Average Error: 10.7 → 1.5
Time: 12.7s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[t \cdot \frac{y - z}{a - z} + x\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
t \cdot \frac{y - z}{a - z} + x
double f(double x, double y, double z, double t, double a) {
        double r629026 = x;
        double r629027 = y;
        double r629028 = z;
        double r629029 = r629027 - r629028;
        double r629030 = t;
        double r629031 = r629029 * r629030;
        double r629032 = a;
        double r629033 = r629032 - r629028;
        double r629034 = r629031 / r629033;
        double r629035 = r629026 + r629034;
        return r629035;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r629036 = t;
        double r629037 = y;
        double r629038 = z;
        double r629039 = r629037 - r629038;
        double r629040 = a;
        double r629041 = r629040 - r629038;
        double r629042 = r629039 / r629041;
        double r629043 = r629036 * r629042;
        double r629044 = x;
        double r629045 = r629043 + r629044;
        return r629045;
}

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

Original10.7
Target0.6
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program 10.7

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

  2. Simplified1.5

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

  4. Applied *-un-lft-identity1.5

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

  5. Applied add-cube-cbrt2.0

    \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \sqrt[3]{y - z}}}{1 \cdot \left(a - z\right)}, t, x\right)\]

  6. Applied times-frac2.0

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{1} \cdot \frac{\sqrt[3]{y - z}}{a - z}}, t, x\right)\]

  7. Simplified2.0

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

  9. Applied fma-udef2.0

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \frac{\sqrt[3]{y - z}}{a - z}\right) \cdot t + x}\]

  10. Simplified1.5

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

  11. Final simplification1.5

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

Reproduce

herbie shell --seed 2020042 +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))))