invcot (example 3.9)

Percentage Accurate: 6.3% → 99.6%

Time: 13.6s

Alternatives: 3

Speedup: 35.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 6.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

2. Taylor expanded in x around 0 99.5%

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

   1. fma-def 99.5%

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

   2. +-commutative 99.5%

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

   3. fma-def 99.5%

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

4. Simplified 99.5%

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

   1. add-cbrt-cube 33.2%

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

   2. pow3 33.2%

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

6. Applied egg-rr 33.2%

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

   1. rem-cbrt-cube 99.5%

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

   2. fma-udef 99.5%

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

   3. flip-+ 51.5%

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

   4. swap-sqr 51.6%

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

   5. metadata-eval 51.6%

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

   6. *-commutative 51.6%

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

   7. associate-*r* 51.5%

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

   8. unpow2 51.5%

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

   9. div-sub 51.5%

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

   10. associate-/l* 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in x around inf 99.8%

   \[\leadsto \frac{x}{\frac{0.3333333333333333 \cdot x - \mathsf{fma}\left(0.022222222222222223, {x}^{3}, 0.0021164021164021165 \cdot {x}^{5}\right)}{x \cdot 0.1111111111111111}} - \color{blue}{-0.0021164021164021165 \cdot {x}^{5}} \]
10. Step-by-step derivation

    1. *-commutative 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

    \[\leadsto \frac{x}{\frac{x \cdot 0.3333333333333333 - \mathsf{fma}\left(0.022222222222222223, {x}^{3}, 0.0021164021164021165 \cdot {x}^{5}\right)}{x \cdot 0.1111111111111111}} - {x}^{5} \cdot -0.0021164021164021165 \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.3333333333333333 + 0.022222222222222223 \cdot {x}^{3} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

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

2. Taylor expanded in x around 0 99.5%

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

3. Final simplification 99.5%

   \[\leadsto x \cdot 0.3333333333333333 + 0.022222222222222223 \cdot {x}^{3} \]

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 6.2%

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

2. Taylor expanded in x around 0 99.1%

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

3. Final simplification 99.1%

   \[\leadsto x \cdot 0.3333333333333333 \]

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 2023207 
(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))))