Average Error: 61.8 → 0.3
Time: 18.0s
Precision: 64
\[0.9 \le t \le 1.1\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)\]
\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)
↓
t \cdot \left(t \cdot 3.9999999999999997 \cdot 10^{-32}\right)
double f(double t) {
        double r3496009 = 1.0;
        double r3496010 = t;
        double r3496011 = 2e-16;
        double r3496012 = r3496010 * r3496011;
        double r3496013 = r3496009 + r3496012;
        double r3496014 = r3496013 * r3496013;
        double r3496015 = -1.0;
        double r3496016 = 2.0;
        double r3496017 = r3496016 * r3496012;
        double r3496018 = r3496015 - r3496017;
        double r3496019 = r3496014 + r3496018;
        return r3496019;
}
↓
double f(double t) {
        double r3496020 = t;
        double r3496021 = 3.9999999999999997e-32;
        double r3496022 = r3496020 * r3496021;
        double r3496023 = r3496020 * r3496022;
        return r3496023;
}

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

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

  4. Applied associate-*l*0.3

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

  5. Taylor expanded around 0 0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(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)))))