Average Error: 10.5 → 1.3
Time: 16.8s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\left(\frac{1}{\frac{a - z}{y}} - \frac{z}{a - z}\right) \cdot t + x\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\left(\frac{1}{\frac{a - z}{y}} - \frac{z}{a - z}\right) \cdot t + x
double f(double x, double y, double z, double t, double a) {
        double r675621 = x;
        double r675622 = y;
        double r675623 = z;
        double r675624 = r675622 - r675623;
        double r675625 = t;
        double r675626 = r675624 * r675625;
        double r675627 = a;
        double r675628 = r675627 - r675623;
        double r675629 = r675626 / r675628;
        double r675630 = r675621 + r675629;
        return r675630;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r675631 = 1.0;
        double r675632 = a;
        double r675633 = z;
        double r675634 = r675632 - r675633;
        double r675635 = y;
        double r675636 = r675634 / r675635;
        double r675637 = r675631 / r675636;
        double r675638 = r675633 / r675634;
        double r675639 = r675637 - r675638;
        double r675640 = t;
        double r675641 = r675639 * r675640;
        double r675642 = x;
        double r675643 = r675641 + r675642;
        return r675643;
}

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

Original10.5
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 10.5

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

  2. Simplified1.2

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

  4. Applied div-sub1.2

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

  6. Applied fma-udef1.2

    \[\leadsto \color{blue}{\left(\frac{y}{a - z} - \frac{z}{a - z}\right) \cdot t + x}\]
  7. Using strategy rm

  8. Applied clear-num1.3

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

  9. Final simplification1.3

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

Reproduce

herbie shell --seed 2019351 +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))))