Average Error: 10.3 → 6.5
Time: 15.4s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.06401190083563283 \cdot 10^{38} \lor \neg \left(z \le 2.50952591182416396 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.06401190083563283 \cdot 10^{38} \lor \neg \left(z \le 2.50952591182416396 \cdot 10^{-42}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t - a \cdot z}{z}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r770132 = x;
        double r770133 = y;
        double r770134 = z;
        double r770135 = r770133 * r770134;
        double r770136 = r770132 - r770135;
        double r770137 = t;
        double r770138 = a;
        double r770139 = r770138 * r770134;
        double r770140 = r770137 - r770139;
        double r770141 = r770136 / r770140;
        return r770141;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r770142 = z;
        double r770143 = -4.064011900835633e+38;
        bool r770144 = r770142 <= r770143;
        double r770145 = 2.509525911824164e-42;
        bool r770146 = r770142 <= r770145;
        double r770147 = !r770146;
        bool r770148 = r770144 || r770147;
        double r770149 = x;
        double r770150 = t;
        double r770151 = a;
        double r770152 = r770151 * r770142;
        double r770153 = r770150 - r770152;
        double r770154 = r770149 / r770153;
        double r770155 = y;
        double r770156 = 1.0;
        double r770157 = r770153 / r770142;
        double r770158 = r770156 / r770157;
        double r770159 = r770155 * r770158;
        double r770160 = r770154 - r770159;
        double r770161 = r770155 * r770142;
        double r770162 = r770161 / r770153;
        double r770163 = r770154 - r770162;
        double r770164 = r770148 ? r770160 : r770163;
        return r770164;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.3
Target1.7
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.064011900835633e+38 or 2.509525911824164e-42 < z

    1. Initial program 20.2

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied div-sub20.2

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

    4. Simplified12.7

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

    6. Applied clear-num12.8

      \[\leadsto \frac{x}{t - a \cdot z} - y \cdot \color{blue}{\frac{1}{\frac{t - a \cdot z}{z}}}\]

    if -4.064011900835633e+38 < z < 2.509525911824164e-42

    1. Initial program 0.3

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied div-sub0.3

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

    4. Simplified2.9

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

    6. Applied associate-*r/0.3

      \[\leadsto \frac{x}{t - a \cdot z} - \color{blue}{\frac{y \cdot z}{t - a \cdot z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.06401190083563283 \cdot 10^{38} \lor \neg \left(z \le 2.50952591182416396 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{1}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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