Average Error: 1.2 → 1.2
Time: 20.6s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\mathsf{fma}\left(\frac{\frac{1}{z - a}}{\frac{1}{z - t}}, y, x\right)\]
x + y \cdot \frac{z - t}{z - a}
↓
\mathsf{fma}\left(\frac{\frac{1}{z - a}}{\frac{1}{z - t}}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r24627514 = x;
        double r24627515 = y;
        double r24627516 = z;
        double r24627517 = t;
        double r24627518 = r24627516 - r24627517;
        double r24627519 = a;
        double r24627520 = r24627516 - r24627519;
        double r24627521 = r24627518 / r24627520;
        double r24627522 = r24627515 * r24627521;
        double r24627523 = r24627514 + r24627522;
        return r24627523;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r24627524 = 1.0;
        double r24627525 = z;
        double r24627526 = a;
        double r24627527 = r24627525 - r24627526;
        double r24627528 = r24627524 / r24627527;
        double r24627529 = t;
        double r24627530 = r24627525 - r24627529;
        double r24627531 = r24627524 / r24627530;
        double r24627532 = r24627528 / r24627531;
        double r24627533 = y;
        double r24627534 = x;
        double r24627535 = fma(r24627532, r24627533, r24627534);
        return r24627535;
}

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 clear-num1.2

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

  6. Applied div-inv1.3

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

  7. Applied associate-/r*1.2

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

  8. Final simplification1.2

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

Reproduce

herbie shell --seed 2019158 +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)))))