Average Error: 0.0 → 0.0
Time: 20.5s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[\sin re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
\sin re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)
double f(double re, double im) {
        double r456005 = 0.5;
        double r456006 = re;
        double r456007 = sin(r456006);
        double r456008 = r456005 * r456007;
        double r456009 = 0.0;
        double r456010 = im;
        double r456011 = r456009 - r456010;
        double r456012 = exp(r456011);
        double r456013 = exp(r456010);
        double r456014 = r456012 + r456013;
        double r456015 = r456008 * r456014;
        return r456015;
}
↓
double f(double re, double im) {
        double r456016 = re;
        double r456017 = sin(r456016);
        double r456018 = im;
        double r456019 = exp(r456018);
        double r456020 = 0.5;
        double r456021 = r456020 / r456019;
        double r456022 = fma(r456019, r456020, r456021);
        double r456023 = r456017 * r456022;
        return r456023;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))