Average Error: 16.5 → 7.8
Time: 22.8s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -4.869256490744422828247013908290630268308 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a - t}, y + x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.581593314547877943010122734240759935837 \cdot 10^{-285}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 4.891049202688294908428058229064567412917 \cdot 10^{301}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -4.869256490744422828247013908290630268308 \cdot 10^{-232}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a - t}, y + x\right)\\

\mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.581593314547877943010122734240759935837 \cdot 10^{-285}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 4.891049202688294908428058229064567412917 \cdot 10^{301}:\\
\;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r21989619 = x;
        double r21989620 = y;
        double r21989621 = r21989619 + r21989620;
        double r21989622 = z;
        double r21989623 = t;
        double r21989624 = r21989622 - r21989623;
        double r21989625 = r21989624 * r21989620;
        double r21989626 = a;
        double r21989627 = r21989626 - r21989623;
        double r21989628 = r21989625 / r21989627;
        double r21989629 = r21989621 - r21989628;
        return r21989629;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r21989630 = y;
        double r21989631 = x;
        double r21989632 = r21989630 + r21989631;
        double r21989633 = z;
        double r21989634 = t;
        double r21989635 = r21989633 - r21989634;
        double r21989636 = r21989635 * r21989630;
        double r21989637 = a;
        double r21989638 = r21989637 - r21989634;
        double r21989639 = r21989636 / r21989638;
        double r21989640 = r21989632 - r21989639;
        double r21989641 = -4.869256490744423e-232;
        bool r21989642 = r21989640 <= r21989641;
        double r21989643 = r21989634 - r21989633;
        double r21989644 = r21989643 / r21989638;
        double r21989645 = fma(r21989630, r21989644, r21989632);
        double r21989646 = 2.581593314547878e-285;
        bool r21989647 = r21989640 <= r21989646;
        double r21989648 = r21989633 / r21989634;
        double r21989649 = fma(r21989648, r21989630, r21989631);
        double r21989650 = 4.891049202688295e+301;
        bool r21989651 = r21989640 <= r21989650;
        double r21989652 = r21989630 / r21989634;
        double r21989653 = fma(r21989652, r21989633, r21989631);
        double r21989654 = r21989651 ? r21989640 : r21989653;
        double r21989655 = r21989647 ? r21989649 : r21989654;
        double r21989656 = r21989642 ? r21989645 : r21989655;
        return r21989656;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.5
Target8.1
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -4.869256490744423e-232

    1. Initial program 12.5

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

    2. Simplified7.0

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

    if -4.869256490744423e-232 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 2.581593314547878e-285

    1. Initial program 58.1

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

    2. Simplified58.0

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

    3. Taylor expanded around inf 18.3

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

    4. Simplified18.5

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

    if 2.581593314547878e-285 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 4.891049202688295e+301

    1. Initial program 1.4

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

    if 4.891049202688295e+301 < (- (+ x y) (/ (* (- z t) y) (- a t)))

    1. Initial program 59.6

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

    2. Simplified25.6

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

    4. Applied add-cube-cbrt25.8

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

    5. Applied associate-/r*25.8

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

    6. Taylor expanded around inf 40.7

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

    7. Simplified28.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -4.869256490744422828247013908290630268308 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a - t}, y + x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.581593314547877943010122734240759935837 \cdot 10^{-285}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 4.891049202688294908428058229064567412917 \cdot 10^{301}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

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