around 0 99.2%

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

   1. unpow2 99.2%

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

5. Applied egg-rr 99.2%

   \[\leadsto x \cdot \left(0.3333333333333333 + 0.022222222222222223 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. flip3-+ 97.8%

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

   3. associate-*l/ 98.1%

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

7. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\left(1.0973936899862826 \cdot 10^{-5} \cdot {x}^{6} + 0.037037037037037035\right) \cdot x}{0.1111111111111111 + \left({x}^{4} \cdot 0.0004938271604938272 - 0.007407407407407408 \cdot {x}^{2}\right)}} \]

8. Final simplification 99.4%

   \[\leadsto \frac{x \cdot \left(1.0973936899862826 \cdot 10^{-5} \cdot {x}^{6} + 0.037037037037037035\right)}{0.1111111111111111 + \left({x}^{4} \cdot 0.0004938271604938272 - 0.007407407407407408 \cdot {x}^{2}\right)} \]
9. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.037037037037037035}{0.1111111111111111 + \left({x}^{4} \cdot 0.0004938271604938272 - 0.007407407407407408 \cdot {x}^{2}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* x 0.037037037037037035)
  (+
   0.1111111111111111
   (-
    (* (pow x 4.0) 0.0004938271604938272)
    (* 0.007407407407407408 (pow x 2.0))))))

C

double code(double x) {
	return (x * 0.037037037037037035) / (0.1111111111111111 + ((pow(x, 4.0) * 0.0004938271604938272) - (0.007407407407407408 * pow(x, 2.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 0.037037037037037035d0) / (0.1111111111111111d0 + (((x ** 4.0d0) * 0.0004938271604938272d0) - (0.007407407407407408d0 * (x ** 2.0d0))))
end function

Java

public static double code(double x) {
	return (x * 0.037037037037037035) / (0.1111111111111111 + ((Math.pow(x, 4.0) * 0.0004938271604938272) - (0.007407407407407408 * Math.pow(x, 2.0))));
}

Python

def code(x):
	return (x * 0.037037037037037035) / (0.1111111111111111 + ((math.pow(x, 4.0) * 0.0004938271604938272) - (0.007407407407407408 * math.pow(x, 2.0))))

Julia

function code(x)
	return Float64(Float64(x * 0.037037037037037035) / Float64(0.1111111111111111 + Float64(Float64((x ^ 4.0) * 0.0004938271604938272) - Float64(0.007407407407407408 * (x ^ 2.0)))))
end

MATLAB

function tmp = code(x)
	tmp = (x * 0.037037037037037035) / (0.1111111111111111 + (((x ^ 4.0) * 0.0004938271604938272) - (0.007407407407407408 * (x ^ 2.0))));
end

Wolfram

code[x_] := N[(N[(x * 0.037037037037037035), $MachinePrecision] / N[(0.1111111111111111 + N[(N[(N[Power[x, 4.0], $MachinePrecision] * 0.0004938271604938272), $MachinePrecision] - N[(0.007407407407407408 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around 0 99.2%

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

   1. unpow2 99.2%

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

5. Applied egg-rr 99.2%

   \[\leadsto x \cdot \left(0.3333333333333333 + 0.022222222222222223 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. flip3-+ 97.8%

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

   3. associate-*l/ 98.1%

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

7. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\left(1.0973936899862826 \cdot 10^{-5} \cdot {x}^{6} + 0.037037037037037035\right) \cdot x}{0.1111111111111111 + \left({x}^{4} \cdot 0.0004938271604938272 - 0.007407407407407408 \cdot {x}^{2}\right)}} \]

8. Taylor expanded in x around 0 99.4%

   \[\leadsto \frac{\color{blue}{0.037037037037037035 \cdot x}}{0.1111111111111111 + \left({x}^{4} \cdot 0.0004938271604938272 - 0.007407407407407408 \cdot {x}^{2}\right)} \]
9. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \frac{\color{blue}{x \cdot 0.037037037037037035}}{0.1111111111111111 + \left({x}^{4} \cdot 0.0004938271604938272 - 0.007407407407407408 \cdot {x}^{2}\right)} \]

