Development.Shake.Progress:decay from shake-0.15.5

Average Error: 23.0 → 18.6

Time: 22.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -6.665018424556654185786159367392459116065 \cdot 10^{158}:\\ \;\;\;\;\frac{0}{\mathsf{fma}\left(z, b - y, y\right)} - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \mathbf{elif}\;z \le -6.978402419817959865349829113622978959547 \cdot 10^{-298}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)} - \frac{1}{\frac{\left(\frac{y}{z} + b\right) - y}{a}}\\ \mathbf{elif}\;z \le 1.405623299971047515023653888156309257696 \cdot 10^{-186}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right) - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(z, b - y, y\right)} - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r476148 = x;
        double r476149 = y;
        double r476150 = r476148 * r476149;
        double r476151 = z;
        double r476152 = t;
        double r476153 = a;
        double r476154 = r476152 - r476153;
        double r476155 = r476151 * r476154;
        double r476156 = r476150 + r476155;
        double r476157 = b;
        double r476158 = r476157 - r476149;
        double r476159 = r476151 * r476158;
        double r476160 = r476149 + r476159;
        double r476161 = r476156 / r476160;
        return r476161;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r476162 = z;
        double r476163 = -6.665018424556654e+158;
        bool r476164 = r476162 <= r476163;
        double r476165 = 0.0;
        double r476166 = b;
        double r476167 = y;
        double r476168 = r476166 - r476167;
        double r476169 = fma(r476162, r476168, r476167);
        double r476170 = r476165 / r476169;
        double r476171 = a;
        double r476172 = r476167 / r476162;
        double r476173 = r476172 + r476166;
        double r476174 = r476173 - r476167;
        double r476175 = r476171 / r476174;
        double r476176 = r476170 - r476175;
        double r476177 = -6.97840241981796e-298;
        bool r476178 = r476162 <= r476177;
        double r476179 = t;
        double r476180 = x;
        double r476181 = r476180 * r476167;
        double r476182 = fma(r476179, r476162, r476181);
        double r476183 = r476182 / r476169;
        double r476184 = 1.0;
        double r476185 = r476174 / r476171;
        double r476186 = r476184 / r476185;
        double r476187 = r476183 - r476186;
        double r476188 = 1.4056232999710475e-186;
        bool r476189 = r476162 <= r476188;
        double r476190 = fma(r476180, r476162, r476180);
        double r476191 = r476190 - r476175;
        double r476192 = r476184 / r476169;
        double r476193 = r476182 * r476192;
        double r476194 = r476193 - r476175;
        double r476195 = r476189 ? r476191 : r476194;
        double r476196 = r476178 ? r476187 : r476195;
        double r476197 = r476164 ? r476176 : r476196;
        return r476197;
}

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	23.0
Target	17.8
Herbie	18.6

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

Derivation

1. Split input into 4 regimes

2. if z < -6.665018424556654e+158

   1. Initial program 49.9

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

   3. Applied sub-neg 49.9

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

   4. Applied distribute-lft-in 49.9

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

   6. Applied distribute-rgt-neg-out 49.9

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

   7. Applied unsub-neg 49.9

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

   8. Applied associate-+r- 49.9

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

   9. Applied div-sub 49.9

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

   10. Simplified 49.9

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

   11. Simplified 44.8

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

   12. Taylor expanded around 0 33.6

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

   13. Taylor expanded around 0 33.6

       \[\leadsto \frac{\color{blue}{0}}{\mathsf{fma}\left(z, b - y, y\right)} - \frac{a}{\left(\frac{y}{z} + b\right) - y}\]

   if -6.665018424556654e+158 < z < -6.97840241981796e-298

   1. Initial program 15.0

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

   3. Applied sub-neg 15.0

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

   4. Applied distribute-lft-in 15.0

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

   6. Applied distribute-rgt-neg-out 15.0

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

   7. Applied unsub-neg 15.0

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

   8. Applied associate-+r- 15.0

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

   9. Applied div-sub 15.0

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

   10. Simplified 15.0

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

   11. Simplified 14.5

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

   12. Taylor expanded around 0 13.0

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

   14. Applied clear-num 13.0

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

   if -6.97840241981796e-298 < z < 1.4056232999710475e-186

   1. Initial program 8.5

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

   3. Applied sub-neg 8.5

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

   4. Applied distribute-lft-in 8.5

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

   6. Applied distribute-rgt-neg-out 8.5

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

   7. Applied unsub-neg 8.5

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

   8. Applied associate-+r- 8.5

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

   9. Applied div-sub 8.5

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

   10. Simplified 8.5

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

   11. Simplified 9.7

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

   12. Taylor expanded around 0 9.7

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

   13. Taylor expanded around 0 17.3

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

   14. Simplified 17.3

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

   if 1.4056232999710475e-186 < z

   1. Initial program 26.3

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

   3. Applied sub-neg 26.3

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

   4. Applied distribute-lft-in 26.3

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

   6. Applied distribute-rgt-neg-out 26.3

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

   7. Applied unsub-neg 26.3

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

   8. Applied associate-+r- 26.3

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

   9. Applied div-sub 26.3

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

   10. Simplified 26.3

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

   11. Simplified 24.5

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

   12. Taylor expanded around 0 19.7

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

   14. Applied div-inv 19.7

       \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(z, b - y, y\right)}} - \frac{a}{\left(\frac{y}{z} + b\right) - y}\]
3. Recombined 4 regimes into one program.

4. Final simplification 18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.665018424556654185786159367392459116065 \cdot 10^{158}:\\ \;\;\;\;\frac{0}{\mathsf{fma}\left(z, b - y, y\right)} - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \mathbf{elif}\;z \le -6.978402419817959865349829113622978959547 \cdot 10^{-298}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)} - \frac{1}{\frac{\left(\frac{y}{z} + b\right) - y}{a}}\\ \mathbf{elif}\;z \le 1.405623299971047515023653888156309257696 \cdot 10^{-186}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right) - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right) \cdot \frac{1}{\mathsf{fma}\left(z, b - y, y\right)} - \frac{a}{\left(\frac{y}{z} + b\right) - y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))))