Average Error: 24.5 → 10.6
Time: 24.0s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.573086372777872649310539157707014574793 \cdot 10^{-148}:\\ \;\;\;\;\left(y - x\right) \cdot \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) + x\\ \mathbf{elif}\;a \le 1.831583919613879386678232165594971224481 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.573086372777872649310539157707014574793 \cdot 10^{-148}:\\
\;\;\;\;\left(y - x\right) \cdot \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) + x\\

\mathbf{elif}\;a \le 1.831583919613879386678232165594971224481 \cdot 10^{-164}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r473232 = x;
        double r473233 = y;
        double r473234 = r473233 - r473232;
        double r473235 = z;
        double r473236 = t;
        double r473237 = r473235 - r473236;
        double r473238 = r473234 * r473237;
        double r473239 = a;
        double r473240 = r473239 - r473236;
        double r473241 = r473238 / r473240;
        double r473242 = r473232 + r473241;
        return r473242;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r473243 = a;
        double r473244 = -1.5730863727778726e-148;
        bool r473245 = r473243 <= r473244;
        double r473246 = y;
        double r473247 = x;
        double r473248 = r473246 - r473247;
        double r473249 = z;
        double r473250 = t;
        double r473251 = r473249 - r473250;
        double r473252 = 1.0;
        double r473253 = r473243 - r473250;
        double r473254 = r473252 / r473253;
        double r473255 = r473251 * r473254;
        double r473256 = r473248 * r473255;
        double r473257 = r473256 + r473247;
        double r473258 = 1.8315839196138794e-164;
        bool r473259 = r473243 <= r473258;
        double r473260 = r473247 / r473250;
        double r473261 = r473249 * r473246;
        double r473262 = r473261 / r473250;
        double r473263 = r473246 - r473262;
        double r473264 = fma(r473260, r473249, r473263);
        double r473265 = r473251 / r473253;
        double r473266 = r473248 * r473265;
        double r473267 = r473266 + r473247;
        double r473268 = r473259 ? r473264 : r473267;
        double r473269 = r473245 ? r473257 : r473268;
        return r473269;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.5
Target9.6
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.5730863727778726e-148

    1. Initial program 23.0

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

    2. Simplified12.1

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

    4. Applied fma-udef12.2

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

    6. Applied div-inv12.2

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

    7. Applied associate-*l*10.0

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

    8. Simplified9.9

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

    10. Applied div-inv10.0

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

    if -1.5730863727778726e-148 < a < 1.8315839196138794e-164

    1. Initial program 29.7

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

    2. Simplified24.1

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

    4. Applied fma-udef24.2

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

    6. Applied div-inv24.3

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

    7. Applied associate-*l*19.2

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

    8. Simplified19.1

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

    9. Taylor expanded around inf 13.8

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

    10. Simplified13.7

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

    if 1.8315839196138794e-164 < a

    1. Initial program 23.1

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

    2. Simplified11.6

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

    4. Applied fma-udef11.6

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

    6. Applied div-inv11.7

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

    7. Applied associate-*l*9.6

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

    8. Simplified9.5

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.573086372777872649310539157707014574793 \cdot 10^{-148}:\\ \;\;\;\;\left(y - x\right) \cdot \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) + x\\ \mathbf{elif}\;a \le 1.831583919613879386678232165594971224481 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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 3.7744031700831742e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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