Average Error: 1.3 → 1.2
Time: 19.2s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.142790662965348752755157353290722225006 \cdot 10^{-181}:\\ \;\;\;\;x + \frac{y}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;t \le 4.070272537557395579919809546399400646441 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{a - t} \cdot \left(y \cdot \left(z - t\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - \frac{t}{a - t}, y, x\right)\\ \end{array}\]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -4.142790662965348752755157353290722225006 \cdot 10^{-181}:\\
\;\;\;\;x + \frac{y}{\frac{a - t}{z - t}}\\

\mathbf{elif}\;t \le 4.070272537557395579919809546399400646441 \cdot 10^{-104}:\\
\;\;\;\;\frac{1}{a - t} \cdot \left(y \cdot \left(z - t\right)\right) + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r536586 = x;
        double r536587 = y;
        double r536588 = z;
        double r536589 = t;
        double r536590 = r536588 - r536589;
        double r536591 = a;
        double r536592 = r536591 - r536589;
        double r536593 = r536590 / r536592;
        double r536594 = r536587 * r536593;
        double r536595 = r536586 + r536594;
        return r536595;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r536596 = t;
        double r536597 = -4.142790662965349e-181;
        bool r536598 = r536596 <= r536597;
        double r536599 = x;
        double r536600 = y;
        double r536601 = a;
        double r536602 = r536601 - r536596;
        double r536603 = z;
        double r536604 = r536603 - r536596;
        double r536605 = r536602 / r536604;
        double r536606 = r536600 / r536605;
        double r536607 = r536599 + r536606;
        double r536608 = 4.0702725375573956e-104;
        bool r536609 = r536596 <= r536608;
        double r536610 = 1.0;
        double r536611 = r536610 / r536602;
        double r536612 = r536600 * r536604;
        double r536613 = r536611 * r536612;
        double r536614 = r536613 + r536599;
        double r536615 = r536603 / r536602;
        double r536616 = r536596 / r536602;
        double r536617 = r536615 - r536616;
        double r536618 = fma(r536617, r536600, r536599);
        double r536619 = r536609 ? r536614 : r536618;
        double r536620 = r536598 ? r536607 : r536619;
        return r536620;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target0.4
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -4.142790662965349e-181

    1. Initial program 1.0

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

    2. Simplified1.0

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

    4. Applied fma-udef1.0

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

    5. Simplified0.8

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

    if -4.142790662965349e-181 < t < 4.0702725375573956e-104

    1. Initial program 3.4

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

    2. Simplified3.4

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

    4. Applied fma-udef3.4

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

    5. Simplified3.1

      \[\leadsto \color{blue}{\frac{y}{\frac{a - t}{z - t}}} + x\]
    6. Using strategy rm

    7. Applied div-inv3.2

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

    8. Applied *-un-lft-identity3.2

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

    9. Applied times-frac3.3

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

    10. Simplified3.3

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

    if 4.0702725375573956e-104 < t

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied div-sub0.1

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.142790662965348752755157353290722225006 \cdot 10^{-181}:\\ \;\;\;\;x + \frac{y}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;t \le 4.070272537557395579919809546399400646441 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{a - t} \cdot \left(y \cdot \left(z - t\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - \frac{t}{a - t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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