Average Error: 61.8 → 0.4
Time: 9.5s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[3.9999999999999997 \cdot 10^{-32} \cdot \left(t \cdot t\right)\]
\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)
↓
3.9999999999999997 \cdot 10^{-32} \cdot \left(t \cdot t\right)
double f(double t) {
        double r4862060 = 1.0;
        double r4862061 = t;
        double r4862062 = 2e-16;
        double r4862063 = r4862061 * r4862062;
        double r4862064 = r4862060 + r4862063;
        double r4862065 = r4862064 * r4862064;
        double r4862066 = -1.0;
        double r4862067 = 2.0;
        double r4862068 = r4862067 * r4862063;
        double r4862069 = r4862066 - r4862068;
        double r4862070 = r4862065 + r4862069;
        return r4862070;
}
↓
double f(double t) {
        double r4862071 = 3.9999999999999997e-32;
        double r4862072 = t;
        double r4862073 = r4862072 * r4862072;
        double r4862074 = r4862071 * r4862073;
        return r4862074;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.6
Herbie0.4
\[(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right))_*\]

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 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019119 
(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)))))