Average Error: 24.5 → 10.4
Time: 8.9s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.5165747926583058 \cdot 10^{-205} \lor \neg \left(a \le 1.1030275495192448 \cdot 10^{-211}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.5165747926583058 \cdot 10^{-205} \lor \neg \left(a \le 1.1030275495192448 \cdot 10^{-211}\right):\\
\;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r627780 = x;
        double r627781 = y;
        double r627782 = r627781 - r627780;
        double r627783 = z;
        double r627784 = t;
        double r627785 = r627783 - r627784;
        double r627786 = r627782 * r627785;
        double r627787 = a;
        double r627788 = r627787 - r627784;
        double r627789 = r627786 / r627788;
        double r627790 = r627780 + r627789;
        return r627790;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r627791 = a;
        double r627792 = -1.5165747926583058e-205;
        bool r627793 = r627791 <= r627792;
        double r627794 = 1.1030275495192448e-211;
        bool r627795 = r627791 <= r627794;
        double r627796 = !r627795;
        bool r627797 = r627793 || r627796;
        double r627798 = y;
        double r627799 = x;
        double r627800 = r627798 - r627799;
        double r627801 = z;
        double r627802 = t;
        double r627803 = r627801 - r627802;
        double r627804 = r627791 - r627802;
        double r627805 = r627803 / r627804;
        double r627806 = r627800 * r627805;
        double r627807 = r627806 + r627799;
        double r627808 = r627799 / r627802;
        double r627809 = r627801 * r627798;
        double r627810 = r627809 / r627802;
        double r627811 = r627798 - r627810;
        double r627812 = fma(r627808, r627801, r627811);
        double r627813 = r627797 ? r627807 : r627812;
        return r627813;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.5
Target9.2
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -1.5165747926583058e-205 or 1.1030275495192448e-211 < a

    1. Initial program 23.7

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

    2. Simplified13.1

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

    4. Applied fma-udef13.2

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

    6. Applied div-inv13.3

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

    7. Applied associate-*l*10.5

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

    8. Simplified10.5

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

    if -1.5165747926583058e-205 < a < 1.1030275495192448e-211

    1. Initial program 29.4

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

    2. Simplified23.3

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

    4. Applied fma-udef23.3

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

    6. Applied div-inv23.4

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

    7. Applied associate-*l*18.4

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

    8. Simplified18.3

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

    9. Taylor expanded around inf 10.7

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

    10. Simplified9.9

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.5165747926583058 \cdot 10^{-205} \lor \neg \left(a \le 1.1030275495192448 \cdot 10^{-211}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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