Average Error: 8.0 → 7.3
Time: 14.7s
Precision: 64
\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\]
\frac{x0}{1 - x1} - x0
↓
\mathsf{fma}\left(\frac{\sqrt{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)
double f(double x0, double x1) {
        double r15360114 = x0;
        double r15360115 = 1.0;
        double r15360116 = x1;
        double r15360117 = r15360115 - r15360116;
        double r15360118 = r15360114 / r15360117;
        double r15360119 = r15360118 - r15360114;
        return r15360119;
}
↓
double f(double x0, double x1) {
        double r15360120 = x0;
        double r15360121 = sqrt(r15360120);
        double r15360122 = 1.0;
        double r15360123 = sqrt(r15360122);
        double r15360124 = x1;
        double r15360125 = sqrt(r15360124);
        double r15360126 = r15360123 + r15360125;
        double r15360127 = r15360121 / r15360126;
        double r15360128 = r15360123 - r15360125;
        double r15360129 = r15360121 / r15360128;
        double r15360130 = -r15360120;
        double r15360131 = fma(r15360127, r15360129, r15360130);
        return r15360131;
}

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 add-sqr-sqrt8.0

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

  5. Applied difference-of-squares8.0

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

  6. Applied add-sqr-sqrt8.0

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

  7. Applied times-frac8.3

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

  8. Applied fma-neg7.3

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

  9. Final simplification7.3

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

Reproduce

herbie shell --seed 2019173 +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) (- 1.0 x1))

  (- (/ x0 (- 1.0 x1)) x0))