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

Average Error: 1.8 → 1.7

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \le -8.828418023450618 \cdot 10^{-208}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{z}{t} \le 5.1135271015432274 \cdot 10^{-55}:\\ \;\;\;\;\left(\frac{z}{\frac{t}{y}} - \frac{x}{\frac{t}{z}}\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r30521193 = x;
        double r30521194 = y;
        double r30521195 = r30521194 - r30521193;
        double r30521196 = z;
        double r30521197 = t;
        double r30521198 = r30521196 / r30521197;
        double r30521199 = r30521195 * r30521198;
        double r30521200 = r30521193 + r30521199;
        return r30521200;
}

↓

double f(double x, double y, double z, double t) {
        double r30521201 = z;
        double r30521202 = t;
        double r30521203 = r30521201 / r30521202;
        double r30521204 = -8.828418023450618e-208;
        bool r30521205 = r30521203 <= r30521204;
        double r30521206 = x;
        double r30521207 = y;
        double r30521208 = r30521207 - r30521206;
        double r30521209 = r30521208 * r30521203;
        double r30521210 = r30521206 + r30521209;
        double r30521211 = 5.1135271015432274e-55;
        bool r30521212 = r30521203 <= r30521211;
        double r30521213 = r30521202 / r30521207;
        double r30521214 = r30521201 / r30521213;
        double r30521215 = r30521202 / r30521201;
        double r30521216 = r30521206 / r30521215;
        double r30521217 = r30521214 - r30521216;
        double r30521218 = r30521217 + r30521206;
        double r30521219 = r30521212 ? r30521218 : r30521210;
        double r30521220 = r30521205 ? r30521210 : r30521219;
        return r30521220;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.8
Target	2.0
Herbie	1.7

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.8867:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ z t) < -8.828418023450618e-208 or 5.1135271015432274e-55 < (/ z t)

   1. Initial program 2.0

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

   if -8.828418023450618e-208 < (/ z t) < 5.1135271015432274e-55

   1. Initial program 1.4

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

   2. Taylor expanded around 0 2.2

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

   4. Applied associate-/l* 1.3

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

   6. Applied associate-/l* 1.2

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \le -8.828418023450618 \cdot 10^{-208}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{z}{t} \le 5.1135271015432274 \cdot 10^{-55}:\\ \;\;\;\;\left(\frac{z}{\frac{t}{y}} - \frac{x}{\frac{t}{z}}\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ 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))))