Average Error: 1.6 → 1.6
Time: 20.3s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(z, \frac{1}{a - t}, \frac{-t}{a - t}\right), y, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(\mathsf{fma}\left(z, \frac{1}{a - t}, \frac{-t}{a - t}\right), y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r325974 = x;
        double r325975 = y;
        double r325976 = z;
        double r325977 = t;
        double r325978 = r325976 - r325977;
        double r325979 = a;
        double r325980 = r325979 - r325977;
        double r325981 = r325978 / r325980;
        double r325982 = r325975 * r325981;
        double r325983 = r325974 + r325982;
        return r325983;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r325984 = z;
        double r325985 = 1.0;
        double r325986 = a;
        double r325987 = t;
        double r325988 = r325986 - r325987;
        double r325989 = r325985 / r325988;
        double r325990 = -r325987;
        double r325991 = r325990 / r325988;
        double r325992 = fma(r325984, r325989, r325991);
        double r325993 = y;
        double r325994 = x;
        double r325995 = fma(r325992, r325993, r325994);
        return r325995;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.6
Target0.4
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.6

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

  2. Simplified1.6

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

  4. Applied div-sub1.6

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

  6. Applied div-inv1.6

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

  7. Applied fma-neg1.6

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

  8. Simplified1.6

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

  9. Final simplification1.6

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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