Average Error: 24.6 → 11.0
Time: 20.8s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -3.543954312374929 \cdot 10^{-257} \lor \neg \left(a \le 1.0935137489299929 \cdot 10^{-133}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -3.543954312374929 \cdot 10^{-257} \lor \neg \left(a \le 1.0935137489299929 \cdot 10^{-133}\right):\\
\;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r671620 = x;
        double r671621 = y;
        double r671622 = r671621 - r671620;
        double r671623 = z;
        double r671624 = t;
        double r671625 = r671623 - r671624;
        double r671626 = r671622 * r671625;
        double r671627 = a;
        double r671628 = r671627 - r671624;
        double r671629 = r671626 / r671628;
        double r671630 = r671620 + r671629;
        return r671630;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r671631 = a;
        double r671632 = -3.543954312374929e-257;
        bool r671633 = r671631 <= r671632;
        double r671634 = 1.093513748929993e-133;
        bool r671635 = r671631 <= r671634;
        double r671636 = !r671635;
        bool r671637 = r671633 || r671636;
        double r671638 = y;
        double r671639 = x;
        double r671640 = r671638 - r671639;
        double r671641 = z;
        double r671642 = t;
        double r671643 = r671641 - r671642;
        double r671644 = r671631 - r671642;
        double r671645 = r671643 / r671644;
        double r671646 = r671640 * r671645;
        double r671647 = r671646 + r671639;
        double r671648 = r671639 * r671641;
        double r671649 = r671648 / r671642;
        double r671650 = r671638 + r671649;
        double r671651 = r671641 * r671638;
        double r671652 = r671651 / r671642;
        double r671653 = r671650 - r671652;
        double r671654 = r671637 ? r671647 : r671653;
        return r671654;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.6
Target9.3
Herbie11.0
\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -3.543954312374929e-257 or 1.093513748929993e-133 < a

    1. Initial program 23.6

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

    2. Simplified12.9

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

    4. Applied fma-udef13.0

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

    6. Applied div-inv13.0

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

    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. Simplified10.4

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

    if -3.543954312374929e-257 < a < 1.093513748929993e-133

    1. Initial program 30.2

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

    2. Simplified24.5

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

    4. Applied fma-udef24.5

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

    6. Applied div-inv24.6

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

    7. Applied associate-*l*19.5

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

    8. Simplified19.5

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

    9. Taylor expanded around inf 13.8

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.543954312374929 \cdot 10^{-257} \lor \neg \left(a \le 1.0935137489299929 \cdot 10^{-133}\right):\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))))