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

Average Error: 7.9 → 7.0

Time: 9.3s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 2.985
  2. x1 = 0.000209

\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]

Math

\[\frac{x0}{1 - x1} - x0\]

↓

\[\mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\]

C

double f(double x0, double x1) {
        double r128746 = x0;
        double r128747 = 1.0;
        double r128748 = x1;
        double r128749 = r128747 - r128748;
        double r128750 = r128746 / r128749;
        double r128751 = r128750 - r128746;
        return r128751;
}

↓

double f(double x0, double x1) {
        double r128752 = x0;
        double r128753 = cbrt(r128752);
        double r128754 = r128753 * r128753;
        double r128755 = 1.0;
        double r128756 = x1;
        double r128757 = r128755 - r128756;
        double r128758 = r128753 / r128757;
        double r128759 = -r128752;
        double r128760 = fma(r128754, r128758, r128759);
        return r128760;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.2
Herbie	7.0

\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

1. Initial program 7.9

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

3. Applied *-un-lft-identity 7.9

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

4. Applied add-cube-cbrt 7.9

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}\right) \cdot \sqrt[3]{x0}}}{1 \cdot \left(1 - x1\right)} - x0\]

5. Applied times-frac 8.2

   \[\leadsto \color{blue}{\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1} \cdot \frac{\sqrt[3]{x0}}{1 - x1}} - x0\]

6. Applied fma-neg 7.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)}\]

7. Final simplification 7.0

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :precision binary64
  :pre (or (and (== x0 1.855) (== x1 2.09000000000000012e-4)) (and (== x0 2.98499999999999988) (== x1 0.018599999999999998)))

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

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