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

Average Error: 23.1 → 10.0

Time: 23.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.576904563487002 \cdot 10^{+105}:\\ \;\;\;\;t + \left(\frac{x}{z} - \frac{t}{z}\right) \cdot y\\ \mathbf{elif}\;z \le 2.4122683507888716 \cdot 10^{+184}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}, \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \left(y - z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(\frac{x}{z} - \frac{t}{z}\right) \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r27710329 = x;
        double r27710330 = y;
        double r27710331 = z;
        double r27710332 = r27710330 - r27710331;
        double r27710333 = t;
        double r27710334 = r27710333 - r27710329;
        double r27710335 = r27710332 * r27710334;
        double r27710336 = a;
        double r27710337 = r27710336 - r27710331;
        double r27710338 = r27710335 / r27710337;
        double r27710339 = r27710329 + r27710338;
        return r27710339;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r27710340 = z;
        double r27710341 = -4.576904563487002e+105;
        bool r27710342 = r27710340 <= r27710341;
        double r27710343 = t;
        double r27710344 = x;
        double r27710345 = r27710344 / r27710340;
        double r27710346 = r27710343 / r27710340;
        double r27710347 = r27710345 - r27710346;
        double r27710348 = y;
        double r27710349 = r27710347 * r27710348;
        double r27710350 = r27710343 + r27710349;
        double r27710351 = 2.4122683507888716e+184;
        bool r27710352 = r27710340 <= r27710351;
        double r27710353 = r27710343 - r27710344;
        double r27710354 = cbrt(r27710353);
        double r27710355 = r27710354 * r27710354;
        double r27710356 = a;
        double r27710357 = r27710356 - r27710340;
        double r27710358 = cbrt(r27710357);
        double r27710359 = r27710358 * r27710358;
        double r27710360 = r27710355 / r27710359;
        double r27710361 = r27710354 / r27710358;
        double r27710362 = r27710348 - r27710340;
        double r27710363 = r27710361 * r27710362;
        double r27710364 = fma(r27710360, r27710363, r27710344);
        double r27710365 = r27710352 ? r27710364 : r27710350;
        double r27710366 = r27710342 ? r27710350 : r27710365;
        return r27710366;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	23.1
Target	11.7
Herbie	10.0

\[\begin{array}{l} \mathbf{if}\;z \lt -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -4.576904563487002e+105 or 2.4122683507888716e+184 < z

   1. Initial program 44.1

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

   2. Simplified 26.7

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

   4. Applied fma-udef 26.8

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

   6. Applied div-inv 26.8

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

   7. Applied associate-*l* 21.8

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

   8. Simplified 21.7

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

   9. Taylor expanded around inf 24.7

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

   10. Simplified 16.8

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

   if -4.576904563487002e+105 < z < 2.4122683507888716e+184

   1. Initial program 14.4

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

   2. Simplified 9.2

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

   4. Applied fma-udef 9.2

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

   6. Applied add-cube-cbrt 9.8

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

   7. Applied add-cube-cbrt 9.9

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

   8. Applied times-frac 9.9

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

   9. Applied associate-*l* 7.1

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

   11. Applied fma-def 7.1

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

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.576904563487002 \cdot 10^{+105}:\\ \;\;\;\;t + \left(\frac{x}{z} - \frac{t}{z}\right) \cdot y\\ \mathbf{elif}\;z \le 2.4122683507888716 \cdot 10^{+184}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}, \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \left(y - z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(\frac{x}{z} - \frac{t}{z}\right) \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"

  :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))))