Average Error: 61.8 → 0.3
Time: 13.7s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(t \cdot \left(t \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right)\right)\right)\]
\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(t \cdot \left(t \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right)\right)\right)
double f(double t) {
        double r27558 = 1.0;
        double r27559 = t;
        double r27560 = 2e-16;
        double r27561 = r27559 * r27560;
        double r27562 = r27558 + r27561;
        double r27563 = r27562 * r27562;
        double r27564 = -1.0;
        double r27565 = 2.0;
        double r27566 = r27565 * r27561;
        double r27567 = r27564 - r27566;
        double r27568 = r27563 + r27567;
        return r27568;
}
↓
double f(double t) {
        double r27569 = t;
        double r27570 = 3.9999999999999997e-32;
        double r27571 = r27569 * r27570;
        double r27572 = r27569 * r27571;
        double r27573 = expm1(r27572);
        double r27574 = log1p(r27573);
        return r27574;
}

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
\[\mathsf{fma}\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

  2. Simplified50.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1\right), \mathsf{fma}\left(t, 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1\right), -1 - \left(1.999999999999999958195573448069207123682 \cdot 10^{-16} \cdot t\right) \cdot 2\right)}\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]

  4. Simplified0.3

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \left(t \cdot t\right)}\]
  5. Using strategy rm

  6. Applied log1p-expm1-u0.3

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot \left(t \cdot t\right)\right)\right)}\]

  7. Simplified0.3

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(t \cdot \left(3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot t\right)\right)}\right)\]

  8. Final simplification0.3

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(t \cdot \left(t \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(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)))))