Average Error: 11.2 → 1.3
Time: 18.8s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\frac{t}{\frac{a - z}{y - z}} + x\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\frac{t}{\frac{a - z}{y - z}} + x
double f(double x, double y, double z, double t, double a) {
        double r405375 = x;
        double r405376 = y;
        double r405377 = z;
        double r405378 = r405376 - r405377;
        double r405379 = t;
        double r405380 = r405378 * r405379;
        double r405381 = a;
        double r405382 = r405381 - r405377;
        double r405383 = r405380 / r405382;
        double r405384 = r405375 + r405383;
        return r405384;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r405385 = t;
        double r405386 = a;
        double r405387 = z;
        double r405388 = r405386 - r405387;
        double r405389 = y;
        double r405390 = r405389 - r405387;
        double r405391 = r405388 / r405390;
        double r405392 = r405385 / r405391;
        double r405393 = x;
        double r405394 = r405392 + r405393;
        return r405394;
}

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

Original11.2
Target0.5
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;t \lt -1.068297449017406694366747246993994850729 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.911094988758637497591020599238553861375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program 11.2

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

  2. Simplified1.4

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

  4. Applied clear-num1.5

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

  6. Applied fma-udef1.5

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

  7. Simplified1.3

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

  8. Final simplification1.3

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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