Average Error: 11.2 → 1.4
Time: 25.4s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\mathsf{fma}\left(\frac{z}{z - a} - \frac{1}{z - a} \cdot t, y, x\right)\]
x + \frac{y \cdot \left(z - t\right)}{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 r23417582 = x;
        double r23417583 = y;
        double r23417584 = z;
        double r23417585 = t;
        double r23417586 = r23417584 - r23417585;
        double r23417587 = r23417583 * r23417586;
        double r23417588 = a;
        double r23417589 = r23417584 - r23417588;
        double r23417590 = r23417587 / r23417589;
        double r23417591 = r23417582 + r23417590;
        return r23417591;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r23417592 = z;
        double r23417593 = a;
        double r23417594 = r23417592 - r23417593;
        double r23417595 = r23417592 / r23417594;
        double r23417596 = 1.0;
        double r23417597 = r23417596 / r23417594;
        double r23417598 = t;
        double r23417599 = r23417597 * r23417598;
        double r23417600 = r23417595 - r23417599;
        double r23417601 = y;
        double r23417602 = x;
        double r23417603 = fma(r23417600, r23417601, r23417602);
        return r23417603;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.2
Target1.2
Herbie1.4
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 11.2

    \[x + \frac{y \cdot \left(z - t\right)}{z - a}\]

  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 2019174 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"

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

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