invcot (example 3.9)

Percentage Accurate: 6.4% → 99.5%

Time: 13.1s

Alternatives: 4

Speedup: 11.9×

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in x around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. flip3-+ 98.1%

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

   3. associate-*l/ 98.3%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

8. Simplified 99.6%

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

   1. unpow2 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

    \[\leadsto \frac{x \cdot \mathsf{fma}\left({x}^{6}, 1.0973936899862826 \cdot 10^{-5}, 0.037037037037037035\right)}{0.1111111111111111 + \left(x \cdot x\right) \cdot -0.007407407407407408} \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in x around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. flip-+ 99.5%

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

   3. associate-*l/ 99.6%

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

   4. metadata-eval 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. swap-sqr 99.6%

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

   8. pow-prod-up 99.6%

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

   9. metadata-eval 99.6%

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

   10. metadata-eval 99.6%

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

   11. cancel-sign-sub-inv 99.6%

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

   12. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

   1. unpow2 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{3 + {x}^{2} \cdot -0.2}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ (+ 3.0 (* (pow x 2.0) -0.2)) x)))

C

double code(double x) {
	return 1.0 / ((3.0 + (pow(x, 2.0) * -0.2)) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((3.0d0 + ((x ** 2.0d0) * (-0.2d0))) / x)
end function

Java

public static double code(double x) {
	return 1.0 / ((3.0 + (Math.pow(x, 2.0) * -0.2)) / x);
}

Python

def code(x):
	return 1.0 / ((3.0 + (math.pow(x, 2.0) * -0.2)) / x)

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(3.0 + Float64((x ^ 2.0) * -0.2)) / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((3.0 + ((x ^ 2.0) * -0.2)) / x);
end

Wolfram

code[x_] := N[(1.0 / N[(N[(3.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.2), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in x around 0 99.5%

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

   1. unpow2 99.6%

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

5. Applied egg-rr 99.5%

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

7. Applied egg-rr 99.3%

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

8. Taylor expanded in x around 0 99.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{3 + -0.2 \cdot {x}^{2}}{x}}} \]
9. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \frac{1}{\frac{3 + \color{blue}{{x}^{2} \cdot -0.2}}{x}} \]

10. Simplified 99.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{3 + {x}^{2} \cdot -0.2}{x}}} \]
11. Add Preprocessing

Alternative 4: 99.4% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.1%

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

3. Taylor expanded in x around 0 99.5%

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

   1. unpow2 99.6%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

   \[\leadsto x \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot 0.022222222222222223\right) \]
7. 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 2024179 
(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))))