10. Simplified 99.4%

    \[\leadsto \frac{\color{blue}{x \cdot 0.037037037037037035}}{0.1111111111111111 + \left({x}^{4} \cdot 0.0004938271604938272 - 0.007407407407407408 \cdot {x}^{2}\right)} \]
11. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right) \cdot \frac{x}{0.3333333333333333 + {x}^{2} \cdot -0.022222222222222223} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (- 0.1111111111111111 (* (pow x 4.0) 0.0004938271604938272))
  (/ x (+ 0.3333333333333333 (* (pow x 2.0) -0.022222222222222223)))))

C

double code(double x) {
	return (0.1111111111111111 - (pow(x, 4.0) * 0.0004938271604938272)) * (x / (0.3333333333333333 + (pow(x, 2.0) * -0.022222222222222223)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.1111111111111111d0 - ((x ** 4.0d0) * 0.0004938271604938272d0)) * (x / (0.3333333333333333d0 + ((x ** 2.0d0) * (-0.022222222222222223d0))))
end function

Java

public static double code(double x) {
	return (0.1111111111111111 - (Math.pow(x, 4.0) * 0.0004938271604938272)) * (x / (0.3333333333333333 + (Math.pow(x, 2.0) * -0.022222222222222223)));
}

Python

def code(x):
	return (0.1111111111111111 - (math.pow(x, 4.0) * 0.0004938271604938272)) * (x / (0.3333333333333333 + (math.pow(x, 2.0) * -0.022222222222222223)))

Julia

function code(x)
	return Float64(Float64(0.1111111111111111 - Float64((x ^ 4.0) * 0.0004938271604938272)) * Float64(x / Float64(0.3333333333333333 + Float64((x ^ 2.0) * -0.022222222222222223))))
end

MATLAB

function tmp = code(x)
	tmp = (0.1111111111111111 - ((x ^ 4.0) * 0.0004938271604938272)) * (x / (0.3333333333333333 + ((x ^ 2.0) * -0.022222222222222223)));
end

Wolfram

code[x_] := N[(N[(0.1111111111111111 - N[(N[Power[x, 4.0], $MachinePrecision] * 0.0004938271604938272), $MachinePrecision]), $MachinePrecision] * N[(x / N[(0.3333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.022222222222222223), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around 0 99.2%

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

   1. unpow2 99.2%

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

5. Applied egg-rr 99.2%

   \[\leadsto x \cdot \left(0.3333333333333333 + 0.022222222222222223 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
6. Step-by-step derivation

   1. flip-+ 99.2%

      \[\leadsto x \cdot \color{blue}{\frac{0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot \left(x \cdot x\right)\right) \cdot \left(0.022222222222222223 \cdot \left(x \cdot x\right)\right)}{0.3333333333333333 - 0.022222222222222223 \cdot \left(x \cdot x\right)}} \]

   2. associate-*r/ 99.3%

      \[\leadsto \color{blue}{\frac{x \cdot \left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot \left(x \cdot x\right)\right) \cdot \left(0.022222222222222223 \cdot \left(x \cdot x\right)\right)\right)}{0.3333333333333333 - 0.022222222222222223 \cdot \left(x \cdot x\right)}} \]

   3. metadata-eval 99.3%

      \[\leadsto \frac{x \cdot \left(\color{blue}{0.1111111111111111} - \left(0.022222222222222223 \cdot \left(x \cdot x\right)\right) \cdot \left(0.022222222222222223 \cdot \left(x \cdot x\right)\right)\right)}{0.3333333333333333 - 0.022222222222222223 \cdot \left(x \cdot x\right)} \]

   4. pow2 99.3%

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

   5. pow2 99.3%

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

   6. *-commutative 99.3%

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

   7. *-commutative 99.3%

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

   8. swap-sqr 99.3%

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

   9. pow-prod-up 99.3%

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

   10. metadata-eval 99.3%

       \[\leadsto \frac{x \cdot \left(0.1111111111111111 - {x}^{\color{blue}{4}} \cdot \left(0.022222222222222223 \cdot 0.022222222222222223\right)\right)}{0.3333333333333333 - 0.022222222222222223 \cdot \left(x \cdot x\right)} \]

