Development.Shake.Progress:decay from shake-0.15.5

Average Error: 23.2 → 15.0

Time: 8.4s

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 r692067 = x;
        double r692068 = y;
        double r692069 = r692067 * r692068;
        double r692070 = z;
        double r692071 = t;
        double r692072 = a;
        double r692073 = r692071 - r692072;
        double r692074 = r692070 * r692073;
        double r692075 = r692069 + r692074;
        double r692076 = b;
        double r692077 = r692076 - r692068;
        double r692078 = r692070 * r692077;
        double r692079 = r692068 + r692078;
        double r692080 = r692075 / r692079;
        return r692080;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r692081 = x;
        double r692082 = y;
        double r692083 = r692081 * r692082;
        double r692084 = z;
        double r692085 = t;
        double r692086 = a;
        double r692087 = r692085 - r692086;
        double r692088 = r692084 * r692087;
        double r692089 = r692083 + r692088;
        double r692090 = b;
        double r692091 = r692090 - r692082;
        double r692092 = r692084 * r692091;
        double r692093 = r692082 + r692092;
        double r692094 = r692089 / r692093;
        double r692095 = -inf.0;
        bool r692096 = r692094 <= r692095;
        double r692097 = -7.096695019497389e-293;
        bool r692098 = r692094 <= r692097;
        double r692099 = 0.0;
        bool r692100 = r692094 <= r692099;
        double r692101 = r692085 / r692090;
        double r692102 = r692086 / r692090;
        double r692103 = r692101 - r692102;
        double r692104 = 3.365170107416158e+290;
        bool r692105 = r692094 <= r692104;
        double r692106 = r692105 ? r692094 : r692103;
        double r692107 = r692100 ? r692103 : r692106;
        double r692108 = r692098 ? r692094 : r692107;
        double r692109 = r692096 ? r692081 : r692108;
        return r692109;
}

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)))))