invcot (example 3.9)

Average Error: 59.8 → 0.1

Time: 21.8s

Precision: binary64

Cost: 6976

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	59.8
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 59.8

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

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

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.022222222222222223, x \cdot x, 0.3333333333333333\right)} \]
   Proof
   (*.f64 x (fma.f64 1/45 (*.f64 x x) 1/3)): 0 points increase in error, 0 points decrease in error
   (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/45 (*.f64 x x)) 1/3))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (Rewrite=> +-commutative_binary64 (+.f64 1/3 (*.f64 1/45 (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 1/3 x) (*.f64 (*.f64 1/45 (*.f64 x x)) x))): 1 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 1/3 x) (Rewrite<= associate-*r*_binary64 (*.f64 1/45 (*.f64 (*.f64 x x) x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 1/3 x) (*.f64 1/45 (Rewrite<= unpow3_binary64 (pow.f64 x 3)))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.4

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

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.7
Cost	192

\[x \cdot 0.3333333333333333 \]

Alternative 2
Error	0.4
Cost	192

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

Reproduce

herbie shell --seed 2022317 
(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))))