Average Error: 61.8 → 0.3
Time: 18.3s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)\]
\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)
↓
t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)
double f(double t) {
        double r1762006 = 1.0;
        double r1762007 = t;
        double r1762008 = 2e-16;
        double r1762009 = r1762007 * r1762008;
        double r1762010 = r1762006 + r1762009;
        double r1762011 = r1762010 * r1762010;
        double r1762012 = -1.0;
        double r1762013 = 2.0;
        double r1762014 = r1762013 * r1762009;
        double r1762015 = r1762012 - r1762014;
        double r1762016 = r1762011 + r1762015;
        return r1762016;
}
↓
double f(double t) {
        double r1762017 = t;
        double r1762018 = 3.9999999999999997e-32;
        double r1762019 = r1762017 * r1762018;
        double r1762020 = r1762017 * r1762019;
        return r1762020;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.5
Herbie0.3
\[\mathsf{fma}\left(\left(1 + t \cdot 2 \cdot 10^{-16}\right), \left(1 + t \cdot 2 \cdot 10^{-16}\right), \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right)\]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\left(2 \cdot 10^{-16} \cdot t\right) \cdot \left(2 \cdot 10^{-16} \cdot t\right)}\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]

  4. Simplified0.3

    \[\leadsto \color{blue}{\left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right) \cdot t}\]

  5. Final simplification0.3

    \[\leadsto t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (t)
  :name "fma_test1"
  :pre (<= 0.9 t 1.1)

  :herbie-target
  (fma (+ 1 (* t 2e-16)) (+ 1 (* t 2e-16)) (- -1 (* 2 (* t 2e-16))))

  (+ (* (+ 1 (* t 2e-16)) (+ 1 (* t 2e-16))) (- -1 (* 2 (* t 2e-16)))))