Average Error: 1.2 → 0.4
Time: 20.8s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.707807104532213458457818164860080687628 \cdot 10^{-46} \lor \neg \left(y \le 162944616184124985573376\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a - t} + x\\ \end{array}\]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.707807104532213458457818164860080687628 \cdot 10^{-46} \lor \neg \left(y \le 162944616184124985573376\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r454574 = x;
        double r454575 = y;
        double r454576 = z;
        double r454577 = t;
        double r454578 = r454576 - r454577;
        double r454579 = a;
        double r454580 = r454579 - r454577;
        double r454581 = r454578 / r454580;
        double r454582 = r454575 * r454581;
        double r454583 = r454574 + r454582;
        return r454583;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r454584 = y;
        double r454585 = -1.7078071045322135e-46;
        bool r454586 = r454584 <= r454585;
        double r454587 = 1.62944616184125e+23;
        bool r454588 = r454584 <= r454587;
        double r454589 = !r454588;
        bool r454590 = r454586 || r454589;
        double r454591 = z;
        double r454592 = t;
        double r454593 = r454591 - r454592;
        double r454594 = a;
        double r454595 = r454594 - r454592;
        double r454596 = r454593 / r454595;
        double r454597 = x;
        double r454598 = fma(r454596, r454584, r454597);
        double r454599 = r454593 * r454584;
        double r454600 = r454599 / r454595;
        double r454601 = r454600 + r454597;
        double r454602 = r454590 ? r454598 : r454601;
        return r454602;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.2
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.7078071045322135e-46 or 1.62944616184125e+23 < y

    1. Initial program 0.5

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

    2. Simplified0.4

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

    if -1.7078071045322135e-46 < y < 1.62944616184125e+23

    1. Initial program 2.0

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

    2. Simplified2.0

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

    4. Applied div-inv2.0

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

    6. Applied add-cube-cbrt2.3

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

    7. Applied associate-/r*2.3

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

    9. Applied fma-udef2.3

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

    10. Simplified3.6

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

    12. Applied associate-*l/0.3

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

    13. Simplified0.3

      \[\leadsto \frac{\color{blue}{\left(z - t\right) \cdot y}}{a - t} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.707807104532213458457818164860080687628 \cdot 10^{-46} \lor \neg \left(y \le 162944616184124985573376\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a - t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +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 -8.50808486055124107e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.8944268627920891e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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