Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.1 → 0.5

Time: 19.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(y + z\right) \cdot x}{z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{elif}\;\frac{\left(y + z\right) \cdot x}{z} \le -1.522718976563493184587437004709758745494 \cdot 10^{-277}:\\ \;\;\;\;\frac{\left(y + z\right) \cdot x}{z}\\ \mathbf{elif}\;\frac{\left(y + z\right) \cdot x}{z} \le -0.0:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{elif}\;\frac{\left(y + z\right) \cdot x}{z} \le 2.129993894531984000374357331184793830462 \cdot 10^{297}:\\ \;\;\;\;\frac{\left(y + z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r17206149 = x;
        double r17206150 = y;
        double r17206151 = z;
        double r17206152 = r17206150 + r17206151;
        double r17206153 = r17206149 * r17206152;
        double r17206154 = r17206153 / r17206151;
        return r17206154;
}

↓

double f(double x, double y, double z) {
        double r17206155 = y;
        double r17206156 = z;
        double r17206157 = r17206155 + r17206156;
        double r17206158 = x;
        double r17206159 = r17206157 * r17206158;
        double r17206160 = r17206159 / r17206156;
        double r17206161 = -inf.0;
        bool r17206162 = r17206160 <= r17206161;
        double r17206163 = r17206158 / r17206156;
        double r17206164 = fma(r17206155, r17206163, r17206158);
        double r17206165 = -1.5227189765634932e-277;
        bool r17206166 = r17206160 <= r17206165;
        double r17206167 = -0.0;
        bool r17206168 = r17206160 <= r17206167;
        double r17206169 = 2.129993894531984e+297;
        bool r17206170 = r17206160 <= r17206169;
        double r17206171 = r17206170 ? r17206160 : r17206164;
        double r17206172 = r17206168 ? r17206164 : r17206171;
        double r17206173 = r17206166 ? r17206160 : r17206172;
        double r17206174 = r17206162 ? r17206164 : r17206173;
        return r17206174;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.1
Target	3.1
Herbie	0.5

\[\frac{x}{\frac{z}{y + z}}\]

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ y z)) z) < -inf.0 or -1.5227189765634932e-277 < (/ (* x (+ y z)) z) < -0.0 or 2.129993894531984e+297 < (/ (* x (+ y z)) z)

   1. Initial program 57.1

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

   2. Simplified 1.3

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

   3. Taylor expanded around 0 21.7

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

   4. Simplified 1.3

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

   if -inf.0 < (/ (* x (+ y z)) z) < -1.5227189765634932e-277 or -0.0 < (/ (* x (+ y z)) z) < 2.129993894531984e+297

   1. Initial program 0.3

      \[\frac{x \cdot \left(y + z\right)}{z}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y + z\right) \cdot x}{z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{elif}\;\frac{\left(y + z\right) \cdot x}{z} \le -1.522718976563493184587437004709758745494 \cdot 10^{-277}:\\ \;\;\;\;\frac{\left(y + z\right) \cdot x}{z}\\ \mathbf{elif}\;\frac{\left(y + z\right) \cdot x}{z} \le -0.0:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{elif}\;\frac{\left(y + z\right) \cdot x}{z} \le 2.129993894531984000374357331184793830462 \cdot 10^{297}:\\ \;\;\;\;\frac{\left(y + z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))