3frac (problem 3.3.3)

Average Error: 10.2 → 0.1

Time: 8.0s

Precision: binary64

Cost: 6976

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (/ 2.0 (fma x x 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 / fma(x, x, 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 / fma(x, x, x)) / Float64(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 / N[(x * x + x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]

Target

Original	10.2
Target	0.3
Herbie	0.1

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

Derivation

1. Initial program 10.2

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

2. Applied egg-rr 25.4

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

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 0.1

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

5. Simplified 0.1

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

   [Start] 0.1	\[ \frac{2}{x + x \cdot x} \cdot \frac{1}{x + -1} \]
   *-commutative [=>] 0.1	\[ \color{blue}{\frac{1}{x + -1} \cdot \frac{2}{x + x \cdot x}} \]
   associate-*l/ [=>] 0.1	\[ \color{blue}{\frac{1 \cdot \frac{2}{x + x \cdot x}}{x + -1}} \]
   *-lft-identity [=>] 0.1	\[ \frac{\color{blue}{\frac{2}{x + x \cdot x}}}{x + -1} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{2}{\color{blue}{x \cdot x + x}}}{x + -1} \]
   fma-def [=>] 0.1	\[ \frac{\frac{2}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}}}{x + -1} \]

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	10.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.65\right):\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 2
Error	1.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 3
Error	15.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -0.4 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	704

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

Alternative 5
Error	0.1
Cost	704

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

Alternative 6
Error	15.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.38 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 7
Error	30.7
Cost	192

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

Reproduce

herbie shell --seed 2023053 
(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))))