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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.206477194529533326393607486873472590929 \cdot 10^{-59} \lor \neg \left(z \le 6.370790812859238783452018396763277563967 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 \cdot \mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}

↓

\begin{array}{l}
\mathbf{if}\;z \le -4.206477194529533326393607486873472590929 \cdot 10^{-59} \lor \neg \left(z \le 6.370790812859238783452018396763277563967 \cdot 10^{-53}\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 \cdot \mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\

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

\end{array}

double f(double x, double y, double z, double t, double a, double b) {
        double r761113 = x;
        double r761114 = y;
        double r761115 = z;
        double r761116 = r761114 * r761115;
        double r761117 = t;
        double r761118 = r761116 / r761117;
        double r761119 = r761113 + r761118;
        double r761120 = a;
        double r761121 = 1.0;
        double r761122 = r761120 + r761121;
        double r761123 = b;
        double r761124 = r761114 * r761123;
        double r761125 = r761124 / r761117;
        double r761126 = r761122 + r761125;
        double r761127 = r761119 / r761126;
        return r761127;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r761128 = z;
        double r761129 = -4.206477194529533e-59;
        bool r761130 = r761128 <= r761129;
        double r761131 = 6.370790812859239e-53;
        bool r761132 = r761128 <= r761131;
        double r761133 = !r761132;
        bool r761134 = r761130 || r761133;
        double r761135 = y;
        double r761136 = t;
        double r761137 = r761135 / r761136;
        double r761138 = x;
        double r761139 = fma(r761137, r761128, r761138);
        double r761140 = 1.0;
        double r761141 = b;
        double r761142 = a;
        double r761143 = 1.0;
        double r761144 = r761142 + r761143;
        double r761145 = fma(r761137, r761141, r761144);
        double r761146 = r761140 * r761145;
        double r761147 = r761139 / r761146;
        double r761148 = r761136 / r761128;
        double r761149 = r761135 / r761148;
        double r761150 = r761138 + r761149;
        double r761151 = r761135 * r761141;
        double r761152 = r761151 / r761136;
        double r761153 = r761144 + r761152;
        double r761154 = r761150 / r761153;
        double r761155 = r761134 ? r761147 : r761154;
        return r761155;
}

Original	16.8
Target	13.4
Herbie	14.0

Derivation

Split input into 2 regimes
if z < -4.206477194529533e-59 or 6.370790812859239e-53 < z
1. Initial program 22.6
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Using strategy rm
3. Applied *-un-lft-identity22.6
  \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 \cdot \left(\left(a + 1\right) + \frac{y \cdot b}{t}\right)}}\]
4. Applied associate-/r*22.6
  \[\leadsto \color{blue}{\frac{\frac{x + \frac{y \cdot z}{t}}{1}}{\left(a + 1\right) + \frac{y \cdot b}{t}}}\]
5. Simplified20.1
  \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
6. Using strategy rm
7. Applied *-un-lft-identity20.1
  \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\left(a + 1\right) + \color{blue}{1 \cdot \frac{y \cdot b}{t}}}\]
8. Applied *-un-lft-identity20.1
  \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 \cdot \left(a + 1\right)} + 1 \cdot \frac{y \cdot b}{t}}\]
9. Applied distribute-lft-out20.1
  \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\color{blue}{1 \cdot \left(\left(a + 1\right) + \frac{y \cdot b}{t}\right)}}\]
10. Simplified17.9
  \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 \cdot \color{blue}{\mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}}\]
if -4.206477194529533e-59 < z < 6.370790812859239e-53
1. Initial program 8.9
  \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
2. Using strategy rm
3. Applied associate-/l*8.8
  \[\leadsto \frac{x + \color{blue}{\frac{y}{\frac{t}{z}}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\]
Recombined 2 regimes into one program.
Final simplification14.0
\[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.206477194529533326393607486873472590929 \cdot 10^{-59} \lor \neg \left(z \le 6.370790812859238783452018396763277563967 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{1 \cdot \mathsf{fma}\left(\frac{y}{t}, b, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t}{z}}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< t -1.3659085366310088e-271) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 3.036967103737246e-130) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b)))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1) (/ (* y b) t))))

Error

Target

Derivation

`if z < -4.206477194529533e-59 or 6.370790812859239e-53 < z`

`if -4.206477194529533e-59 < z < 6.370790812859239e-53`

Reproduce