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

Average Error: 24.0 → 8.7

Time: 23.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -8.143669236760688 \cdot 10^{-278}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\left(\frac{z \cdot x}{t} + y\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r32014440 = x;
        double r32014441 = y;
        double r32014442 = r32014441 - r32014440;
        double r32014443 = z;
        double r32014444 = t;
        double r32014445 = r32014443 - r32014444;
        double r32014446 = r32014442 * r32014445;
        double r32014447 = a;
        double r32014448 = r32014447 - r32014444;
        double r32014449 = r32014446 / r32014448;
        double r32014450 = r32014440 + r32014449;
        return r32014450;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r32014451 = x;
        double r32014452 = y;
        double r32014453 = r32014452 - r32014451;
        double r32014454 = z;
        double r32014455 = t;
        double r32014456 = r32014454 - r32014455;
        double r32014457 = r32014453 * r32014456;
        double r32014458 = a;
        double r32014459 = r32014458 - r32014455;
        double r32014460 = r32014457 / r32014459;
        double r32014461 = r32014451 + r32014460;
        double r32014462 = -8.143669236760688e-278;
        bool r32014463 = r32014461 <= r32014462;
        double r32014464 = r32014456 / r32014459;
        double r32014465 = r32014453 * r32014464;
        double r32014466 = r32014465 + r32014451;
        double r32014467 = 0.0;
        bool r32014468 = r32014461 <= r32014467;
        double r32014469 = r32014454 * r32014451;
        double r32014470 = r32014469 / r32014455;
        double r32014471 = r32014470 + r32014452;
        double r32014472 = r32014454 * r32014452;
        double r32014473 = r32014472 / r32014455;
        double r32014474 = r32014471 - r32014473;
        double r32014475 = r32014468 ? r32014474 : r32014466;
        double r32014476 = r32014463 ? r32014466 : r32014475;
        return r32014476;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.0
Target	9.5
Herbie	8.7

\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -8.143669236760688e-278 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 20.5

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

   3. Applied *-un-lft-identity 20.5

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

   4. Applied times-frac 7.4

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

   5. Simplified 7.4

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

   if -8.143669236760688e-278 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 58.7

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

   2. Taylor expanded around inf 21.4

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -8.143669236760688 \cdot 10^{-278}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\left(\frac{z \cdot x}{t} + y\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \end{array}\]

Reproduce

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

  :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))))