invcot (example 3.9)

Percentage Accurate: 6.5% → 99.6%

Time: 16.5s

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: 6.5% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right)\\ \left(0.037037037037037035 + {x}^{6} \cdot {t\_0}^{3}\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{2} \cdot t\_0, \mathsf{fma}\left({x}^{2}, t\_0, -0.3333333333333333\right), 0.1111111111111111\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (pow x 2.0) 0.0021164021164021165 0.022222222222222223)))
   (*
    (+ 0.037037037037037035 (* (pow x 6.0) (pow t_0 3.0)))
    (/
     x
     (fma
      (* (pow x 2.0) t_0)
      (fma (pow x 2.0) t_0 -0.3333333333333333)
      0.1111111111111111)))))

C

double code(double x) {
	double t_0 = fma(pow(x, 2.0), 0.0021164021164021165, 0.022222222222222223);
	return (0.037037037037037035 + (pow(x, 6.0) * pow(t_0, 3.0))) * (x / fma((pow(x, 2.0) * t_0), fma(pow(x, 2.0), t_0, -0.3333333333333333), 0.1111111111111111));
}

Julia

function code(x)
	t_0 = fma((x ^ 2.0), 0.0021164021164021165, 0.022222222222222223)
	return Float64(Float64(0.037037037037037035 + Float64((x ^ 6.0) * (t_0 ^ 3.0))) * Float64(x / fma(Float64((x ^ 2.0) * t_0), fma((x ^ 2.0), t_0, -0.3333333333333333), 0.1111111111111111)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 2.0], $MachinePrecision] * 0.0021164021164021165 + 0.022222222222222223), $MachinePrecision]}, N[(N[(0.037037037037037035 + N[(N[Power[x, 6.0], $MachinePrecision] * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(N[Power[x, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Power[x, 2.0], $MachinePrecision] * t$95$0 + -0.3333333333333333), $MachinePrecision] + 0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.2%

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

3. Taylor expanded in x around 0 99.5%

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

   1. *-commutative 99.5%

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

5. Simplified 99.5%

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

   1. flip3-+ 98.0%

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

   2. associate-*r/ 98.3%

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

7. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{x \cdot \left({\left(\mathsf{fma}\left(0.022222222222222223, {x}^{2}, 0.0021164021164021165 \cdot {x}^{4}\right)\right)}^{3} + 0.037037037037037035\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, {x}^{2}, 0.0021164021164021165 \cdot {x}^{4}\right), \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right), -0.3333333333333333\right), 0.1111111111111111\right)}} \]
8. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(0.022222222222222223, {x}^{2}, 0.0021164021164021165 \cdot {x}^{4}\right)\right)}^{3} + 0.037037037037037035\right) \cdot x}}{\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, {x}^{2}, 0.0021164021164021165 \cdot {x}^{4}\right), \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right), -0.3333333333333333\right), 0.1111111111111111\right)} \]

   2. associate-/l* 99.6%

      \[\leadsto \color{blue}{\left({\left(\mathsf{fma}\left(0.022222222222222223, {x}^{2}, 0.0021164021164021165 \cdot {x}^{4}\right)\right)}^{3} + 0.037037037037037035\right) \cdot \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, {x}^{2}, 0.0021164021164021165 \cdot {x}^{4}\right), \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right), -0.3333333333333333\right), 0.1111111111111111\right)}} \]

9. Simplified 99.6%

   \[\leadsto \color{blue}{\left(0.037037037037037035 + {x}^{6} \cdot {\left(\mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right)\right)}^{3}\right) \cdot \frac{x}{\mathsf{fma}\left({x}^{2} \cdot \mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right), \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, 0.0021164021164021165, 0.022222222222222223\right), -0.3333333333333333\right), 0.1111111111111111\right)}} \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.1111111111111111 \cdot \frac{x}{0.3333333333333333 + {x}^{2} \cdot -0.022222222222222223} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

3. Taylor expanded in x around 0 99.4%

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

   1. flip-+ 99.4%

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

   2. associate-*r/ 99.5%

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

   3. metadata-eval 99.5%

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

   4. *-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. swap-sqr 99.5%

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

   7. pow-prod-up 99.5%

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

   8. metadata-eval 99.5%

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

   9. metadata-eval 99.5%

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

   10. cancel-sign-sub-inv 99.5%

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

   11. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-/l* 99.5%

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

   3. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 99.6%

   \[\leadsto \color{blue}{0.1111111111111111} \cdot \frac{x}{0.3333333333333333 + {x}^{2} \cdot -0.022222222222222223} \]
9. Add Preprocessing

Alternative 3: 99.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.1111111111111111 \cdot \frac{1}{0.3333333333333333 + -0.022222222222222223 \cdot \left(x \cdot x\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   0.1111111111111111
   (/ 1.0 (+ 0.3333333333333333 (* -0.022222222222222223 (* x x)))))))

C

double code(double x) {
	return x * (0.1111111111111111 * (1.0 / (0.3333333333333333 + (-0.022222222222222223 * (x * x)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.1111111111111111d0 * (1.0d0 / (0.3333333333333333d0 + ((-0.022222222222222223d0) * (x * x)))))
end function

Java

public static double code(double x) {
	return x * (0.1111111111111111 * (1.0 / (0.3333333333333333 + (-0.022222222222222223 * (x * x)))));
}

