Average Error: 10.9 → 0.4
Time: 4.3s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -168171556.617986828:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{elif}\;t \le 232.138274777626037:\\ \;\;\;\;\frac{t \cdot \left(y - z\right)}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t, x\right)\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;t \le -168171556.617986828:\\
\;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\

\mathbf{elif}\;t \le 232.138274777626037:\\
\;\;\;\;\frac{t \cdot \left(y - z\right)}{a - z} + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r2061207 = x;
        double r2061208 = y;
        double r2061209 = z;
        double r2061210 = r2061208 - r2061209;
        double r2061211 = t;
        double r2061212 = r2061210 * r2061211;
        double r2061213 = a;
        double r2061214 = r2061213 - r2061209;
        double r2061215 = r2061212 / r2061214;
        double r2061216 = r2061207 + r2061215;
        return r2061216;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r2061217 = t;
        double r2061218 = -168171556.61798683;
        bool r2061219 = r2061217 <= r2061218;
        double r2061220 = y;
        double r2061221 = z;
        double r2061222 = r2061220 - r2061221;
        double r2061223 = a;
        double r2061224 = r2061223 - r2061221;
        double r2061225 = r2061222 / r2061224;
        double r2061226 = x;
        double r2061227 = fma(r2061225, r2061217, r2061226);
        double r2061228 = 232.13827477762604;
        bool r2061229 = r2061217 <= r2061228;
        double r2061230 = r2061217 * r2061222;
        double r2061231 = r2061230 / r2061224;
        double r2061232 = r2061231 + r2061226;
        double r2061233 = 1.0;
        double r2061234 = r2061233 / r2061224;
        double r2061235 = r2061222 * r2061234;
        double r2061236 = fma(r2061235, r2061217, r2061226);
        double r2061237 = r2061229 ? r2061232 : r2061236;
        double r2061238 = r2061219 ? r2061227 : r2061237;
        return r2061238;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.9
Target0.5
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -168171556.61798683

    1. Initial program 23.9

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

    2. Simplified0.5

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

    if -168171556.61798683 < t < 232.13827477762604

    1. Initial program 0.3

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

    2. Simplified1.9

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

    4. Applied add-cube-cbrt2.2

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

    5. Applied associate-/l*2.2

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

    7. Applied fma-udef2.2

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

    8. Simplified0.3

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

    if 232.13827477762604 < t

    1. Initial program 23.1

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

    2. Simplified0.5

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

    4. Applied div-inv0.6

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y - z\right) \cdot \frac{1}{a - z}}, t, x\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -168171556.617986828:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{elif}\;t \le 232.138274777626037:\\ \;\;\;\;\frac{t \cdot \left(y - z\right)}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{a - z}, t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))))