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

Average Error: 24.4 → 11.1

Time: 23.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.861395718698591849252320273553934613349 \cdot 10^{239} \lor \neg \left(t \le 1.25588328312964090805240113551723241526 \cdot 10^{172}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y - x}{\sqrt[3]{a - t}}\right) + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r468276 = x;
        double r468277 = y;
        double r468278 = r468277 - r468276;
        double r468279 = z;
        double r468280 = t;
        double r468281 = r468279 - r468280;
        double r468282 = r468278 * r468281;
        double r468283 = a;
        double r468284 = r468283 - r468280;
        double r468285 = r468282 / r468284;
        double r468286 = r468276 + r468285;
        return r468286;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r468287 = t;
        double r468288 = -1.8613957186985918e+239;
        bool r468289 = r468287 <= r468288;
        double r468290 = 1.2558832831296409e+172;
        bool r468291 = r468287 <= r468290;
        double r468292 = !r468291;
        bool r468293 = r468289 || r468292;
        double r468294 = x;
        double r468295 = r468294 / r468287;
        double r468296 = z;
        double r468297 = y;
        double r468298 = fma(r468295, r468296, r468297);
        double r468299 = r468296 * r468297;
        double r468300 = r468299 / r468287;
        double r468301 = r468298 - r468300;
        double r468302 = r468296 - r468287;
        double r468303 = cbrt(r468302);
        double r468304 = r468303 * r468303;
        double r468305 = a;
        double r468306 = r468305 - r468287;
        double r468307 = cbrt(r468306);
        double r468308 = r468304 / r468307;
        double r468309 = r468303 / r468307;
        double r468310 = r468297 - r468294;
        double r468311 = r468310 / r468307;
        double r468312 = r468309 * r468311;
        double r468313 = r468308 * r468312;
        double r468314 = r468313 + r468294;
        double r468315 = r468293 ? r468301 : r468314;
        return r468315;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.4
Target	9.3
Herbie	11.1

\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -1.8613957186985918e+239 or 1.2558832831296409e+172 < t

   1. Initial program 48.8

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

   2. Simplified 30.4

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

   4. Applied div-inv 30.5

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

   6. Applied fma-udef 30.6

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

   7. Simplified 30.5

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

   9. Applied add-cube-cbrt 31.1

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

   10. Applied *-un-lft-identity 31.1

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

   11. Applied times-frac 31.1

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

   12. Applied associate-*r* 26.4

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

   13. Simplified 26.4

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

   14. Taylor expanded around inf 24.9

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

   15. Simplified 20.3

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

   if -1.8613957186985918e+239 < t < 1.2558832831296409e+172

   1. Initial program 18.4

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

   2. Simplified 10.8

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

   4. Applied div-inv 10.9

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

   6. Applied fma-udef 10.9

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

   7. Simplified 10.9

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

   9. Applied add-cube-cbrt 11.5

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

   10. Applied *-un-lft-identity 11.5

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

   11. Applied times-frac 11.4

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

   12. Applied associate-*r* 9.2

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

   13. Simplified 9.2

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

   15. Applied add-cube-cbrt 9.1

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

   16. Applied times-frac 9.2

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

   17. Applied associate-*l* 8.9

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.861395718698591849252320273553934613349 \cdot 10^{239} \lor \neg \left(t \le 1.25588328312964090805240113551723241526 \cdot 10^{172}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y - x}{\sqrt[3]{a - t}}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 3.7744031700831742e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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