Average Error: 24.3 → 10.8
Time: 25.0s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.322021561305451746656429473091041729273 \cdot 10^{-228}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \mathbf{elif}\;a \le 1.474978086208166197165144421451213606011 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) + x\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.322021561305451746656429473091041729273 \cdot 10^{-228}:\\
\;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\

\mathbf{elif}\;a \le 1.474978086208166197165144421451213606011 \cdot 10^{-72}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(y - x\right) \cdot \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r500086 = x;
        double r500087 = y;
        double r500088 = r500087 - r500086;
        double r500089 = z;
        double r500090 = t;
        double r500091 = r500089 - r500090;
        double r500092 = r500088 * r500091;
        double r500093 = a;
        double r500094 = r500093 - r500090;
        double r500095 = r500092 / r500094;
        double r500096 = r500086 + r500095;
        return r500096;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r500097 = a;
        double r500098 = -1.3220215613054517e-228;
        bool r500099 = r500097 <= r500098;
        double r500100 = y;
        double r500101 = x;
        double r500102 = r500100 - r500101;
        double r500103 = z;
        double r500104 = t;
        double r500105 = r500103 - r500104;
        double r500106 = r500097 - r500104;
        double r500107 = r500105 / r500106;
        double r500108 = r500102 * r500107;
        double r500109 = r500108 + r500101;
        double r500110 = 1.4749780862081662e-72;
        bool r500111 = r500097 <= r500110;
        double r500112 = r500101 / r500104;
        double r500113 = r500103 * r500100;
        double r500114 = r500113 / r500104;
        double r500115 = r500100 - r500114;
        double r500116 = fma(r500112, r500103, r500115);
        double r500117 = 1.0;
        double r500118 = r500117 / r500106;
        double r500119 = r500105 * r500118;
        double r500120 = r500102 * r500119;
        double r500121 = r500120 + r500101;
        double r500122 = r500111 ? r500116 : r500121;
        double r500123 = r500099 ? r500109 : r500122;
        return r500123;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.3
Target9.5
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.3220215613054517e-228

    1. Initial program 23.4

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

    2. Simplified13.5

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

    4. Applied fma-udef13.6

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

    6. Applied div-inv13.6

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

    7. Applied associate-*l*10.9

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

    8. Simplified10.8

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

    if -1.3220215613054517e-228 < a < 1.4749780862081662e-72

    1. Initial program 29.5

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

    2. Simplified24.7

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

    4. Applied fma-udef24.7

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

    6. Applied div-inv24.8

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

    7. Applied associate-*l*20.4

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

    8. Simplified20.3

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

    9. Taylor expanded around inf 15.4

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

    10. Simplified14.9

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

    if 1.4749780862081662e-72 < a

    1. Initial program 21.9

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

    2. Simplified9.8

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

    4. Applied fma-udef9.9

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

    6. Applied div-inv10.0

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

    7. Applied associate-*l*8.0

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

    8. Simplified7.9

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

    10. Applied div-inv8.0

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.322021561305451746656429473091041729273 \cdot 10^{-228}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \mathbf{elif}\;a \le 1.474978086208166197165144421451213606011 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))