Average Error: 8.0 → 7.3
Time: 9.8s
Precision: 64
\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\mathsf{fma}\left(\frac{\sqrt{x0}}{1 + \sqrt{x1}}, \frac{\sqrt{x0}}{1 - \sqrt{x1}}, -x0\right)\]
\frac{x0}{1 - x1} - x0
↓
\mathsf{fma}\left(\frac{\sqrt{x0}}{1 + \sqrt{x1}}, \frac{\sqrt{x0}}{1 - \sqrt{x1}}, -x0\right)
double f(double x0, double x1) {
        double r18322995 = x0;
        double r18322996 = 1.0;
        double r18322997 = x1;
        double r18322998 = r18322996 - r18322997;
        double r18322999 = r18322995 / r18322998;
        double r18323000 = r18322999 - r18322995;
        return r18323000;
}
↓
double f(double x0, double x1) {
        double r18323001 = x0;
        double r18323002 = sqrt(r18323001);
        double r18323003 = 1.0;
        double r18323004 = x1;
        double r18323005 = sqrt(r18323004);
        double r18323006 = r18323003 + r18323005;
        double r18323007 = r18323002 / r18323006;
        double r18323008 = r18323003 - r18323005;
        double r18323009 = r18323002 / r18323008;
        double r18323010 = -r18323001;
        double r18323011 = fma(r18323007, r18323009, r18323010);
        return r18323011;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original8.0
Target0.2
Herbie7.3
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 8.0

    \[\frac{x0}{1 - x1} - x0\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt8.0

    \[\leadsto \frac{x0}{1 - \color{blue}{\sqrt{x1} \cdot \sqrt{x1}}} - x0\]

  4. Applied *-un-lft-identity8.0

    \[\leadsto \frac{x0}{\color{blue}{1 \cdot 1} - \sqrt{x1} \cdot \sqrt{x1}} - x0\]

  5. Applied difference-of-squares8.0

    \[\leadsto \frac{x0}{\color{blue}{\left(1 + \sqrt{x1}\right) \cdot \left(1 - \sqrt{x1}\right)}} - x0\]

  6. Applied add-sqr-sqrt8.0

    \[\leadsto \frac{\color{blue}{\sqrt{x0} \cdot \sqrt{x0}}}{\left(1 + \sqrt{x1}\right) \cdot \left(1 - \sqrt{x1}\right)} - x0\]

  7. Applied times-frac8.3

    \[\leadsto \color{blue}{\frac{\sqrt{x0}}{1 + \sqrt{x1}} \cdot \frac{\sqrt{x0}}{1 - \sqrt{x1}}} - x0\]

  8. Applied fma-neg7.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt{x0}}{1 + \sqrt{x1}}, \frac{\sqrt{x0}}{1 - \sqrt{x1}}, -x0\right)}\]

  9. Final simplification7.3

    \[\leadsto \mathsf{fma}\left(\frac{\sqrt{x0}}{1 + \sqrt{x1}}, \frac{\sqrt{x0}}{1 - \sqrt{x1}}, -x0\right)\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :pre (or (and (== x0 1.855) (== x1 0.000209)) (and (== x0 2.985) (== x1 0.0186)))

  :herbie-target
  (/ (* x0 x1) (- 1 x1))

  (- (/ x0 (- 1 x1)) x0))