Average Error: 24.6 → 9.6
Time: 8.5s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -5.0386299408790279 \cdot 10^{-180} \lor \neg \left(a \le 4.41666153153843721 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \mathsf{fma}\left(\frac{t}{a - t}, y - x, -1 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -5.0386299408790279 \cdot 10^{-180} \lor \neg \left(a \le 4.41666153153843721 \cdot 10^{-92}\right):\\
\;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \mathsf{fma}\left(\frac{t}{a - t}, y - x, -1 \cdot x\right)\\

\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 r648376 = x;
        double r648377 = y;
        double r648378 = r648377 - r648376;
        double r648379 = z;
        double r648380 = t;
        double r648381 = r648379 - r648380;
        double r648382 = r648378 * r648381;
        double r648383 = a;
        double r648384 = r648383 - r648380;
        double r648385 = r648382 / r648384;
        double r648386 = r648376 + r648385;
        return r648386;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r648387 = a;
        double r648388 = -5.038629940879028e-180;
        bool r648389 = r648387 <= r648388;
        double r648390 = 4.416661531538437e-92;
        bool r648391 = r648387 <= r648390;
        double r648392 = !r648391;
        bool r648393 = r648389 || r648392;
        double r648394 = z;
        double r648395 = t;
        double r648396 = r648387 - r648395;
        double r648397 = y;
        double r648398 = x;
        double r648399 = r648397 - r648398;
        double r648400 = r648396 / r648399;
        double r648401 = r648394 / r648400;
        double r648402 = r648395 / r648396;
        double r648403 = -1.0;
        double r648404 = r648403 * r648398;
        double r648405 = fma(r648402, r648399, r648404);
        double r648406 = r648401 - r648405;
        double r648407 = r648398 / r648395;
        double r648408 = r648394 * r648397;
        double r648409 = r648408 / r648395;
        double r648410 = r648397 - r648409;
        double r648411 = fma(r648407, r648394, r648410);
        double r648412 = r648393 ? r648406 : r648411;
        return r648412;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.6
Target9.1
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -5.038629940879028e-180 or 4.416661531538437e-92 < a

    1. Initial program 23.0

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

    2. Simplified11.1

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

    4. Applied clear-num11.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y - x}}}, z - t, x\right)\]
    5. Using strategy rm

    6. Applied fma-udef11.5

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

    7. Simplified11.2

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

    9. Applied div-sub11.2

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

    10. Applied associate-+l-9.8

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

    12. Applied associate-/r/8.3

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

    13. Applied fma-neg8.3

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

    14. Simplified8.3

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

    if -5.038629940879028e-180 < a < 4.416661531538437e-92

    1. Initial program 29.4

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

    2. Simplified24.9

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

    4. Applied clear-num25.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y - x}}}, z - t, x\right)\]
    5. Using strategy rm

    6. Applied fma-udef25.3

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

    7. Simplified24.9

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

    8. Taylor expanded around inf 14.6

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

    9. Simplified13.8

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -5.0386299408790279 \cdot 10^{-180} \lor \neg \left(a \le 4.41666153153843721 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \mathsf{fma}\left(\frac{t}{a - t}, y - x, -1 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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))))