Python

def code(x):
	return x * (0.1111111111111111 * (1.0 / (0.3333333333333333 + (-0.022222222222222223 * (x * x)))))

Julia

function code(x)
	return Float64(x * Float64(0.1111111111111111 * Float64(1.0 / Float64(0.3333333333333333 + Float64(-0.022222222222222223 * Float64(x * x))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.1111111111111111 * (1.0 / (0.3333333333333333 + (-0.022222222222222223 * (x * x)))));
end

Wolfram

code[x_] := N[(x * N[(0.1111111111111111 * N[(1.0 / N[(0.3333333333333333 + N[(-0.022222222222222223 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.2%

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

3. Taylor expanded in x around 0 99.4%

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

   1. flip-+ 99.4%

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

   2. div-inv 99.4%

      \[\leadsto x \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot 0.3333333333333333 - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right)\right) \cdot \frac{1}{0.3333333333333333 - 0.022222222222222223 \cdot {x}^{2}}\right)} \]

   3. metadata-eval 99.4%

      \[\leadsto x \cdot \left(\left(\color{blue}{0.1111111111111111} - \left(0.022222222222222223 \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot {x}^{2}\right)\right) \cdot \frac{1}{0.3333333333333333 - 0.022222222222222223 \cdot {x}^{2}}\right) \]

   4. *-commutative 99.4%

      \[\leadsto x \cdot \left(\left(0.1111111111111111 - \color{blue}{\left({x}^{2} \cdot 0.022222222222222223\right)} \cdot \left(0.022222222222222223 \cdot {x}^{2}\right)\right) \cdot \frac{1}{0.3333333333333333 - 0.022222222222222223 \cdot {x}^{2}}\right) \]

   5. *-commutative 99.4%

      \[\leadsto x \cdot \left(\left(0.1111111111111111 - \left({x}^{2} \cdot 0.022222222222222223\right) \cdot \color{blue}{\left({x}^{2} \cdot 0.022222222222222223\right)}\right) \cdot \frac{1}{0.3333333333333333 - 0.022222222222222223 \cdot {x}^{2}}\right) \]

   6. swap-sqr 99.4%

      \[\leadsto x \cdot \left(\left(0.1111111111111111 - \color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(0.022222222222222223 \cdot 0.022222222222222223\right)}\right) \cdot \frac{1}{0.3333333333333333 - 0.022222222222222223 \cdot {x}^{2}}\right) \]

   7. pow-prod-up 99.4%

      \[\leadsto x \cdot \left(\left(0.1111111111111111 - \color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(0.022222222222222223 \cdot 0.022222222222222223\right)\right) \cdot \frac{1}{0.3333333333333333 - 0.022222222222222223 \cdot {x}^{2}}\right) \]

   8. metadata-eval 99.4%

      \[\leadsto x \cdot \left(\left(0.1111111111111111 - {x}^{\color{blue}{4}} \cdot \left(0.022222222222222223 \cdot 0.022222222222222223\right)\right) \cdot \frac{1}{0.3333333333333333 - 0.022222222222222223 \cdot {x}^{2}}\right) \]

   9. metadata-eval 99.4%

      \[\leadsto x \cdot \left(\left(0.1111111111111111 - {x}^{4} \cdot \color{blue}{0.0004938271604938272}\right) \cdot \frac{1}{0.3333333333333333 - 0.022222222222222223 \cdot {x}^{2}}\right) \]

   10. cancel-sign-sub-inv 99.4%

       \[\leadsto x \cdot \left(\left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right) \cdot \frac{1}{\color{blue}{0.3333333333333333 + \left(-0.022222222222222223\right) \cdot {x}^{2}}}\right) \]

   11. metadata-eval 99.4%

       \[\leadsto x \cdot \left(\left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right) \cdot \frac{1}{0.3333333333333333 + \color{blue}{-0.022222222222222223} \cdot {x}^{2}}\right) \]

5. Applied egg-rr 99.4%

   \[\leadsto x \cdot \color{blue}{\left(\left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right) \cdot \frac{1}{0.3333333333333333 + -0.022222222222222223 \cdot {x}^{2}}\right)} \]
6. Step-by-step derivation

   1. unpow2 99.4%

      \[\leadsto x \cdot \left(\left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right) \cdot \frac{1}{0.3333333333333333 + -0.022222222222222223 \cdot \color{blue}{\left(x \cdot x\right)}}\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto x \cdot \left(\left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right) \cdot \frac{1}{0.3333333333333333 + -0.022222222222222223 \cdot \color{blue}{\left(x \cdot x\right)}}\right) \]

8. Taylor expanded in x around 0 99.4%

   \[\leadsto x \cdot \left(\color{blue}{0.1111111111111111} \cdot \frac{1}{0.3333333333333333 + -0.022222222222222223 \cdot \left(x \cdot x\right)}\right) \]
9. Add Preprocessing

Alternative 4: 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 6.2%

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

3. Taylor expanded in x around 0 99.4%

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

   1. unpow2 99.4%

      \[\leadsto x \cdot \left(\left(0.1111111111111111 - {x}^{4} \cdot 0.0004938271604938272\right) \cdot \frac{1}{0.3333333333333333 + -0.022222222222222223 \cdot \color{blue}{\left(x \cdot x\right)}}\right) \]

5. Applied egg-rr 99.4%

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

Alternative 5: 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 6.2%

   \[\frac{1}{x} - \frac{1}{\tan x} \]
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 2024185 
(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))))