Average Error: 61.8 → 0.3
Time: 1.3m
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t\]
\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)
↓
\left(3.9999999999999997 \cdot 10^{-32} \cdot t\right) \cdot t
double f(double t) {
        double r2209146 = 1.0;
        double r2209147 = t;
        double r2209148 = 2e-16;
        double r2209149 = r2209147 * r2209148;
        double r2209150 = r2209146 + r2209149;
        double r2209151 = r2209150 * r2209150;
        double r2209152 = -1.0;
        double r2209153 = 2.0;
        double r2209154 = r2209153 * r2209149;
        double r2209155 = r2209152 - r2209154;
        double r2209156 = r2209151 + r2209155;
        return r2209156;
}
↓
double f(double t) {
        double r2209157 = 3.9999999999999997e-32;
        double r2209158 = t;
        double r2209159 = r2209157 * r2209158;
        double r2209160 = r2209159 * r2209158;
        return r2209160;
}

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. Simplified0.4

    \[\leadsto \color{blue}{\left(2 \cdot 10^{-16} \cdot t\right) \cdot \left(2 \cdot 10^{-16} \cdot t\right)}\]
  3. Using strategy rm

  4. Applied associate-*r*0.3

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

  5. Taylor expanded around -inf 0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019151 +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)))))