Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Average Error: 24.2 → 9.7

Time: 20.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -4.82516022252417878 \cdot 10^{-304}:\\ \;\;\;\;x + \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \left(\sqrt[3]{y - z} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right)\right) \cdot \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \left(\sqrt[3]{y - z} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right)\right) \cdot \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r598267 = x;
        double r598268 = y;
        double r598269 = z;
        double r598270 = r598268 - r598269;
        double r598271 = t;
        double r598272 = r598271 - r598267;
        double r598273 = r598270 * r598272;
        double r598274 = a;
        double r598275 = r598274 - r598269;
        double r598276 = r598273 / r598275;
        double r598277 = r598267 + r598276;
        return r598277;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r598278 = x;
        double r598279 = y;
        double r598280 = z;
        double r598281 = r598279 - r598280;
        double r598282 = t;
        double r598283 = r598282 - r598278;
        double r598284 = r598281 * r598283;
        double r598285 = a;
        double r598286 = r598285 - r598280;
        double r598287 = r598284 / r598286;
        double r598288 = r598278 + r598287;
        double r598289 = -4.825160222524179e-304;
        bool r598290 = r598288 <= r598289;
        double r598291 = cbrt(r598281);
        double r598292 = cbrt(r598286);
        double r598293 = r598292 * r598292;
        double r598294 = cbrt(r598293);
        double r598295 = r598291 / r598294;
        double r598296 = 1.0;
        double r598297 = r598296 / r598293;
        double r598298 = cbrt(r598297);
        double r598299 = r598291 * r598298;
        double r598300 = r598295 * r598299;
        double r598301 = r598281 / r598293;
        double r598302 = cbrt(r598301);
        double r598303 = r598283 / r598292;
        double r598304 = r598302 * r598303;
        double r598305 = r598300 * r598304;
        double r598306 = r598278 + r598305;
        double r598307 = 0.0;
        bool r598308 = r598288 <= r598307;
        double r598309 = r598278 * r598279;
        double r598310 = r598309 / r598280;
        double r598311 = r598310 + r598282;
        double r598312 = r598282 * r598279;
        double r598313 = r598312 / r598280;
        double r598314 = r598311 - r598313;
        double r598315 = r598299 * r598299;
        double r598316 = r598315 * r598304;
        double r598317 = r598278 + r598316;
        double r598318 = r598308 ? r598314 : r598317;
        double r598319 = r598290 ? r598306 : r598318;
        return r598319;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.2
Target	11.7
Herbie	9.7

\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y z) (- t x)) (- a z))) < -4.825160222524179e-304

   1. Initial program 21.1

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

   3. Applied add-cube-cbrt 21.6

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

   4. Applied times-frac 8.7

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 8.9

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

   7. Applied associate-*l* 8.9

      \[\leadsto x + \color{blue}{\left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)}\]
   8. Using strategy rm

   9. Applied div-inv 8.9

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

   10. Applied cbrt-prod 8.9

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

   12. Applied cbrt-div 8.8

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

   if -4.825160222524179e-304 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 0.0

   1. Initial program 60.8

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

   2. Taylor expanded around inf 19.3

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

   if 0.0 < (+ x (/ (* (- y z) (- t x)) (- a z)))

   1. Initial program 20.7

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

   3. Applied add-cube-cbrt 21.2

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

   4. Applied times-frac 8.6

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 8.8

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

   7. Applied associate-*l* 8.8

      \[\leadsto x + \color{blue}{\left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)}\]
   8. Using strategy rm

   9. Applied div-inv 8.8

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

   10. Applied cbrt-prod 8.7

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

   12. Applied div-inv 8.7

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

   13. Applied cbrt-prod 8.8

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

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -4.82516022252417878 \cdot 10^{-304}:\\ \;\;\;\;x + \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \left(\sqrt[3]{y - z} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right)\right) \cdot \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \left(\sqrt[3]{y - z} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right)\right) \cdot \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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