Average Error: 1.4 → 1.4
Time: 29.7s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\mathsf{fma}\left(\frac{z}{z - a} - \frac{1}{z - a} \cdot t, y, x\right)\]
x + y \cdot \frac{z - t}{z - a}
↓
\mathsf{fma}\left(\frac{z}{z - a} - \frac{1}{z - a} \cdot t, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r26442500 = x;
        double r26442501 = y;
        double r26442502 = z;
        double r26442503 = t;
        double r26442504 = r26442502 - r26442503;
        double r26442505 = a;
        double r26442506 = r26442502 - r26442505;
        double r26442507 = r26442504 / r26442506;
        double r26442508 = r26442501 * r26442507;
        double r26442509 = r26442500 + r26442508;
        return r26442509;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r26442510 = z;
        double r26442511 = a;
        double r26442512 = r26442510 - r26442511;
        double r26442513 = r26442510 / r26442512;
        double r26442514 = 1.0;
        double r26442515 = r26442514 / r26442512;
        double r26442516 = t;
        double r26442517 = r26442515 * r26442516;
        double r26442518 = r26442513 - r26442517;
        double r26442519 = y;
        double r26442520 = x;
        double r26442521 = fma(r26442518, r26442519, r26442520);
        return r26442521;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

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

Derivation

  1. Initial program 1.4

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

  2. Simplified1.3

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

  4. Applied div-sub1.3

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

  6. Applied div-inv1.4

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

  7. Final simplification1.4

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"

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

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