Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4

Average Error: 2.1 → 1.4

Time: 22.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \le -1.096294202967875894367897162265744320298 \cdot 10^{84} \lor \neg \left(\frac{z}{t} \le -4.353292471597975087888797752093698096964 \cdot 10^{-154}\right) \land \frac{z}{t} \le 0.0:\\ \;\;\;\;\frac{y - x}{t} \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r309462 = x;
        double r309463 = y;
        double r309464 = r309463 - r309462;
        double r309465 = z;
        double r309466 = t;
        double r309467 = r309465 / r309466;
        double r309468 = r309464 * r309467;
        double r309469 = r309462 + r309468;
        return r309469;
}

↓

double f(double x, double y, double z, double t) {
        double r309470 = z;
        double r309471 = t;
        double r309472 = r309470 / r309471;
        double r309473 = -1.096294202967876e+84;
        bool r309474 = r309472 <= r309473;
        double r309475 = -4.353292471597975e-154;
        bool r309476 = r309472 <= r309475;
        double r309477 = !r309476;
        double r309478 = 0.0;
        bool r309479 = r309472 <= r309478;
        bool r309480 = r309477 && r309479;
        bool r309481 = r309474 || r309480;
        double r309482 = y;
        double r309483 = x;
        double r309484 = r309482 - r309483;
        double r309485 = r309484 / r309471;
        double r309486 = r309485 * r309470;
        double r309487 = r309486 + r309483;
        double r309488 = fma(r309484, r309472, r309483);
        double r309489 = r309481 ? r309487 : r309488;
        return r309489;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.1
Target	2.3
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.88671875:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ z t) < -1.096294202967876e+84 or -4.353292471597975e-154 < (/ z t) < 0.0

   1. Initial program 4.0

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

   2. Simplified 4.0

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

   4. Applied fma-udef 4.0

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

   5. Simplified 3.8

      \[\leadsto \color{blue}{\frac{y - x}{\frac{t}{z}}} + x\]
   6. Using strategy rm

   7. Applied associate-/r/ 2.7

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

   if -1.096294202967876e+84 < (/ z t) < -4.353292471597975e-154 or 0.0 < (/ z t)

   1. Initial program 1.3

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

   2. Simplified 1.3

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \le -1.096294202967875894367897162265744320298 \cdot 10^{84} \lor \neg \left(\frac{z}{t} \le -4.353292471597975087888797752093698096964 \cdot 10^{-154}\right) \land \frac{z}{t} \le 0.0:\\ \;\;\;\;\frac{y - x}{t} \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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