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

Average Error: 7.9 → 2.1

Time: 12.3s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 1.855
  2. x1 = 0.0186

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861321705391922876060562 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

C

double f(double x0, double x1) {
        double r102372 = x0;
        double r102373 = 1.0;
        double r102374 = x1;
        double r102375 = r102373 - r102374;
        double r102376 = r102372 / r102375;
        double r102377 = r102376 - r102372;
        return r102377;
}

↓

double f(double x0, double x1) {
        double r102378 = x1;
        double r102379 = 0.00021208908081054686;
        bool r102380 = r102378 <= r102379;
        double r102381 = x0;
        double r102382 = 3.0;
        double r102383 = pow(r102381, r102382);
        double r102384 = 1.0;
        double r102385 = r102384 - r102378;
        double r102386 = 6.0;
        double r102387 = pow(r102385, r102386);
        double r102388 = r102383 / r102387;
        double r102389 = r102388 - r102383;
        double r102390 = exp(r102389);
        double r102391 = sqrt(r102390);
        double r102392 = log(r102391);
        double r102393 = r102392 + r102392;
        double r102394 = r102381 * r102381;
        double r102395 = r102385 * r102385;
        double r102396 = r102381 / r102395;
        double r102397 = r102396 + r102381;
        double r102398 = r102397 * r102396;
        double r102399 = r102394 + r102398;
        double r102400 = r102393 / r102399;
        double r102401 = r102381 * r102400;
        double r102402 = r102381 / r102385;
        double r102403 = r102402 + r102381;
        double r102404 = r102401 / r102403;
        double r102405 = sqrt(r102384);
        double r102406 = sqrt(r102378);
        double r102407 = r102405 + r102406;
        double r102408 = pow(r102407, r102386);
        double r102409 = r102405 - r102406;
        double r102410 = pow(r102409, r102386);
        double r102411 = r102408 * r102410;
        double r102412 = r102383 / r102411;
        double r102413 = r102412 - r102383;
        double r102414 = exp(r102413);
        double r102415 = log(r102414);
        double r102416 = r102415 / r102399;
        double r102417 = r102381 * r102416;
        double r102418 = r102417 / r102403;
        double r102419 = r102380 ? r102404 : r102418;
        return r102419;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.2
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if x1 < 0.00021208908081054686

   1. Initial program 11.2

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

   3. Applied flip-- 11.4

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

   4. Simplified 8.7

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

   6. Applied flip3-- 6.0

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

   7. Simplified 6.0

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

   9. Applied add-log-exp 6.0

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

   10. Applied add-log-exp 6.0

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

   11. Applied diff-log 5.2

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

   12. Simplified 5.2

       \[\leadsto \frac{x0 \cdot \frac{\log \color{blue}{\left(e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt 3.5

       \[\leadsto \frac{x0 \cdot \frac{\log \color{blue}{\left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}} \cdot \sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   15. Applied log-prod 2.6

       \[\leadsto \frac{x0 \cdot \frac{\color{blue}{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   if 0.00021208908081054686 < x1

   1. Initial program 4.5

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

   3. Applied flip-- 3.1

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

   4. Simplified 3.8

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

   6. Applied flip3-- 3.9

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

   7. Simplified 3.9

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

   9. Applied add-log-exp 3.9

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

   10. Applied add-log-exp 3.9

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

   11. Applied diff-log 4.0

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

   12. Simplified 3.8

       \[\leadsto \frac{x0 \cdot \frac{\log \color{blue}{\left(e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt 3.8

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(1 - \color{blue}{\sqrt{x1} \cdot \sqrt{x1}}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   15. Applied add-sqr-sqrt 3.8

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\color{blue}{\sqrt{1} \cdot \sqrt{1}} - \sqrt{x1} \cdot \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   16. Applied difference-of-squares 3.8

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\color{blue}{\left(\left(\sqrt{1} + \sqrt{x1}\right) \cdot \left(\sqrt{1} - \sqrt{x1}\right)\right)}}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   17. Applied unpow-prod-down 1.6

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{\color{blue}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861321705391922876060562 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :precision binary64
  :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))