Average Error: 23.4 → 14.5
Time: 8.4s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -1.9851788359 \cdot 10^{-314}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -0.0:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 7.65441169402193301 \cdot 10^{305}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\
\;\;\;\;x\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -1.9851788359 \cdot 10^{-314}:\\
\;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -0.0:\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 7.65441169402193301 \cdot 10^{305}:\\
\;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{t}{b} - \frac{a}{b}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r744113 = x;
        double r744114 = y;
        double r744115 = r744113 * r744114;
        double r744116 = z;
        double r744117 = t;
        double r744118 = a;
        double r744119 = r744117 - r744118;
        double r744120 = r744116 * r744119;
        double r744121 = r744115 + r744120;
        double r744122 = b;
        double r744123 = r744122 - r744114;
        double r744124 = r744116 * r744123;
        double r744125 = r744114 + r744124;
        double r744126 = r744121 / r744125;
        return r744126;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r744127 = x;
        double r744128 = y;
        double r744129 = r744127 * r744128;
        double r744130 = z;
        double r744131 = t;
        double r744132 = a;
        double r744133 = r744131 - r744132;
        double r744134 = r744130 * r744133;
        double r744135 = r744129 + r744134;
        double r744136 = b;
        double r744137 = r744136 - r744128;
        double r744138 = r744130 * r744137;
        double r744139 = r744128 + r744138;
        double r744140 = r744135 / r744139;
        double r744141 = -inf.0;
        bool r744142 = r744140 <= r744141;
        double r744143 = -1.9851788358795e-314;
        bool r744144 = r744140 <= r744143;
        double r744145 = -0.0;
        bool r744146 = r744140 <= r744145;
        double r744147 = r744131 / r744136;
        double r744148 = r744132 / r744136;
        double r744149 = r744147 - r744148;
        double r744150 = 7.654411694021933e+305;
        bool r744151 = r744140 <= r744150;
        double r744152 = r744151 ? r744140 : r744149;
        double r744153 = r744146 ? r744149 : r744152;
        double r744154 = r744144 ? r744140 : r744153;
        double r744155 = r744142 ? r744127 : r744154;
        return r744155;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original23.4
Target17.9
Herbie14.5
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < -inf.0

    1. Initial program 64.0

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
    2. Using strategy rm

    3. Applied clear-num64.0

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

    4. Simplified64.0

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

    5. Taylor expanded around 0 33.7

      \[\leadsto \color{blue}{x}\]

    if -inf.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < -1.9851788358795e-314 or -0.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 7.654411694021933e+305

    1. Initial program 0.3

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

    if -1.9851788358795e-314 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < -0.0 or 7.654411694021933e+305 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))

    1. Initial program 58.6

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
    2. Using strategy rm

    3. Applied clear-num58.6

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

    4. Simplified58.6

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

    5. Taylor expanded around inf 37.5

      \[\leadsto \color{blue}{\frac{t}{b} - \frac{a}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification14.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -1.9851788359 \cdot 10^{-314}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -0.0:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 7.65441169402193301 \cdot 10^{305}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :herbie-target
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

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