Average Error: 9.8 → 0.3
Time: 9.2s
Precision: binary64
Cost: 6976
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
↓
\[\frac{-1}{\left(x \cdot -0.5\right) \cdot \mathsf{fma}\left(x, x, -1\right)} \]
(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x) :precision binary64 (/ -1.0 (* (* x -0.5) (fma x x -1.0))))
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 * -0.5) * fma(x, x, -1.0));
}

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

\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{-1}{\left(x \cdot -0.5\right) \cdot \mathsf{fma}\left(x, x, -1\right)}

Error

Target

Original9.8
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

  1. Initial program 9.8

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

  2. Simplified9.8

    \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)} \]
    Proof
    (+.f64 (/.f64 1 (+.f64 1 x)) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (/.f64 (Rewrite<= metadata-eval (neg.f64 2)) x))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 x))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 1 (+.f64 x 1)) (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (/.f64 2 x)) (/.f64 1 (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (/.f64 1 (+.f64 x 1)) (neg.f64 (/.f64 2 x))) (/.f64 1 (-.f64 x 1)))): 1 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x))) (/.f64 1 (-.f64 x 1))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr9.8

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

  4. Applied egg-rr9.8

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

  5. Applied egg-rr25.3

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

  6. Taylor expanded in x around 0 0.3

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

  7. Final simplification0.3

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

Alternatives

Alternative 1
Error0.7
Cost8712
\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := \frac{1}{-1 + x}\\ t_2 := \left(\frac{-2}{x} + t_0\right) + t_1\\ \mathbf{if}\;t_2 \leq -5000:\\ \;\;\;\;\frac{-2}{x} + \left(t_0 + t_1\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{x \cdot -0.5 + 1}{x \cdot -0.5 + x \cdot \left(x \cdot 0.5\right)}\\ \end{array} \]
Alternative 2
Error9.8
Cost960
\[\frac{-2}{x} + \left(\frac{1}{x + 1} + \frac{1}{-1 + x}\right) \]
Alternative 3
Error9.8
Cost960
\[\frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{1}{-1 + x}\right) \]
Alternative 4
Error10.7
Cost712
\[\begin{array}{l} t_0 := \frac{2}{x} + \frac{-2}{x}\\ \mathbf{if}\;x \leq -291147832.0563465:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.00037218625360065295:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error10.6
Cost712
\[\begin{array}{l} t_0 := \frac{2}{x} + \frac{-2}{x}\\ \mathbf{if}\;x \leq -291147832.0563465:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.00037218625360065295:\\ \;\;\;\;\frac{-2}{x} + x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error15.5
Cost584
\[\begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -291147832.0563465:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.00037218625360065295:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error15.5
Cost584
\[\begin{array}{l} t_0 := \frac{-1}{x \cdot x}\\ \mathbf{if}\;x \leq -291147832.0563465:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.00037218625360065295:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error10.7
Cost448
\[1 + \left(-1 + \frac{-2}{x}\right) \]
Alternative 9
Error56.4
Cost192
\[\frac{-1}{x} \]
Alternative 10
Error30.7
Cost192
\[\frac{-2}{x} \]
Alternative 11
Error61.9
Cost64
\[-1 \]

Error

Reproduce

herbie shell --seed 2022317 
(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))))