Average Error: 0.0 → 0.0
Time: 6.9s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{\mathsf{fma}\left(\frac{2}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 1\right)}{\mathsf{fma}\left(\frac{2}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 2\right)}\]
\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}
↓
\frac{\mathsf{fma}\left(\frac{2}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 1\right)}{\mathsf{fma}\left(\frac{2}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 2\right)}
double f(double t) {
        double r1675059 = 1.0;
        double r1675060 = 2.0;
        double r1675061 = t;
        double r1675062 = r1675060 * r1675061;
        double r1675063 = r1675059 + r1675061;
        double r1675064 = r1675062 / r1675063;
        double r1675065 = r1675064 * r1675064;
        double r1675066 = r1675059 + r1675065;
        double r1675067 = r1675060 + r1675065;
        double r1675068 = r1675066 / r1675067;
        return r1675068;
}
↓
double f(double t) {
        double r1675069 = 2.0;
        double r1675070 = 1.0;
        double r1675071 = t;
        double r1675072 = r1675070 + r1675071;
        double r1675073 = r1675072 / r1675071;
        double r1675074 = r1675069 / r1675073;
        double r1675075 = fma(r1675074, r1675074, r1675070);
        double r1675076 = fma(r1675074, r1675074, r1675069);
        double r1675077 = r1675075 / r1675076;
        return r1675077;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{2}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 1\right)}{\mathsf{fma}\left(\frac{2}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 2\right)}}\]

  3. Final simplification0.0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 1\right)}{\mathsf{fma}\left(\frac{2}{\frac{1 + t}{t}}, \frac{2}{\frac{1 + t}{t}}, 2\right)}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 1"
  (/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))))