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

Average Error: 7.8 → 6.1

Time: 12.3s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 1.855
  2. x1 = 0.0186

\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]

Math

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

↓

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

C

double f(double x0, double x1) {
        double r6527211 = x0;
        double r6527212 = 1.0;
        double r6527213 = x1;
        double r6527214 = r6527212 - r6527213;
        double r6527215 = r6527211 / r6527214;
        double r6527216 = r6527215 - r6527211;
        return r6527216;
}

↓

double f(double x0, double x1) {
        double r6527217 = x0;
        double r6527218 = 1.0;
        double r6527219 = x1;
        double r6527220 = r6527218 - r6527219;
        double r6527221 = r6527217 / r6527220;
        double r6527222 = r6527221 * r6527221;
        double r6527223 = r6527222 * r6527217;
        double r6527224 = r6527223 / r6527220;
        double r6527225 = r6527217 * r6527217;
        double r6527226 = r6527217 * r6527225;
        double r6527227 = r6527224 - r6527226;
        double r6527228 = r6527221 * r6527222;
        double r6527229 = cbrt(r6527228);
        double r6527230 = r6527229 * r6527217;
        double r6527231 = r6527230 + r6527225;
        double r6527232 = r6527222 + r6527231;
        double r6527233 = r6527227 / r6527232;
        return r6527233;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.8
Target	0.2
Herbie	6.1

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

Derivation

1. Initial program 7.8

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

3. Applied flip3-- 7.6

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

4. Simplified 7.2

   \[\leadsto \frac{\color{blue}{\frac{x0}{1 - x1} \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) - x0 \cdot \left(x0 \cdot x0\right)}}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \frac{x0}{1 - x1} \cdot x0\right)}\]
5. Using strategy rm

6. Applied associate-*l/ 6.1

   \[\leadsto \frac{\color{blue}{\frac{x0 \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right)}{1 - x1}} - x0 \cdot \left(x0 \cdot x0\right)}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \frac{x0}{1 - x1} \cdot x0\right)}\]
7. Using strategy rm

8. Applied add-cbrt-cube 6.1

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

9. Applied add-cbrt-cube 6.1

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

10. Applied cbrt-undiv 6.1

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

11. Simplified 6.1

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

12. Final simplification 6.1

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

Reproduce

herbie shell --seed 2019168 
(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))