Average Error: 43.4 → 0.8
Time: 12.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r54463 = x;
        double r54464 = exp(r54463);
        double r54465 = -r54463;
        double r54466 = exp(r54465);
        double r54467 = r54464 + r54466;
        double r54468 = 2.0;
        double r54469 = r54467 / r54468;
        double r54470 = y;
        double r54471 = cos(r54470);
        double r54472 = r54469 * r54471;
        double r54473 = r54464 - r54466;
        double r54474 = r54473 / r54468;
        double r54475 = sin(r54470);
        double r54476 = r54474 * r54475;
        double r54477 = /* ERROR: no complex support in C */;
        double r54478 = /* ERROR: no complex support in C */;
        return r54478;
}
↓
double f(double x, double y) {
        double r54479 = x;
        double r54480 = exp(r54479);
        double r54481 = -r54479;
        double r54482 = exp(r54481);
        double r54483 = r54480 + r54482;
        double r54484 = 2.0;
        double r54485 = r54483 / r54484;
        double r54486 = y;
        double r54487 = cos(r54486);
        double r54488 = r54485 * r54487;
        double r54489 = 0.3333333333333333;
        double r54490 = 3.0;
        double r54491 = pow(r54479, r54490);
        double r54492 = 0.016666666666666666;
        double r54493 = 5.0;
        double r54494 = pow(r54479, r54493);
        double r54495 = 2.0;
        double r54496 = r54495 * r54479;
        double r54497 = fma(r54492, r54494, r54496);
        double r54498 = fma(r54489, r54491, r54497);
        double r54499 = r54498 / r54484;
        double r54500 = sin(r54486);
        double r54501 = r54499 * r54500;
        double r54502 = /* ERROR: no complex support in C */;
        double r54503 = /* ERROR: no complex support in C */;
        return r54503;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.4

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Taylor expanded around 0 0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2} \cdot \sin y i\right))\]

  3. Simplified0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}}{2} \cdot \sin y i\right))\]

  4. Final simplification0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))