Average Error: 14.7 → 11.2
Time: 11.1s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 4.54856210553534213 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{elif}\;z \le 9.2575140458516465 \cdot 10^{270}:\\ \;\;\;\;\frac{y}{\frac{a - z}{t - x}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le 4.54856210553534213 \cdot 10^{-5}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\

\mathbf{elif}\;z \le 9.2575140458516465 \cdot 10^{270}:\\
\;\;\;\;\frac{y}{\frac{a - z}{t - x}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r106292 = x;
        double r106293 = y;
        double r106294 = z;
        double r106295 = r106293 - r106294;
        double r106296 = t;
        double r106297 = r106296 - r106292;
        double r106298 = a;
        double r106299 = r106298 - r106294;
        double r106300 = r106297 / r106299;
        double r106301 = r106295 * r106300;
        double r106302 = r106292 + r106301;
        return r106302;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r106303 = z;
        double r106304 = 4.548562105535342e-05;
        bool r106305 = r106303 <= r106304;
        double r106306 = y;
        double r106307 = r106306 - r106303;
        double r106308 = a;
        double r106309 = r106308 - r106303;
        double r106310 = r106307 / r106309;
        double r106311 = t;
        double r106312 = x;
        double r106313 = r106311 - r106312;
        double r106314 = fma(r106310, r106313, r106312);
        double r106315 = 9.257514045851647e+270;
        bool r106316 = r106303 <= r106315;
        double r106317 = r106309 / r106313;
        double r106318 = r106306 / r106317;
        double r106319 = r106303 / r106317;
        double r106320 = r106319 - r106312;
        double r106321 = r106318 - r106320;
        double r106322 = r106312 * r106306;
        double r106323 = r106322 / r106303;
        double r106324 = r106323 + r106311;
        double r106325 = r106311 * r106306;
        double r106326 = r106325 / r106303;
        double r106327 = r106324 - r106326;
        double r106328 = r106316 ? r106321 : r106327;
        double r106329 = r106305 ? r106314 : r106328;
        return r106329;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Split input into 3 regimes

  2. if z < 4.548562105535342e-05

    1. Initial program 12.1

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

    2. Simplified12.1

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

    4. Applied clear-num12.3

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

    6. Applied fma-udef12.3

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

    7. Simplified12.0

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

    9. Applied associate-/r/9.2

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

    10. Applied fma-def9.2

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

    if 4.548562105535342e-05 < z < 9.257514045851647e+270

    1. Initial program 18.8

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

    2. Simplified18.8

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

    4. Applied clear-num19.1

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

    6. Applied fma-udef19.1

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

    7. Simplified19.0

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

    9. Applied div-sub19.0

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

    10. Applied associate-+l-15.1

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

    if 9.257514045851647e+270 < z

    1. Initial program 35.3

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

    2. Simplified35.3

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

    4. Applied clear-num35.9

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

    6. Applied fma-udef35.8

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

    7. Simplified35.7

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

    8. Taylor expanded around inf 21.5

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 4.54856210553534213 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{elif}\;z \le 9.2575140458516465 \cdot 10^{270}:\\ \;\;\;\;\frac{y}{\frac{a - z}{t - x}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))