Average Error: 9.9 → 0.1
Time: 3.0s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
↓
\[\frac{2}{x + -1} \cdot \frac{1}{\mathsf{fma}\left(x, x, x\right)} \]
(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)) (/ 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)) * (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(Float64(2.0 / Float64(x + -1.0)) * Float64(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[(N[(2.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Target

Original9.9
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

  1. Initial program 9.9

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

  2. Applied egg-rr25.9

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

  3. Taylor expanded in x around 0 0.3

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

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022165 
(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))))