3frac (problem 3.3.3)

Average Error: 9.8 → 0.1

Time: 12.2s

Precision: binary64

Cost: 6976

Math

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

↓

\[\frac{\frac{1}{x}}{\mathsf{fma}\left(0.5, x \cdot x, -0.5\right)} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (/ 1.0 x) (fma 0.5 (* x x) -0.5)))

C

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

↓

double code(double x) {
	return (1.0 / x) / fma(0.5, (x * x), -0.5);
}

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(1.0 / x) / fma(0.5, Float64(x * x), -0.5))
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[(1.0 / x), $MachinePrecision] / N[(0.5 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]

Target

Original	9.8
Target	0.3
Herbie	0.1

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

Derivation

1. Initial program 9.8

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

2. Simplified 9.8

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

   [Start] 9.8	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 9.8	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 9.8	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 9.8	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 9.8	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 9.8	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 9.8	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 9.8	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 9.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 9.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 9.8

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

4. Taylor expanded in x around 0 0.3

   \[\leadsto \frac{1}{\color{blue}{-0.5 \cdot x + 0.5 \cdot {x}^{3}}} \]

5. Applied egg-rr 0.3

   \[\leadsto \frac{1}{\color{blue}{x \cdot \left(0.5 \cdot \left(x \cdot x\right) + -0.5\right)}} \]

6. Applied egg-rr 0.1

   \[\leadsto \color{blue}{-\frac{\frac{-1}{x}}{\mathsf{fma}\left(0.5, x \cdot x, -0.5\right)}} \]

7. Final simplification 0.1

   \[\leadsto \frac{\frac{1}{x}}{\mathsf{fma}\left(0.5, x \cdot x, -0.5\right)} \]

Alternatives

Alternative 1
Error	0.3
Cost	832

\[\frac{1}{\frac{x}{\frac{-1}{0.5 + \left(x \cdot x\right) \cdot -0.5}}} \]

Alternative 2
Error	0.3
Cost	704

\[\frac{1}{x \cdot \left(-0.5 + 0.5 \cdot \left(x \cdot x\right)\right)} \]

Alternative 3
Error	15.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.53\right):\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 4
Error	15.1
Cost	584

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

Alternative 5
Error	10.7
Cost	448

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

Alternative 6
Error	30.8
Cost	192

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

Alternative 7
Error	61.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023083 
(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))))