Average Error: 1.2 → 1.1
Time: 4.9s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\frac{y}{\frac{a - t}{z - t}} + x\]
x + y \cdot \frac{z - t}{a - t}
↓
\frac{y}{\frac{a - t}{z - t}} + x
double f(double x, double y, double z, double t, double a) {
        double r503470 = x;
        double r503471 = y;
        double r503472 = z;
        double r503473 = t;
        double r503474 = r503472 - r503473;
        double r503475 = a;
        double r503476 = r503475 - r503473;
        double r503477 = r503474 / r503476;
        double r503478 = r503471 * r503477;
        double r503479 = r503470 + r503478;
        return r503479;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r503480 = y;
        double r503481 = a;
        double r503482 = t;
        double r503483 = r503481 - r503482;
        double r503484 = z;
        double r503485 = r503484 - r503482;
        double r503486 = r503483 / r503485;
        double r503487 = r503480 / r503486;
        double r503488 = x;
        double r503489 = r503487 + r503488;
        return r503489;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.2
Target0.5
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.2

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

  2. Simplified1.2

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

  4. Applied clear-num1.2

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

  6. Applied fma-udef1.2

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

  7. Simplified1.1

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

  8. Final simplification1.1

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

Reproduce

herbie shell --seed 2020001 +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)))))