Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A

Average Error: 11.0 → 0.3

Time: 5.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 1.18240911569920632079989379387111191087 \cdot 10^{268}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z} - \frac{z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r642750 = x;
        double r642751 = y;
        double r642752 = z;
        double r642753 = r642751 - r642752;
        double r642754 = t;
        double r642755 = r642753 * r642754;
        double r642756 = a;
        double r642757 = r642756 - r642752;
        double r642758 = r642755 / r642757;
        double r642759 = r642750 + r642758;
        return r642759;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r642760 = y;
        double r642761 = z;
        double r642762 = r642760 - r642761;
        double r642763 = t;
        double r642764 = r642762 * r642763;
        double r642765 = a;
        double r642766 = r642765 - r642761;
        double r642767 = r642764 / r642766;
        double r642768 = -inf.0;
        bool r642769 = r642767 <= r642768;
        double r642770 = 1.1824091156992063e+268;
        bool r642771 = r642767 <= r642770;
        double r642772 = !r642771;
        bool r642773 = r642769 || r642772;
        double r642774 = r642760 / r642766;
        double r642775 = r642761 / r642766;
        double r642776 = r642774 - r642775;
        double r642777 = x;
        double r642778 = fma(r642776, r642763, r642777);
        double r642779 = r642777 + r642767;
        double r642780 = r642773 ? r642778 : r642779;
        return r642780;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	11.0
Target	0.6
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;t \lt -1.068297449017406694366747246993994850729 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.911094988758637497591020599238553861375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (* (- y z) t) (- a z)) < -inf.0 or 1.1824091156992063e+268 < (/ (* (- y z) t) (- a z))

   1. Initial program 60.9

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

   2. Simplified 0.5

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

   4. Applied div-sub 0.5

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

   if -inf.0 < (/ (* (- y z) t) (- a z)) < 1.1824091156992063e+268

   1. Initial program 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} = -\infty \lor \neg \left(\frac{\left(y - z\right) \cdot t}{a - z} \le 1.18240911569920632079989379387111191087 \cdot 10^{268}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z} - \frac{z}{a - z}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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