Average Error: 61.8 → 0.3
Time: 10.2s
Precision: 64
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[3.9999999999999997 \cdot 10^{-32} \cdot \left(t \cdot t\right)\]
double f(double t) {
        double r7828092 = 1.0;
        double r7828093 = t;
        double r7828094 = 2e-16;
        double r7828095 = r7828093 * r7828094;
        double r7828096 = r7828092 + r7828095;
        double r7828097 = r7828096 * r7828096;
        double r7828098 = -1.0;
        double r7828099 = 2.0;
        double r7828100 = r7828099 * r7828095;
        double r7828101 = r7828098 - r7828100;
        double r7828102 = r7828097 + r7828101;
        return r7828102;
}
↓
double f(double t) {
        double r7828103 = 3.9999999999999997e-32;
        double r7828104 = t;
        double r7828105 = r7828104 * r7828104;
        double r7828106 = r7828103 * r7828105;
        return r7828106;
}


\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)

Error

Bits error versus t

Target

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

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

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

  3. Taylor expanded around inf 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 2019102 +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)))))