   11. metadata-eval 99.3%

       \[\leadsto \frac{x \cdot \left(0.1111111111111111 - {x}^{4} \cdot \color{blue}{0.0004938271604938272}\right)}{0.3333333333333333 - 0.022222222222222223 \cdot \left(x \cdot x\right)} \]

   12. pow2 99.3%

       \[\leadsto \frac{x \cdot \left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right)}{0.3333333333333333 - 0.022222222222222223 \cdot \color{blue}{{x}^{2}}} \]

   13. cancel-sign-sub-inv 99.3%

       \[\leadsto \frac{x \cdot \left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right)}{\color{blue}{0.3333333333333333 + \left(-0.022222222222222223\right) \cdot {x}^{2}}} \]

   14. metadata-eval 99.3%

       \[\leadsto \frac{x \cdot \left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right)}{0.3333333333333333 + \color{blue}{-0.022222222222222223} \cdot {x}^{2}} \]

7. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{x \cdot \left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right)}{0.3333333333333333 + -0.022222222222222223 \cdot {x}^{2}}} \]
8. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \frac{\color{blue}{\left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right) \cdot x}}{0.3333333333333333 + -0.022222222222222223 \cdot {x}^{2}} \]

   2. associate-/l* 99.4%

      \[\leadsto \color{blue}{\left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right) \cdot \frac{x}{0.3333333333333333 + -0.022222222222222223 \cdot {x}^{2}}} \]

   3. *-commutative 99.4%

      \[\leadsto \left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right) \cdot \frac{x}{0.3333333333333333 + \color{blue}{{x}^{2} \cdot -0.022222222222222223}} \]

9. Simplified 99.4%

   \[\leadsto \color{blue}{\left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right) \cdot \frac{x}{0.3333333333333333 + {x}^{2} \cdot -0.022222222222222223}} \]
10. Add Preprocessing

Alternative 4: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + \left(x \cdot x\right) \cdot \left(0.0021164021164021165 + \left(x \cdot x\right) \cdot 0.00021164021164021165\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   0.3333333333333333
   (*
    (pow x 2.0)
    (+
     0.022222222222222223
     (*
      (* x x)
      (+ 0.0021164021164021165 (* (* x x) 0.00021164021164021165))))))))

C

double code(double x) {
	return x * (0.3333333333333333 + (pow(x, 2.0) * (0.022222222222222223 + ((x * x) * (0.0021164021164021165 + ((x * x) * 0.00021164021164021165))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.3333333333333333d0 + ((x ** 2.0d0) * (0.022222222222222223d0 + ((x * x) * (0.0021164021164021165d0 + ((x * x) * 0.00021164021164021165d0))))))
end function

Java

public static double code(double x) {
	return x * (0.3333333333333333 + (Math.pow(x, 2.0) * (0.022222222222222223 + ((x * x) * (0.0021164021164021165 + ((x * x) * 0.00021164021164021165))))));
}

Python

def code(x):
	return x * (0.3333333333333333 + (math.pow(x, 2.0) * (0.022222222222222223 + ((x * x) * (0.0021164021164021165 + ((x * x) * 0.00021164021164021165))))))

Julia

function code(x)
	return Float64(x * Float64(0.3333333333333333 + Float64((x ^ 2.0) * Float64(0.022222222222222223 + Float64(Float64(x * x) * Float64(0.0021164021164021165 + Float64(Float64(x * x) * 0.00021164021164021165)))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.3333333333333333 + ((x ^ 2.0) * (0.022222222222222223 + ((x * x) * (0.0021164021164021165 + ((x * x) * 0.00021164021164021165))))));
end

