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

Average Error: 25.0 → 10.9

Time: 20.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -6.35262688856634336 \cdot 10^{-298}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{1}, \frac{\sqrt[3]{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{y - x}}}{\sqrt[3]{a - t}} \cdot \left(z - t\right)\right), x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 1.96010087509 \cdot 10^{-248}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{1}, \frac{\sqrt[3]{y - x}}{a - t} \cdot \left(z - t\right), x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r2342 = x;
        double r2343 = y;
        double r2344 = r2343 - r2342;
        double r2345 = z;
        double r2346 = t;
        double r2347 = r2345 - r2346;
        double r2348 = r2344 * r2347;
        double r2349 = a;
        double r2350 = r2349 - r2346;
        double r2351 = r2348 / r2350;
        double r2352 = r2342 + r2351;
        return r2352;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r2353 = x;
        double r2354 = y;
        double r2355 = r2354 - r2353;
        double r2356 = z;
        double r2357 = t;
        double r2358 = r2356 - r2357;
        double r2359 = r2355 * r2358;
        double r2360 = a;
        double r2361 = r2360 - r2357;
        double r2362 = r2359 / r2361;
        double r2363 = r2353 + r2362;
        double r2364 = -6.352626888566343e-298;
        bool r2365 = r2363 <= r2364;
        double r2366 = cbrt(r2355);
        double r2367 = r2366 * r2366;
        double r2368 = 1.0;
        double r2369 = r2367 / r2368;
        double r2370 = cbrt(r2367);
        double r2371 = cbrt(r2361);
        double r2372 = r2371 * r2371;
        double r2373 = r2370 / r2372;
        double r2374 = cbrt(r2366);
        double r2375 = r2374 / r2371;
        double r2376 = r2375 * r2358;
        double r2377 = r2373 * r2376;
        double r2378 = fma(r2369, r2377, r2353);
        double r2379 = 1.960100875085522e-248;
        bool r2380 = r2363 <= r2379;
        double r2381 = r2366 / r2361;
        double r2382 = r2381 * r2358;
        double r2383 = fma(r2369, r2382, r2353);
        double r2384 = r2380 ? r2354 : r2383;
        double r2385 = r2365 ? r2378 : r2384;
        return r2385;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.0
Target	9.3
Herbie	10.9

\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -6.352626888566343e-298

   1. Initial program 21.9

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

   2. Simplified 10.9

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

   4. Applied fma-udef 11.0

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

   6. Applied *-un-lft-identity 11.0

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

   7. Applied add-cube-cbrt 11.6

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

   8. Applied times-frac 11.6

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

   9. Applied associate-*l* 8.6

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

   11. Applied add-cube-cbrt 8.8

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

   12. Applied add-cube-cbrt 8.8

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

   13. Applied cbrt-prod 8.9

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

   14. Applied times-frac 8.9

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

   15. Applied associate-*l* 8.2

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

   17. Applied fma-def 8.2

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

   if -6.352626888566343e-298 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 1.960100875085522e-248

   1. Initial program 55.8

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

   2. Simplified 56.6

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

   3. Taylor expanded around 0 36.2

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

   if 1.960100875085522e-248 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 21.9

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

   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 fma-udef 10.8

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

   6. Applied *-un-lft-identity 10.8

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

   7. Applied add-cube-cbrt 11.4

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

   8. Applied times-frac 11.4

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

   9. Applied associate-*l* 8.5

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

   11. Applied fma-def 8.5

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

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -6.35262688856634336 \cdot 10^{-298}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{1}, \frac{\sqrt[3]{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{y - x}}}{\sqrt[3]{a - t}} \cdot \left(z - t\right)\right), x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 1.96010087509 \cdot 10^{-248}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{1}, \frac{\sqrt[3]{y - x}}{a - t} \cdot \left(z - t\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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