Average Error: 0.0 → 0.0
Time: 24.3s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[\left(\sin re \cdot \sqrt{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)}\right) \cdot \sqrt{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)}\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
\left(\sin re \cdot \sqrt{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)}\right) \cdot \sqrt{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)}
double f(double re, double im) {
        double r583448 = 0.5;
        double r583449 = re;
        double r583450 = sin(r583449);
        double r583451 = r583448 * r583450;
        double r583452 = 0.0;
        double r583453 = im;
        double r583454 = r583452 - r583453;
        double r583455 = exp(r583454);
        double r583456 = exp(r583453);
        double r583457 = r583455 + r583456;
        double r583458 = r583451 * r583457;
        return r583458;
}
↓
double f(double re, double im) {
        double r583459 = re;
        double r583460 = sin(r583459);
        double r583461 = im;
        double r583462 = exp(r583461);
        double r583463 = 0.5;
        double r583464 = r583463 / r583462;
        double r583465 = fma(r583462, r583463, r583464);
        double r583466 = sqrt(r583465);
        double r583467 = r583460 * r583466;
        double r583468 = r583467 * r583466;
        return r583468;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right) \cdot \sin re}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.0

    \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)} \cdot \sqrt{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)}\right)} \cdot \sin re\]

  5. Applied associate-*l*0.0

    \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)} \cdot \sin re\right)}\]

  6. Final simplification0.0

    \[\leadsto \left(\sin re \cdot \sqrt{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)}\right) \cdot \sqrt{\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))