Average Error: 61.8 → 0.3
Time: 2.0s
Precision: 64
\[0.900000000000000022 \le t \le 1.1000000000000001\]
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(3.9999999999999997 \cdot 10^{-32} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\]
\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)
↓
\left(3.9999999999999997 \cdot 10^{-32} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}
double f(double t) {
        double r79485 = 1.0;
        double r79486 = t;
        double r79487 = 2e-16;
        double r79488 = r79486 * r79487;
        double r79489 = r79485 + r79488;
        double r79490 = r79489 * r79489;
        double r79491 = -1.0;
        double r79492 = 2.0;
        double r79493 = r79492 * r79488;
        double r79494 = r79491 - r79493;
        double r79495 = r79490 + r79494;
        return r79495;
}
↓
double f(double t) {
        double r79496 = 3.9999999999999997e-32;
        double r79497 = t;
        double r79498 = fabs(r79497);
        double r79499 = r79496 * r79498;
        double r79500 = 2.0;
        double r79501 = pow(r79497, r79500);
        double r79502 = sqrt(r79501);
        double r79503 = r79499 * r79502;
        return r79503;
}

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. Using strategy rm

  4. Applied add-sqr-sqrt0.3

    \[\leadsto 3.9999999999999997 \cdot 10^{-32} \cdot \color{blue}{\left(\sqrt{{t}^{2}} \cdot \sqrt{{t}^{2}}\right)}\]

  5. Applied associate-*r*0.3

    \[\leadsto \color{blue}{\left(3.9999999999999997 \cdot 10^{-32} \cdot \sqrt{{t}^{2}}\right) \cdot \sqrt{{t}^{2}}}\]

  6. Simplified0.3

    \[\leadsto \color{blue}{\left(3.9999999999999997 \cdot 10^{-32} \cdot \left|t\right|\right)} \cdot \sqrt{{t}^{2}}\]

  7. Final simplification0.3

    \[\leadsto \left(3.9999999999999997 \cdot 10^{-32} \cdot \left|t\right|\right) \cdot \sqrt{{t}^{2}}\]

Reproduce

herbie shell --seed 2020062 
(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)))))