Average Error: 1.3 → 1.4
Time: 22.8s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(\frac{1}{\frac{a - t}{z - t}}, y, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(\frac{1}{\frac{a - t}{z - t}}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r27198448 = x;
        double r27198449 = y;
        double r27198450 = z;
        double r27198451 = t;
        double r27198452 = r27198450 - r27198451;
        double r27198453 = a;
        double r27198454 = r27198453 - r27198451;
        double r27198455 = r27198452 / r27198454;
        double r27198456 = r27198449 * r27198455;
        double r27198457 = r27198448 + r27198456;
        return r27198457;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r27198458 = 1.0;
        double r27198459 = a;
        double r27198460 = t;
        double r27198461 = r27198459 - r27198460;
        double r27198462 = z;
        double r27198463 = r27198462 - r27198460;
        double r27198464 = r27198461 / r27198463;
        double r27198465 = r27198458 / r27198464;
        double r27198466 = y;
        double r27198467 = x;
        double r27198468 = fma(r27198465, r27198466, r27198467);
        return r27198468;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target0.4
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.3

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

  2. Simplified1.3

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

  4. Applied clear-num1.4

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

  5. Final simplification1.4

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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