Average Error: 1.3 → 1.4
Time: 7.9s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\mathsf{fma}\left(y, \frac{z}{z - a} - \frac{1}{\frac{z - a}{t}}, x\right)\]
x + y \cdot \frac{z - t}{z - a}
↓
\mathsf{fma}\left(y, \frac{z}{z - a} - \frac{1}{\frac{z - a}{t}}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r592473 = x;
        double r592474 = y;
        double r592475 = z;
        double r592476 = t;
        double r592477 = r592475 - r592476;
        double r592478 = a;
        double r592479 = r592475 - r592478;
        double r592480 = r592477 / r592479;
        double r592481 = r592474 * r592480;
        double r592482 = r592473 + r592481;
        return r592482;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r592483 = y;
        double r592484 = z;
        double r592485 = a;
        double r592486 = r592484 - r592485;
        double r592487 = r592484 / r592486;
        double r592488 = 1.0;
        double r592489 = t;
        double r592490 = r592486 / r592489;
        double r592491 = r592488 / r592490;
        double r592492 = r592487 - r592491;
        double r592493 = x;
        double r592494 = fma(r592483, r592492, r592493);
        return r592494;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target1.2
Herbie1.4
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 1.3

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

  2. Simplified1.3

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

  4. Applied div-sub1.3

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

  6. Applied clear-num1.4

    \[\leadsto \mathsf{fma}\left(y, \frac{z}{z - a} - \color{blue}{\frac{1}{\frac{z - a}{t}}}, x\right)\]

  7. Final simplification1.4

    \[\leadsto \mathsf{fma}\left(y, \frac{z}{z - a} - \frac{1}{\frac{z - a}{t}}, x\right)\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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