Average Error: 9.7 → 0.2
Time: 2.6s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
↓
\[\frac{2}{\left(x + -1\right) \cdot \mathsf{fma}\left(x, x, x\right)} \]
(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x) :precision binary64 (/ 2.0 (* (+ x -1.0) (fma x x x))))
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 + -1.0) * fma(x, x, x));
}

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(2.0 / Float64(Float64(x + -1.0) * fma(x, x, x)))
end

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[(2.0 / N[(N[(x + -1.0), $MachinePrecision] * N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Target

Original9.7
Target0.2
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

  1. Initial program 9.7

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

  2. Applied egg-rr25.7

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

  3. Taylor expanded in x around 0 0.2

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

  4. Taylor expanded in x around 0 0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2022192 
(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))))