invcot (example 3.9)

Percentage Accurate: 6.3% → 99.9%

Time: 15.6s

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: 99.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

   \[\frac{1}{x} - \frac{1}{\tan x} \]

2. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{0.022222222222222223 \cdot {x}^{3} + 0.3333333333333333 \cdot x} \]
3. Step-by-step derivation

   1. unpow3 99.5%

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

   2. associate-*r* 99.5%

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

   3. distribute-rgt-out 99.5%

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

   4. pow2 99.5%

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

4. Applied egg-rr 99.5%

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

   1. flip3-+ 98.0%

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

   2. associate-*r/ 98.3%

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

   3. metadata-eval 99.5%

      \[\leadsto \frac{x \cdot \left({\left(0.022222222222222223 \cdot {x}^{2}\right)}^{3} + \color{blue}{0.037037037037037035}\right)}{\left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right) + \left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot 0.3333333333333333\right)} \]

   4. +-commutative 99.5%

      \[\leadsto \frac{x \cdot \color{blue}{\left(0.037037037037037035 + {\left(0.022222222222222223 \cdot {x}^{2}\right)}^{3}\right)}}{\left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right) + \left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot 0.3333333333333333\right)} \]

   5. *-commutative 99.5%

      \[\leadsto \frac{x \cdot \left(0.037037037037037035 + {\color{blue}{\left({x}^{2} \cdot 0.022222222222222223\right)}}^{3}\right)}{\left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right) + \left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot 0.3333333333333333\right)} \]

   6. cube-prod 99.5%

      \[\leadsto \frac{x \cdot \left(0.037037037037037035 + \color{blue}{{\left({x}^{2}\right)}^{3} \cdot {0.022222222222222223}^{3}}\right)}{\left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right) + \left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot 0.3333333333333333\right)} \]

   7. unpow2 99.5%

      \[\leadsto \frac{x \cdot \left(0.037037037037037035 + {\color{blue}{\left(x \cdot x\right)}}^{3} \cdot {0.022222222222222223}^{3}\right)}{\left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right) + \left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot 0.3333333333333333\right)} \]

   8. pow-prod-down 99.5%

      \[\leadsto \frac{x \cdot \left(0.037037037037037035 + \color{blue}{\left({x}^{3} \cdot {x}^{3}\right)} \cdot {0.022222222222222223}^{3}\right)}{\left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right) + \left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot 0.3333333333333333\right)} \]

   9. pow-prod-up 99.5%

      \[\leadsto \frac{x \cdot \left(0.037037037037037035 + \color{blue}{{x}^{\left(3 + 3\right)}} \cdot {0.022222222222222223}^{3}\right)}{\left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right) + \left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot 0.3333333333333333\right)} \]

   10. metadata-eval 99.5%

       \[\leadsto \frac{x \cdot \left(0.037037037037037035 + {x}^{\color{blue}{6}} \cdot {0.022222222222222223}^{3}\right)}{\left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right) + \left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot 0.3333333333333333\right)} \]

   11. metadata-eval 99.5%

       \[\leadsto \frac{x \cdot \left(0.037037037037037035 + {x}^{6} \cdot \color{blue}{1.0973936899862826 \cdot 10^{-5}}\right)}{\left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right) + \left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot 0.3333333333333333\right)} \]

   12. +-commutative 99.5%

       \[\leadsto \frac{x \cdot \left(0.037037037037037035 + {x}^{6} \cdot 1.0973936899862826 \cdot 10^{-5}\right)}{\color{blue}{\left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot 0.3333333333333333\right) + \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right)}} \]

   13. metadata-eval 99.5%

       \[\leadsto \frac{x \cdot \left(0.037037037037037035 + {x}^{6} \cdot 1.0973936899862826 \cdot 10^{-5}\right)}{\left(\color{blue}{0.1111111111111111} - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot 0.3333333333333333\right) + \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right)} \]

6. Applied egg-rr 99.5%

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

   1. associate-/l* 99.9%

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

   2. associate--r- 99.9%

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

   3. *-commutative 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{3}{x} + x \cdot -0.2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ (/ 3.0 x) (* x -0.2))))

C

double code(double x) {
	return 1.0 / ((3.0 / x) + (x * -0.2));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((3.0d0 / x) + (x * (-0.2d0)))
end function

