Development.Shake.Progress:decay from shake-0.15.5

Average Error: 22.6 → 18.9

Time: 18.0s

Precision: 64

Math

\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + \left(b - y\right) \cdot z} = -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + \left(b - y\right) \cdot z} \le 9.318271836675295 \cdot 10^{+307}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + \left(b - y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r41592653 = x;
        double r41592654 = y;
        double r41592655 = r41592653 * r41592654;
        double r41592656 = z;
        double r41592657 = t;
        double r41592658 = a;
        double r41592659 = r41592657 - r41592658;
        double r41592660 = r41592656 * r41592659;
        double r41592661 = r41592655 + r41592660;
        double r41592662 = b;
        double r41592663 = r41592662 - r41592654;
        double r41592664 = r41592656 * r41592663;
        double r41592665 = r41592654 + r41592664;
        double r41592666 = r41592661 / r41592665;
        return r41592666;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r41592667 = z;
        double r41592668 = t;
        double r41592669 = a;
        double r41592670 = r41592668 - r41592669;
        double r41592671 = r41592667 * r41592670;
        double r41592672 = x;
        double r41592673 = y;
        double r41592674 = r41592672 * r41592673;
        double r41592675 = r41592671 + r41592674;
        double r41592676 = b;
        double r41592677 = r41592676 - r41592673;
        double r41592678 = r41592677 * r41592667;
        double r41592679 = r41592673 + r41592678;
        double r41592680 = r41592675 / r41592679;
        double r41592681 = -inf.0;
        bool r41592682 = r41592680 <= r41592681;
        double r41592683 = 9.318271836675295e+307;
        bool r41592684 = r41592680 <= r41592683;
        double r41592685 = r41592684 ? r41592680 : r41592672;
        double r41592686 = r41592682 ? r41592672 : r41592685;
        return r41592686;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	22.6
Target	17.4
Herbie	18.9

\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

1. Split input into 2 regimes

2. if (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < -inf.0 or 9.318271836675295e+307 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))

   1. Initial program 61.3

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
   2. Using strategy rm

   3. Applied clear-num 61.3

      \[\leadsto \color{blue}{\frac{1}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 61.3

      \[\leadsto \frac{1}{\frac{y + z \cdot \left(b - y\right)}{\color{blue}{1 \cdot \left(x \cdot y + z \cdot \left(t - a\right)\right)}}}\]

   6. Applied *-un-lft-identity 61.3

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \left(y + z \cdot \left(b - y\right)\right)}}{1 \cdot \left(x \cdot y + z \cdot \left(t - a\right)\right)}}\]

   7. Applied times-frac 61.3

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{1} \cdot \frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}}\]

   8. Applied add-cube-cbrt 61.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}\]

   9. Applied times-frac 61.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}}\]

   10. Simplified 61.3

       \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}\]

   11. Simplified 61.3

       \[\leadsto 1 \cdot \color{blue}{\frac{\left(t - a\right) \cdot z + x \cdot y}{y + \left(b - y\right) \cdot z}}\]

   12. Taylor expanded around 0 48.9

       \[\leadsto 1 \cdot \color{blue}{x}\]

   if -inf.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 9.318271836675295e+307

   1. Initial program 6.0

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
   2. Using strategy rm

   3. Applied clear-num 6.2

      \[\leadsto \color{blue}{\frac{1}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 6.2

      \[\leadsto \frac{1}{\frac{y + z \cdot \left(b - y\right)}{\color{blue}{1 \cdot \left(x \cdot y + z \cdot \left(t - a\right)\right)}}}\]

   6. Applied *-un-lft-identity 6.2

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot \left(y + z \cdot \left(b - y\right)\right)}}{1 \cdot \left(x \cdot y + z \cdot \left(t - a\right)\right)}}\]

   7. Applied times-frac 6.2

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{1} \cdot \frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}}\]

   8. Applied add-cube-cbrt 6.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}\]

   9. Applied times-frac 6.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}}\]

   10. Simplified 6.2

       \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{y + z \cdot \left(b - y\right)}{x \cdot y + z \cdot \left(t - a\right)}}\]

   11. Simplified 6.0

       \[\leadsto 1 \cdot \color{blue}{\frac{\left(t - a\right) \cdot z + x \cdot y}{y + \left(b - y\right) \cdot z}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 18.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + \left(b - y\right) \cdot z} = -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + \left(b - y\right) \cdot z} \le 9.318271836675295 \cdot 10^{+307}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + \left(b - y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"

  :herbie-target
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))