Average Error: 24.5 → 10.6
Time: 24.2s
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 r1063120 = x;
        double r1063121 = y;
        double r1063122 = r1063121 - r1063120;
        double r1063123 = z;
        double r1063124 = t;
        double r1063125 = r1063123 - r1063124;
        double r1063126 = r1063122 * r1063125;
        double r1063127 = a;
        double r1063128 = r1063127 - r1063124;
        double r1063129 = r1063126 / r1063128;
        double r1063130 = r1063120 + r1063129;
        return r1063130;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r1063131 = a;
        double r1063132 = -1.5730863727778726e-148;
        bool r1063133 = r1063131 <= r1063132;
        double r1063134 = y;
        double r1063135 = x;
        double r1063136 = r1063134 - r1063135;
        double r1063137 = z;
        double r1063138 = t;
        double r1063139 = r1063137 - r1063138;
        double r1063140 = 1.0;
        double r1063141 = r1063131 - r1063138;
        double r1063142 = r1063140 / r1063141;
        double r1063143 = r1063139 * r1063142;
        double r1063144 = r1063136 * r1063143;
        double r1063145 = r1063144 + r1063135;
        double r1063146 = 1.8315839196138794e-164;
        bool r1063147 = r1063131 <= r1063146;
        double r1063148 = r1063135 / r1063138;
        double r1063149 = r1063137 * r1063134;
        double r1063150 = r1063149 / r1063138;
        double r1063151 = r1063134 - r1063150;
        double r1063152 = fma(r1063148, r1063137, r1063151);
        double r1063153 = r1063139 / r1063141;
        double r1063154 = r1063136 * r1063153;
        double r1063155 = r1063154 + r1063135;
        double r1063156 = r1063147 ? r1063152 : r1063155;
        double r1063157 = r1063133 ? r1063145 : r1063156;
        return r1063157;
}

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))))