Average Error: 1.2 → 1.2
Time: 23.5s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\mathsf{fma}\left(\frac{z}{z - a} - \frac{1}{\frac{z - a}{t}}, y, x\right)\]
x + y \cdot \frac{z - t}{z - a}
↓
\mathsf{fma}\left(\frac{z}{z - a} - \frac{1}{\frac{z - a}{t}}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r414495 = x;
        double r414496 = y;
        double r414497 = z;
        double r414498 = t;
        double r414499 = r414497 - r414498;
        double r414500 = a;
        double r414501 = r414497 - r414500;
        double r414502 = r414499 / r414501;
        double r414503 = r414496 * r414502;
        double r414504 = r414495 + r414503;
        return r414504;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r414505 = z;
        double r414506 = a;
        double r414507 = r414505 - r414506;
        double r414508 = r414505 / r414507;
        double r414509 = 1.0;
        double r414510 = t;
        double r414511 = r414507 / r414510;
        double r414512 = r414509 / r414511;
        double r414513 = r414508 - r414512;
        double r414514 = y;
        double r414515 = x;
        double r414516 = fma(r414513, r414514, r414515);
        return r414516;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.2
Target1.1
Herbie1.2
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 1.2

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

  2. Simplified1.2

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

  4. Applied div-sub1.2

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

  6. Applied clear-num1.2

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

  7. Final simplification1.2

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

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, A"
  :precision binary64

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

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