invcot (example 3.9)

Average Error: 60.0 → 0.1

Time: 15.0s

Precision: binary64

Cost: 6848

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return x / fma(x, (x * -0.2), 3.0);
}

Julia

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

↓

function code(x)
	return Float64(x / fma(x, Float64(x * -0.2), 3.0))
end

Wolfram

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

↓

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

Target

Original	60.0
Target	0.1
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;\left|x\right| < 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array} \]

Derivation

1. Initial program 60.0

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

2. Taylor expanded in x around 0 0.4

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

3. Simplified 0.4

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

4. Applied egg-rr 0.4

   \[\leadsto x \cdot \color{blue}{\frac{{\left(x \cdot \left(x \cdot 0.022222222222222223\right)\right)}^{3} + 0.037037037037037035}{\left(x \cdot \left(x \cdot 0.022222222222222223\right)\right) \cdot \left(x \cdot \left(x \cdot 0.022222222222222223\right)\right) + \left(0.1111111111111111 - \left(x \cdot \left(x \cdot 0.022222222222222223\right)\right) \cdot 0.3333333333333333\right)}} \]

5. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{2}, 0.0004938271604938272, 0.1111111111111111\right) - \left(x \cdot x\right) \cdot 0.007407407407407408}{\mathsf{fma}\left({\left(x \cdot x\right)}^{3}, 1.0973936899862826 \cdot 10^{-5}, 0.037037037037037035\right)}}} \]

6. Taylor expanded in x around 0 0.1

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

7. Simplified 0.1

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

8. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.4
Cost	576

\[x \cdot \left(x \cdot \left(x \cdot 0.022222222222222223\right) + 0.3333333333333333\right) \]

Alternative 2
Error	0.3
Cost	192

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

Reproduce

herbie shell --seed 2022228 
(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))))