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

Average Error: 7.8 → 6.1

Time: 24.6s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 1.855
  2. x1 = 0.0186

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

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 r8065816 = x0;
        double r8065817 = 1.0;
        double r8065818 = x1;
        double r8065819 = r8065817 - r8065818;
        double r8065820 = r8065816 / r8065819;
        double r8065821 = r8065820 - r8065816;
        return r8065821;
}

↓

double f(double x0, double x1) {
        double r8065822 = x0;
        double r8065823 = 1.0;
        double r8065824 = x1;
        double r8065825 = r8065823 - r8065824;
        double r8065826 = r8065822 / r8065825;
        double r8065827 = r8065826 * r8065826;
        double r8065828 = r8065827 * r8065822;
        double r8065829 = r8065828 / r8065825;
        double r8065830 = r8065822 * r8065822;
        double r8065831 = r8065822 * r8065830;
        double r8065832 = r8065829 - r8065831;
        double r8065833 = r8065826 * r8065827;
        double r8065834 = cbrt(r8065833);
        double r8065835 = r8065834 * r8065822;
        double r8065836 = r8065835 + r8065830;
        double r8065837 = r8065827 + r8065836;
        double r8065838 = r8065832 / r8065837;
        return r8065838;
}

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

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

6. Applied associate-*r/ 6.1

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

8. Applied add-cbrt-cube 6.1

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

9. Applied add-cbrt-cube 6.1

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

10. Applied cbrt-undiv 6.1

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

11. Simplified 6.1

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

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) (- 1.0 x1))

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