Average Error: 23.9 → 11.8
Time: 27.7s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -8.93223352667291861 \cdot 10^{-192}:\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)\\ \mathbf{elif}\;a \le -4.36811409341494865 \cdot 10^{-253}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -8.93223352667291861 \cdot 10^{-192}:\\
\;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)\\

\mathbf{elif}\;a \le -4.36811409341494865 \cdot 10^{-253}:\\
\;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r439271 = x;
        double r439272 = y;
        double r439273 = r439272 - r439271;
        double r439274 = z;
        double r439275 = t;
        double r439276 = r439274 - r439275;
        double r439277 = r439273 * r439276;
        double r439278 = a;
        double r439279 = r439278 - r439275;
        double r439280 = r439277 / r439279;
        double r439281 = r439271 + r439280;
        return r439281;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r439282 = a;
        double r439283 = -8.932233526672919e-192;
        bool r439284 = r439282 <= r439283;
        double r439285 = z;
        double r439286 = t;
        double r439287 = r439282 - r439286;
        double r439288 = y;
        double r439289 = x;
        double r439290 = r439288 - r439289;
        double r439291 = r439287 / r439290;
        double r439292 = r439285 / r439291;
        double r439293 = r439286 / r439291;
        double r439294 = r439293 - r439289;
        double r439295 = r439292 - r439294;
        double r439296 = -4.3681140934149486e-253;
        bool r439297 = r439282 <= r439296;
        double r439298 = r439289 * r439285;
        double r439299 = r439298 / r439286;
        double r439300 = r439288 + r439299;
        double r439301 = r439285 * r439288;
        double r439302 = r439301 / r439286;
        double r439303 = r439300 - r439302;
        double r439304 = r439285 - r439286;
        double r439305 = r439304 / r439287;
        double r439306 = fma(r439305, r439290, r439289);
        double r439307 = r439297 ? r439303 : r439306;
        double r439308 = r439284 ? r439295 : r439307;
        return r439308;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original23.9
Target9.0
Herbie11.8
\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -8.932233526672919e-192

    1. Initial program 23.2

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

    2. Simplified13.0

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

    4. Applied clear-num13.1

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

    6. Applied fma-udef13.2

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

    7. Simplified13.0

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

    9. Applied div-sub13.0

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

    10. Applied associate-+l-11.3

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

    if -8.932233526672919e-192 < a < -4.3681140934149486e-253

    1. Initial program 27.4

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

    2. Simplified24.1

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

    4. Applied clear-num24.3

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

    6. Applied fma-udef24.4

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

    7. Simplified23.8

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

    8. Taylor expanded around inf 12.9

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

    if -4.3681140934149486e-253 < a

    1. Initial program 24.2

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

    2. Simplified15.2

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

    4. Applied clear-num15.4

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

    6. Applied fma-udef15.5

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

    7. Simplified15.1

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

    9. Applied *-un-lft-identity15.1

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

    10. Applied *-un-lft-identity15.1

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

    11. Applied distribute-lft-out15.1

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

    12. Simplified12.1

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -8.93223352667291861 \cdot 10^{-192}:\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)\\ \mathbf{elif}\;a \le -4.36811409341494865 \cdot 10^{-253}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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