Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 20.9 → 0.3

Time: 10.3s

Precision: 64

Math

\[\sqrt{x \cdot x + y}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.369499690488931210940314873390714060144 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.496275485111779573801266363185889396502 \cdot 10^{83}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r20663155 = x;
        double r20663156 = r20663155 * r20663155;
        double r20663157 = y;
        double r20663158 = r20663156 + r20663157;
        double r20663159 = sqrt(r20663158);
        return r20663159;
}

↓

double f(double x, double y) {
        double r20663160 = x;
        double r20663161 = -1.3694996904889312e+154;
        bool r20663162 = r20663160 <= r20663161;
        double r20663163 = -0.5;
        double r20663164 = y;
        double r20663165 = r20663160 / r20663164;
        double r20663166 = r20663163 / r20663165;
        double r20663167 = r20663166 - r20663160;
        double r20663168 = 1.4962754851117796e+83;
        bool r20663169 = r20663160 <= r20663168;
        double r20663170 = fma(r20663160, r20663160, r20663164);
        double r20663171 = sqrt(r20663170);
        double r20663172 = r20663164 / r20663160;
        double r20663173 = 0.5;
        double r20663174 = fma(r20663172, r20663173, r20663160);
        double r20663175 = r20663169 ? r20663171 : r20663174;
        double r20663176 = r20663162 ? r20663167 : r20663175;
        return r20663176;
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.9
Target	0.5
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.3694996904889312e+154

   1. Initial program 64.0

      \[\sqrt{x \cdot x + y}\]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}}\]

   3. Taylor expanded around -inf 0

      \[\leadsto \color{blue}{-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)}\]

   4. Simplified 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{\frac{x}{y}} - x}\]

   if -1.3694996904889312e+154 < x < 1.4962754851117796e+83

   1. Initial program 0.0

      \[\sqrt{x \cdot x + y}\]

   2. Simplified 0.0

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}}\]

   if 1.4962754851117796e+83 < x

   1. Initial program 43.9

      \[\sqrt{x \cdot x + y}\]

   2. Simplified 43.9

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}}\]

   3. Taylor expanded around inf 1.3

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \frac{y}{x}}\]

   4. Simplified 1.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.369499690488931210940314873390714060144 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.496275485111779573801266363185889396502 \cdot 10^{83}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"

  :herbie-target
  (if (< x -1.5097698010472593e+153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.582399551122541e+57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

  (sqrt (+ (* x x) y)))