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

Average Error: 2.1 → 1.4

Time: 22.5s

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 r372078 = x;
        double r372079 = y;
        double r372080 = r372079 - r372078;
        double r372081 = z;
        double r372082 = t;
        double r372083 = r372081 / r372082;
        double r372084 = r372080 * r372083;
        double r372085 = r372078 + r372084;
        return r372085;
}

↓

double f(double x, double y, double z, double t) {
        double r372086 = z;
        double r372087 = t;
        double r372088 = r372086 / r372087;
        double r372089 = -1.096294202967876e+84;
        bool r372090 = r372088 <= r372089;
        double r372091 = -4.353292471597975e-154;
        bool r372092 = r372088 <= r372091;
        double r372093 = !r372092;
        double r372094 = 0.0;
        bool r372095 = r372088 <= r372094;
        bool r372096 = r372093 && r372095;
        bool r372097 = r372090 || r372096;
        double r372098 = y;
        double r372099 = x;
        double r372100 = r372098 - r372099;
        double r372101 = r372100 / r372087;
        double r372102 = r372101 * r372086;
        double r372103 = r372102 + r372099;
        double r372104 = fma(r372100, r372088, r372099);
        double r372105 = r372097 ? r372103 : r372104;
        return r372105;
}

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 3.9

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

   2. Simplified 3.9

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

   4. Applied fma-udef 3.9

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

   5. Simplified 3.8

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

   7. Applied associate-/r/ 2.2

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

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

   1. Initial program 1.0

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

   2. Simplified 1.0

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