Average Error: 25.4 → 7.9
Time: 15.5s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.065570213468218434425365539194697957465 \cdot 10^{-134} \lor \neg \left(a \le 2.084757909820724198119172407300139325836 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{z - t}{a - t} \cdot y + \mathsf{fma}\left(\frac{1}{\frac{a - t}{z - t}}, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot y + \frac{x \cdot z}{t}\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.065570213468218434425365539194697957465 \cdot 10^{-134} \lor \neg \left(a \le 2.084757909820724198119172407300139325836 \cdot 10^{-79}\right):\\
\;\;\;\;\frac{z - t}{a - t} \cdot y + \mathsf{fma}\left(\frac{1}{\frac{a - t}{z - t}}, -x, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r455742 = x;
        double r455743 = y;
        double r455744 = r455743 - r455742;
        double r455745 = z;
        double r455746 = t;
        double r455747 = r455745 - r455746;
        double r455748 = r455744 * r455747;
        double r455749 = a;
        double r455750 = r455749 - r455746;
        double r455751 = r455748 / r455750;
        double r455752 = r455742 + r455751;
        return r455752;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r455753 = a;
        double r455754 = -1.0655702134682184e-134;
        bool r455755 = r455753 <= r455754;
        double r455756 = 2.0847579098207242e-79;
        bool r455757 = r455753 <= r455756;
        double r455758 = !r455757;
        bool r455759 = r455755 || r455758;
        double r455760 = z;
        double r455761 = t;
        double r455762 = r455760 - r455761;
        double r455763 = r455753 - r455761;
        double r455764 = r455762 / r455763;
        double r455765 = y;
        double r455766 = r455764 * r455765;
        double r455767 = 1.0;
        double r455768 = r455763 / r455762;
        double r455769 = r455767 / r455768;
        double r455770 = x;
        double r455771 = -r455770;
        double r455772 = fma(r455769, r455771, r455770);
        double r455773 = r455766 + r455772;
        double r455774 = r455770 * r455760;
        double r455775 = r455774 / r455761;
        double r455776 = r455766 + r455775;
        double r455777 = r455759 ? r455773 : r455776;
        return r455777;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original25.4
Target9.8
Herbie7.9
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -1.0655702134682184e-134 or 2.0847579098207242e-79 < a

    1. Initial program 23.3

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

    2. Simplified11.0

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

    4. Applied fma-udef11.1

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

    5. Simplified9.2

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

    7. Applied sub-neg9.2

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

    8. Applied distribute-lft-in9.2

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

    9. Applied associate-+l+6.6

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

    10. Simplified6.6

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

    12. Applied clear-num6.6

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

    if -1.0655702134682184e-134 < a < 2.0847579098207242e-79

    1. Initial program 30.4

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

    2. Simplified26.5

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

    4. Applied fma-udef26.5

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

    5. Simplified20.7

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

    7. Applied sub-neg20.7

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

    8. Applied distribute-lft-in20.7

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

    9. Applied associate-+l+12.2

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

    10. Simplified12.2

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

    11. Taylor expanded around inf 10.9

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

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.065570213468218434425365539194697957465 \cdot 10^{-134} \lor \neg \left(a \le 2.084757909820724198119172407300139325836 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{z - t}{a - t} \cdot y + \mathsf{fma}\left(\frac{1}{\frac{a - t}{z - t}}, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a - t} \cdot y + \frac{x \cdot z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 3.7744031700831742e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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