invcot (example 3.9)

Percentage Accurate: 6.4% → 99.9%

Time: 17.6s

Alternatives: 5

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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{1}{\mathsf{fma}\left(0.022222222222222223, x \cdot x, 0.3333333333333333\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ x (/ 1.0 (fma 0.022222222222222223 (* x x) 0.3333333333333333))))

C

double code(double x) {
	return x / (1.0 / fma(0.022222222222222223, (x * x), 0.3333333333333333));
}

Julia

function code(x)
	return Float64(x / Float64(1.0 / fma(0.022222222222222223, Float64(x * x), 0.3333333333333333)))
end

Wolfram

code[x_] := N[(x / N[(1.0 / N[(0.022222222222222223 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in x around 0 99.4%

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

   1. remove-double-neg 99.4%

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

   2. sub-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. cube-mult 99.4%

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

   5. associate-*l* 99.4%

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

   6. distribute-lft-neg-in 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-lft-out-- 99.4%

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

   9. associate-*l* 99.4%

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

   10. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

   1. flip-- 99.4%

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

   2. clear-num 99.4%

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

   3. un-div-inv 100.0%

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

   4. clear-num 100.0%

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

   5. flip-- 100.0%

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

   6. associate-*r* 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-neg 100.0%

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

   9. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.5% 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 6.7%

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

2. Taylor expanded in x around 0 99.4%

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

   1. flip3-+ 31.0%

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

   2. clear-num 31.0%

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

   3. clear-num 31.0%

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

   4. flip3-+ 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-def 99.3%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in x around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. flip-+ 50.4%

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

   3. sqr-neg 50.4%

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

   4. sub-neg 50.4%

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

   5. flip-- 99.5%

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

   6. div-inv 99.5%

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

   7. *-commutative 99.5%

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

   8. distribute-rgt-neg-in 99.5%

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

   9. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 99.1% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{3}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 3.0 x)))

C

double code(double x) {
	return 1.0 / (3.0 / x);
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (3.0 / x);
}

Python

def code(x):
	return 1.0 / (3.0 / x)

Julia

function code(x)
	return Float64(1.0 / Float64(3.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (3.0 / x);
end

Wolfram

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

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in x around 0 99.4%

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

   1. flip3-+ 31.0%

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

   2. clear-num 31.0%

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

   3. clear-num 31.0%

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

   4. flip3-+ 99.3%

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

   5. +-commutative 99.3%

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

   6. fma-def 99.3%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in x around 0 98.9%

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

6. Final simplification 98.9%

   \[\leadsto \frac{1}{\frac{3}{x}} \]

Alternative 4: 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 6.7%

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

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 5: 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 6.7%

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

2. Taylor expanded in x around 0 99.4%

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

   1. remove-double-neg 99.4%

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

   2. sub-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. cube-mult 99.4%

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

   5. associate-*l* 99.4%

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

   6. distribute-lft-neg-in 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-lft-out-- 99.4%

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

   9. associate-*l* 99.4%

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

   10. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

   1. flip-- 99.4%

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

   2. clear-num 99.4%

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

   3. un-div-inv 100.0%

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

   4. clear-num 100.0%

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

   5. flip-- 100.0%

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

   6. associate-*r* 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-neg 100.0%

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

   9. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 99.4%

   \[\leadsto \frac{x}{\frac{1}{\color{blue}{0.3333333333333333}}} \]

8. Final simplification 99.4%

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

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 2023297 
(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))))