Average Error: 24.5 → 10.1
Time: 4.4s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.10252611137827803 \cdot 10^{112} \lor \neg \left(z \le 1.30984479346046122 \cdot 10^{176}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.10252611137827803 \cdot 10^{112} \lor \neg \left(z \le 1.30984479346046122 \cdot 10^{176}\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r636128 = x;
        double r636129 = y;
        double r636130 = z;
        double r636131 = r636129 - r636130;
        double r636132 = t;
        double r636133 = r636132 - r636128;
        double r636134 = r636131 * r636133;
        double r636135 = a;
        double r636136 = r636135 - r636130;
        double r636137 = r636134 / r636136;
        double r636138 = r636128 + r636137;
        return r636138;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r636139 = z;
        double r636140 = -3.102526111378278e+112;
        bool r636141 = r636139 <= r636140;
        double r636142 = 1.3098447934604612e+176;
        bool r636143 = r636139 <= r636142;
        double r636144 = !r636143;
        bool r636145 = r636141 || r636144;
        double r636146 = y;
        double r636147 = x;
        double r636148 = r636147 / r636139;
        double r636149 = t;
        double r636150 = r636149 / r636139;
        double r636151 = r636148 - r636150;
        double r636152 = fma(r636146, r636151, r636149);
        double r636153 = r636146 - r636139;
        double r636154 = a;
        double r636155 = r636154 - r636139;
        double r636156 = r636153 / r636155;
        double r636157 = r636149 - r636147;
        double r636158 = fma(r636156, r636157, r636147);
        double r636159 = r636145 ? r636152 : r636158;
        return r636159;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.5
Target11.7
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.102526111378278e+112 or 1.3098447934604612e+176 < z

    1. Initial program 46.6

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

    2. Simplified22.1

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

    3. Taylor expanded around inf 26.2

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

    4. Simplified16.8

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

    if -3.102526111378278e+112 < z < 1.3098447934604612e+176

    1. Initial program 15.2

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

    2. Simplified7.2

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

    4. Applied div-inv7.3

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

    6. Applied un-div-inv7.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.10252611137827803 \cdot 10^{112} \lor \neg \left(z \le 1.30984479346046122 \cdot 10^{176}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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