Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B

Average Error: 1.3 → 1.3

Time: 21.5s

Precision: 64

Math

\[x + y \cdot \frac{z - t}{a - t}\]

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r394064 = x;
        double r394065 = y;
        double r394066 = z;
        double r394067 = t;
        double r394068 = r394066 - r394067;
        double r394069 = a;
        double r394070 = r394069 - r394067;
        double r394071 = r394068 / r394070;
        double r394072 = r394065 * r394071;
        double r394073 = r394064 + r394072;
        return r394073;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r394074 = z;
        double r394075 = a;
        double r394076 = t;
        double r394077 = r394075 - r394076;
        double r394078 = r394074 / r394077;
        double r394079 = r394076 / r394077;
        double r394080 = r394078 - r394079;
        double r394081 = y;
        double r394082 = x;
        double r394083 = fma(r394080, r394081, r394082);
        return r394083;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.3
Target	0.5
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Initial program 1.3

   \[x + y \cdot \frac{z - t}{a - t}\]

2. Simplified 1.3

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

4. Applied div-sub 1.3

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

5. Final simplification 1.3

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.50808486055124107e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.8944268627920891e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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