Average Error: 1.4 → 1.3
Time: 13.3s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\frac{y}{\mathsf{fma}\left(a, \frac{1}{z - t}, -\frac{t}{z - t}\right)} + x\]
x + y \cdot \frac{z - t}{a - t}
↓
\frac{y}{\mathsf{fma}\left(a, \frac{1}{z - t}, -\frac{t}{z - t}\right)} + x
double f(double x, double y, double z, double t, double a) {
        double r803932 = x;
        double r803933 = y;
        double r803934 = z;
        double r803935 = t;
        double r803936 = r803934 - r803935;
        double r803937 = a;
        double r803938 = r803937 - r803935;
        double r803939 = r803936 / r803938;
        double r803940 = r803933 * r803939;
        double r803941 = r803932 + r803940;
        return r803941;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r803942 = y;
        double r803943 = a;
        double r803944 = 1.0;
        double r803945 = z;
        double r803946 = t;
        double r803947 = r803945 - r803946;
        double r803948 = r803944 / r803947;
        double r803949 = r803946 / r803947;
        double r803950 = -r803949;
        double r803951 = fma(r803943, r803948, r803950);
        double r803952 = r803942 / r803951;
        double r803953 = x;
        double r803954 = r803952 + r803953;
        return r803954;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.4
Target0.5
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.4

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

  2. Simplified1.4

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

  4. Applied clear-num1.5

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

  6. Applied fma-udef1.5

    \[\leadsto \color{blue}{\frac{1}{\frac{a - t}{z - t}} \cdot y + x}\]

  7. Simplified1.3

    \[\leadsto \color{blue}{\frac{y}{\frac{a - t}{z - t}}} + x\]
  8. Using strategy rm

  9. Applied div-sub1.3

    \[\leadsto \frac{y}{\color{blue}{\frac{a}{z - t} - \frac{t}{z - t}}} + x\]
  10. Using strategy rm

  11. Applied div-inv1.3

    \[\leadsto \frac{y}{\color{blue}{a \cdot \frac{1}{z - t}} - \frac{t}{z - t}} + x\]

  12. Applied fma-neg1.3

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

  13. Final simplification1.3

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

Reproduce

herbie shell --seed 2019350 +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)))))