Average Error: 61.8 → 0.3
Time: 11.0s
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 r5393209 = 1.0;
        double r5393210 = t;
        double r5393211 = 2e-16;
        double r5393212 = r5393210 * r5393211;
        double r5393213 = r5393209 + r5393212;
        double r5393214 = r5393213 * r5393213;
        double r5393215 = -1.0;
        double r5393216 = 2.0;
        double r5393217 = r5393216 * r5393212;
        double r5393218 = r5393215 - r5393217;
        double r5393219 = r5393214 + r5393218;
        return r5393219;
}
↓
double f(double t) {
        double r5393220 = 3.9999999999999997e-32;
        double r5393221 = t;
        double r5393222 = r5393221 * r5393221;
        double r5393223 = r5393220 * r5393222;
        return r5393223;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.6
Herbie0.3
\[\mathsf{fma}\left(\left(1 + t \cdot 2 \cdot 10^{-16}\right), \left(1 + t \cdot 2 \cdot 10^{-16}\right), \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right)\]

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. Simplified50.3

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019124 +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)))))