Average Error: 61.8 → 0.3
Time: 11.8s
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 r43331 = 1.0;
        double r43332 = t;
        double r43333 = 2e-16;
        double r43334 = r43332 * r43333;
        double r43335 = r43331 + r43334;
        double r43336 = r43335 * r43335;
        double r43337 = -1.0;
        double r43338 = 2.0;
        double r43339 = r43338 * r43334;
        double r43340 = r43337 - r43339;
        double r43341 = r43336 + r43340;
        return r43341;
}
↓
double f(double t) {
        double r43342 = 3.9999999999999997e-32;
        double r43343 = t;
        double r43344 = fabs(r43343);
        double r43345 = r43342 * r43344;
        double r43346 = 2.0;
        double r43347 = pow(r43343, r43346);
        double r43348 = sqrt(r43347);
        double r43349 = r43345 * r43348;
        return r43349;
}

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 2019195 
(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)))))