Wolfram

code[x_] := N[(x * N[(0.3333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.022222222222222223 + N[(N[(x * x), $MachinePrecision] * N[(0.0021164021164021165 + N[(N[(x * x), $MachinePrecision] * 0.00021164021164021165), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + {x}^{2} \cdot \left(0.0021164021164021165 + 0.00021164021164021165 \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + {x}^{2} \cdot \left(0.0021164021164021165 + \color{blue}{{x}^{2} \cdot 0.00021164021164021165}\right)\right)\right) \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + {x}^{2} \cdot \left(0.0021164021164021165 + {x}^{2} \cdot 0.00021164021164021165\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 99.2%

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

7. Applied egg-rr 99.4%

   \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + {x}^{2} \cdot \left(0.0021164021164021165 + \color{blue}{\left(x \cdot x\right)} \cdot 0.00021164021164021165\right)\right)\right) \]
8. Step-by-step derivation

   1. unpow2 99.2%

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

9. Applied egg-rr 99.4%

   \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0021164021164021165 + \left(x \cdot x\right) \cdot 0.00021164021164021165\right)\right)\right) \]
10. Add Preprocessing

Alternative 5: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + \left(x \cdot x\right) \cdot 0.0021164021164021165\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   0.3333333333333333
   (*
    (pow x 2.0)
    (+ 0.022222222222222223 (* (* x x) 0.0021164021164021165))))))

C

double code(double x) {
	return x * (0.3333333333333333 + (pow(x, 2.0) * (0.022222222222222223 + ((x * x) * 0.0021164021164021165))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.3333333333333333d0 + ((x ** 2.0d0) * (0.022222222222222223d0 + ((x * x) * 0.0021164021164021165d0))))
end function

Java

public static double code(double x) {
	return x * (0.3333333333333333 + (Math.pow(x, 2.0) * (0.022222222222222223 + ((x * x) * 0.0021164021164021165))));
}

Python

def code(x):
	return x * (0.3333333333333333 + (math.pow(x, 2.0) * (0.022222222222222223 + ((x * x) * 0.0021164021164021165))))

Julia

function code(x)
	return Float64(x * Float64(0.3333333333333333 + Float64((x ^ 2.0) * Float64(0.022222222222222223 + Float64(Float64(x * x) * 0.0021164021164021165)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.3333333333333333 + ((x ^ 2.0) * (0.022222222222222223 + ((x * x) * 0.0021164021164021165))));
end

Wolfram

code[x_] := N[(x * N[(0.3333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.022222222222222223 + N[(N[(x * x), $MachinePrecision] * 0.0021164021164021165), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around 0 99.4%

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

   1. *-commutative 99.4%

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

5. Simplified 99.4%

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

   1. unpow2 99.2%

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

7. Applied egg-rr 99.4%

   \[\leadsto x \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.022222222222222223 + \color{blue}{\left(x \cdot x\right)} \cdot 0.0021164021164021165\right)\right) \]
8. Add Preprocessing

Alternative 6: 99.4% accurate, 11.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 0.3333333333333333 (* 0.022222222222222223 (* x x)))))

C

double code(double x) {
	return x * (0.3333333333333333 + (0.022222222222222223 * (x * x)));
}

Fortran

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

Java

public static double code(double x) {
	return x * (0.3333333333333333 + (0.022222222222222223 * (x * x)));
}

Python

def code(x):
	return x * (0.3333333333333333 + (0.022222222222222223 * (x * x)))

Julia

function code(x)
	return Float64(x * Float64(0.3333333333333333 + Float64(0.022222222222222223 * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.3333333333333333 + (0.022222222222222223 * (x * x)));
end

Wolfram

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

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around 0 99.2%

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

   1. unpow2 99.2%

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

5. Applied egg-rr 99.2%

   \[\leadsto x \cdot \left(0.3333333333333333 + 0.022222222222222223 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
6. Add Preprocessing

Alternative 7: 98.9% 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 6.0%

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

3. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.9%

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

Developer Target 1: 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 2024172 
(FPCore (x)
  :name "invcot (example 3.9)"
  :precision binary64
  :pre (and (< -0.026 x) (< x 0.026))

  :alt
  (! :herbie-platform default (if (< (fabs x) 13/500) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x)))))

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