3frac (problem 3.3.3)

Average Error: 9.7 → 0.1

Time: 8.4s

Precision: binary64

Cost: 6848

Math

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

↓

\[\frac{\frac{2}{x}}{\mathsf{fma}\left(x, x, -1\right)} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

↓

(FPCore (x) :precision binary64 (/ (/ 2.0 x) (fma x x -1.0)))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

double code(double x) {
	return (2.0 / x) / fma(x, x, -1.0);
}

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

↓

function code(x)
	return Float64(Float64(2.0 / x) / fma(x, x, -1.0))
end

Wolfram

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

↓

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

Target

Original	9.7
Target	0.2
Herbie	0.1

\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

1. Initial program 9.7

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

2. Applied egg-rr 25.7

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x - \left(1 + x\right) \cdot 2, x + -1, \left(x \cdot \left(1 + x\right)\right) \cdot 1\right)}{\left(x \cdot \left(1 + x\right)\right) \cdot \left(x + -1\right)}} \]

3. Taylor expanded in x around 0 0.3

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

4. Applied egg-rr 0.2

   \[\leadsto \color{blue}{0 + \frac{2}{x \cdot \mathsf{fma}\left(x, x, -1\right)}} \]

5. Simplified 0.1

   \[\leadsto \color{blue}{\frac{\frac{2}{x}}{\mathsf{fma}\left(x, x, -1\right)}} \]
   Proof
   (/.f64 (/.f64 2 x) (fma.f64 x x -1)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 x (fma.f64 x x -1)))): 9 points increase in error, 8 points decrease in error
   (Rewrite<= +-lft-identity_binary64 (+.f64 0 (/.f64 2 (*.f64 x (fma.f64 x x -1))))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.1

   \[\leadsto \frac{\frac{2}{x}}{\mathsf{fma}\left(x, x, -1\right)} \]

Alternatives

Alternative 1
Error	0.8
Cost	712

\[\begin{array}{l} t_0 := \frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.7
Cost	712

\[\begin{array}{l} t_0 := \frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot -2 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.3
Cost	704

\[\frac{2}{\left(x \cdot \left(x + 1\right)\right) \cdot \left(x + -1\right)} \]

Alternative 4
Error	15.3
Cost	584

\[\begin{array}{l} t_0 := \frac{1}{x \cdot x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.3
Cost	584

\[\begin{array}{l} t_0 := \frac{1}{x \cdot x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	56.4
Cost	192

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

Alternative 7
Error	30.6
Cost	192

\[\frac{-2}{x} \]

Alternative 8
Error	61.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022318 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))