Development.Shake.Progress:decay from shake-0.15.5

Average Error: 23.2 → 15.0

Time: 8.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -7.09669501949738936499941568471402397685 \cdot 10^{-293}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 3.365170107416157809932424091751188734223 \cdot 10^{290}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r789759 = x;
        double r789760 = y;
        double r789761 = r789759 * r789760;
        double r789762 = z;
        double r789763 = t;
        double r789764 = a;
        double r789765 = r789763 - r789764;
        double r789766 = r789762 * r789765;
        double r789767 = r789761 + r789766;
        double r789768 = b;
        double r789769 = r789768 - r789760;
        double r789770 = r789762 * r789769;
        double r789771 = r789760 + r789770;
        double r789772 = r789767 / r789771;
        return r789772;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r789773 = x;
        double r789774 = y;
        double r789775 = r789773 * r789774;
        double r789776 = z;
        double r789777 = t;
        double r789778 = a;
        double r789779 = r789777 - r789778;
        double r789780 = r789776 * r789779;
        double r789781 = r789775 + r789780;
        double r789782 = b;
        double r789783 = r789782 - r789774;
        double r789784 = r789776 * r789783;
        double r789785 = r789774 + r789784;
        double r789786 = r789781 / r789785;
        double r789787 = -inf.0;
        bool r789788 = r789786 <= r789787;
        double r789789 = -7.096695019497389e-293;
        bool r789790 = r789786 <= r789789;
        double r789791 = 0.0;
        bool r789792 = r789786 <= r789791;
        double r789793 = r789777 / r789782;
        double r789794 = r789778 / r789782;
        double r789795 = r789793 - r789794;
        double r789796 = 3.365170107416158e+290;
        bool r789797 = r789786 <= r789796;
        double r789798 = r789797 ? r789786 : r789795;
        double r789799 = r789792 ? r789795 : r789798;
        double r789800 = r789790 ? r789786 : r789799;
        double r789801 = r789788 ? r789773 : r789800;
        return r789801;
}

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	23.2
Target	18.0
Herbie	15.0

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   3. Applied clear-num 64.0

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

   5. Applied div-inv 64.0

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

   6. Applied add-cube-cbrt 64.0

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

   7. Applied times-frac 64.0

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

   8. Simplified 64.0

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

   9. Simplified 64.0

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

   10. Taylor expanded around 0 37.6

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

   if -inf.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < -7.096695019497389e-293 or 0.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 3.365170107416158e+290

   1. Initial program 3.6

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

   if -7.096695019497389e-293 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 0.0 or 3.365170107416158e+290 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))

   1. Initial program 59.7

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

   3. Applied clear-num 59.7

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

   5. Applied div-inv 59.7

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

   6. Applied add-cube-cbrt 59.7

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

   7. Applied times-frac 59.7

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

   8. Simplified 59.7

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

   9. Simplified 59.7

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

   10. Taylor expanded around inf 39.7

       \[\leadsto \color{blue}{\frac{t}{b} - \frac{a}{b}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -7.09669501949738936499941568471402397685 \cdot 10^{-293}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 3.365170107416157809932424091751188734223 \cdot 10^{290}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

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

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