Java

public static double code(double x) {
	return 1.0 / ((3.0 / x) + (x * -0.2));
}

Python

def code(x):
	return 1.0 / ((3.0 / x) + (x * -0.2))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(3.0 / x) + Float64(x * -0.2)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((3.0 / x) + (x * -0.2));
end

Wolfram

code[x_] := N[(1.0 / N[(N[(3.0 / x), $MachinePrecision] + N[(x * -0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

   \[\frac{1}{x} - \frac{1}{\tan x} \]

2. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{0.022222222222222223 \cdot {x}^{3} + 0.3333333333333333 \cdot x} \]
3. Step-by-step derivation

   1. flip3-+ 33.4%

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

   2. clear-num 33.4%

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

   3. clear-num 33.4%

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

   4. flip3-+ 99.4%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{0.022222222222222223 \cdot {x}^{3} + 0.3333333333333333 \cdot x}}} \]

   5. +-commutative 99.4%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{0.3333333333333333 \cdot x + 0.022222222222222223 \cdot {x}^{3}}}} \]

   6. fma-def 99.4%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 0.022222222222222223 \cdot {x}^{3}\right)}}} \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(0.3333333333333333, x, 0.022222222222222223 \cdot {x}^{3}\right)}}} \]

5. Taylor expanded in x around 0 99.3%

   \[\leadsto \frac{1}{\color{blue}{-0.2 \cdot x + 3 \cdot \frac{1}{x}}} \]
6. Step-by-step derivation

   1. fma-def 99.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-0.2, x, 3 \cdot \frac{1}{x}\right)}} \]

   2. associate-*r/ 99.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(-0.2, x, \color{blue}{\frac{3 \cdot 1}{x}}\right)} \]

   3. metadata-eval 99.5%

      \[\leadsto \frac{1}{\mathsf{fma}\left(-0.2, x, \frac{\color{blue}{3}}{x}\right)} \]

7. Simplified 99.5%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-0.2, x, \frac{3}{x}\right)}} \]
8. Step-by-step derivation

   1. fma-udef 99.5%

      \[\leadsto \frac{1}{\color{blue}{-0.2 \cdot x + \frac{3}{x}}} \]

   2. +-commutative 99.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{3}{x} + -0.2 \cdot x}} \]

   3. *-commutative 99.5%

      \[\leadsto \frac{1}{\frac{3}{x} + \color{blue}{x \cdot -0.2}} \]

9. Applied egg-rr 99.5%

   \[\leadsto \frac{1}{\color{blue}{\frac{3}{x} + x \cdot -0.2}} \]

10. Final simplification 99.5%

    \[\leadsto \frac{1}{\frac{3}{x} + x \cdot -0.2} \]

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 6.8%

   \[\frac{1}{x} - \frac{1}{\tan x} \]

2. Taylor expanded in x around 0 98.8%

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

3. Final simplification 98.8%

   \[\leadsto x \cdot 0.3333333333333333 \]

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 6.8%

   \[\frac{1}{x} - \frac{1}{\tan x} \]

2. Taylor expanded in x around 0 98.8%

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

   1. add-cbrt-cube 36.1%

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

   2. unpow3 36.1%

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

   3. pow1/3 19.7%

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

   4. *-commutative 19.7%

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

   5. cube-prod 19.7%

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

   6. metadata-eval 19.7%

      \[\leadsto {\left({x}^{3} \cdot \color{blue}{0.037037037037037035}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 19.7%

   \[\leadsto \color{blue}{{\left({x}^{3} \cdot 0.037037037037037035\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 36.1%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{3} \cdot 0.037037037037037035}} \]

6. Simplified 36.1%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{3} \cdot 0.037037037037037035}} \]
7. Step-by-step derivation

   1. metadata-eval 36.1%

      \[\leadsto \sqrt[3]{{x}^{3} \cdot \color{blue}{{0.3333333333333333}^{3}}} \]

   2. cube-prod 36.1%

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

   3. *-commutative 36.1%

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

   4. unpow3 36.1%

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

   5. add-cbrt-cube 98.8%

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

   6. add-sqr-sqrt 48.1%

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

   7. pow2 48.1%

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

   8. *-commutative 48.1%

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

8. Applied egg-rr 48.1%

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

   1. unpow2 48.1%

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

   2. add-sqr-sqrt 98.8%

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

   3. metadata-eval 98.8%

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

   4. div-inv 99.2%

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

10. Applied egg-rr 99.2%

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

11. Final simplification 99.2%

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

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 2023305 
(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))))