Average Error: 61.8 → 0.3
Time: 14.6s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}\]
\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 {t}^{2}
double f(double t) {
        double r65192 = 1.0;
        double r65193 = t;
        double r65194 = 2e-16;
        double r65195 = r65193 * r65194;
        double r65196 = r65192 + r65195;
        double r65197 = r65196 * r65196;
        double r65198 = -1.0;
        double r65199 = 2.0;
        double r65200 = r65199 * r65195;
        double r65201 = r65198 - r65200;
        double r65202 = r65197 + r65201;
        return r65202;
}
↓
double f(double t) {
        double r65203 = 3.9999999999999997e-32;
        double r65204 = t;
        double r65205 = 2.0;
        double r65206 = pow(r65204, r65205);
        double r65207 = r65203 * r65206;
        return r65207;
}

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

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. Taylor expanded around 0 0.3

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2020042 
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :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)))))