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

Average Error: 7.9 → 5.9

Time: 57.7s

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

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

↓

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

C

double f(double x0, double x1) {
        double r8463399 = x0;
        double r8463400 = 1.0;
        double r8463401 = x1;
        double r8463402 = r8463400 - r8463401;
        double r8463403 = r8463399 / r8463402;
        double r8463404 = r8463403 - r8463399;
        return r8463404;
}

↓

double f(double x0, double x1) {
        double r8463405 = x0;
        double r8463406 = 1.0;
        double r8463407 = x1;
        double r8463408 = r8463406 - r8463407;
        double r8463409 = r8463405 / r8463408;
        double r8463410 = r8463408 * r8463408;
        double r8463411 = r8463409 / r8463410;
        double r8463412 = r8463411 - r8463405;
        double r8463413 = r8463405 * r8463405;
        double r8463414 = r8463412 * r8463413;
        double r8463415 = r8463413 * r8463413;
        double r8463416 = r8463409 * r8463405;
        double r8463417 = r8463416 * r8463416;
        double r8463418 = r8463415 - r8463417;
        double r8463419 = r8463408 * r8463418;
        double r8463420 = r8463413 - r8463416;
        double r8463421 = r8463416 * r8463420;
        double r8463422 = r8463419 + r8463421;
        double r8463423 = r8463420 * r8463408;
        double r8463424 = r8463422 / r8463423;
        double r8463425 = r8463414 / r8463424;
        return r8463425;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.3
Herbie	5.9

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

Derivation

1. Initial program 7.9

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

3. Applied flip3-- 7.7

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

4. Simplified 6.1

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

6. Applied flip-+ 6.2

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

7. Applied associate-*r/ 6.2

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

8. Applied frac-add 5.9

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

9. Final simplification 5.9

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

Reproduce

herbie shell --seed 2019125 
(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))