Average Error: 61.8 → 0.3
Time: 36.3s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1.0 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1.0 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1.0 - 2.0 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)\]
\left(1.0 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1.0 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1.0 - 2.0 \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 r4234219 = 1.0;
        double r4234220 = t;
        double r4234221 = 2e-16;
        double r4234222 = r4234220 * r4234221;
        double r4234223 = r4234219 + r4234222;
        double r4234224 = r4234223 * r4234223;
        double r4234225 = -1.0;
        double r4234226 = 2.0;
        double r4234227 = r4234226 * r4234222;
        double r4234228 = r4234225 - r4234227;
        double r4234229 = r4234224 + r4234228;
        return r4234229;
}
↓
double f(double t) {
        double r4234230 = t;
        double r4234231 = 3.9999999999999997e-32;
        double r4234232 = r4234230 * r4234231;
        double r4234233 = r4234230 * r4234232;
        return r4234233;
}

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
\[\mathsf{fma}\left(1.0 + t \cdot 2 \cdot 10^{-16}, 1.0 + t \cdot 2 \cdot 10^{-16}, -1.0 - 2.0 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

Derivation

  1. Initial program 61.8

    \[\left(1.0 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1.0 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1.0 - 2.0 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Taylor expanded around 0 0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019165 
(FPCore (t)
  :name "fma_test1"
  :pre (<= 0.9 t 1.1)

  :herbie-target
  (fma (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16)) (- -1.0 (* 2.0 (* t 2e-16))))

  (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))