Average Error: 0.0 → 0.1
Time: 47.1s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[\left(\sin re \cdot 0.5\right) \cdot e^{im} + \frac{\frac{\sin re}{\sqrt{\sqrt[3]{e^{im}}}}}{\sqrt{\sqrt{e^{im}}}} \cdot \frac{\frac{0.5}{\sqrt{\sqrt{e^{im}}}}}{\left|\sqrt[3]{e^{im}}\right|}\]
double f(double re, double im) {
        double r1620398 = 0.5;
        double r1620399 = re;
        double r1620400 = sin(r1620399);
        double r1620401 = r1620398 * r1620400;
        double r1620402 = 0.0;
        double r1620403 = im;
        double r1620404 = r1620402 - r1620403;
        double r1620405 = exp(r1620404);
        double r1620406 = exp(r1620403);
        double r1620407 = r1620405 + r1620406;
        double r1620408 = r1620401 * r1620407;
        return r1620408;
}
↓
double f(double re, double im) {
        double r1620409 = re;
        double r1620410 = sin(r1620409);
        double r1620411 = 0.5;
        double r1620412 = r1620410 * r1620411;
        double r1620413 = im;
        double r1620414 = exp(r1620413);
        double r1620415 = r1620412 * r1620414;
        double r1620416 = cbrt(r1620414);
        double r1620417 = sqrt(r1620416);
        double r1620418 = r1620410 / r1620417;
        double r1620419 = sqrt(r1620414);
        double r1620420 = sqrt(r1620419);
        double r1620421 = r1620418 / r1620420;
        double r1620422 = r1620411 / r1620420;
        double r1620423 = fabs(r1620416);
        double r1620424 = r1620422 / r1620423;
        double r1620425 = r1620421 * r1620424;
        double r1620426 = r1620415 + r1620425;
        return r1620426;
}


\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
\left(\sin re \cdot 0.5\right) \cdot e^{im} + \frac{\frac{\sin re}{\sqrt{\sqrt[3]{e^{im}}}}}{\sqrt{\sqrt{e^{im}}}} \cdot \frac{\frac{0.5}{\sqrt{\sqrt{e^{im}}}}}{\left|\sqrt[3]{e^{im}}\right|}

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

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-/r*0.0

    \[\leadsto e^{im} \cdot \left(0.5 \cdot \sin re\right) + \color{blue}{\frac{\frac{0.5 \cdot \sin re}{\sqrt{e^{im}}}}{\sqrt{e^{im}}}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.0

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

  8. Applied sqrt-prod0.0

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

  9. Applied add-cube-cbrt0.1

    \[\leadsto e^{im} \cdot \left(0.5 \cdot \sin re\right) + \frac{\frac{0.5 \cdot \sin re}{\sqrt{\color{blue}{\left(\sqrt[3]{e^{im}} \cdot \sqrt[3]{e^{im}}\right) \cdot \sqrt[3]{e^{im}}}}}}{\sqrt{\sqrt{e^{im}}} \cdot \sqrt{\sqrt{e^{im}}}}\]

  10. Applied sqrt-prod0.1

    \[\leadsto e^{im} \cdot \left(0.5 \cdot \sin re\right) + \frac{\frac{0.5 \cdot \sin re}{\color{blue}{\sqrt{\sqrt[3]{e^{im}} \cdot \sqrt[3]{e^{im}}} \cdot \sqrt{\sqrt[3]{e^{im}}}}}}{\sqrt{\sqrt{e^{im}}} \cdot \sqrt{\sqrt{e^{im}}}}\]

  11. Applied times-frac0.1

    \[\leadsto e^{im} \cdot \left(0.5 \cdot \sin re\right) + \frac{\color{blue}{\frac{0.5}{\sqrt{\sqrt[3]{e^{im}} \cdot \sqrt[3]{e^{im}}}} \cdot \frac{\sin re}{\sqrt{\sqrt[3]{e^{im}}}}}}{\sqrt{\sqrt{e^{im}}} \cdot \sqrt{\sqrt{e^{im}}}}\]

  12. Applied times-frac0.1

    \[\leadsto e^{im} \cdot \left(0.5 \cdot \sin re\right) + \color{blue}{\frac{\frac{0.5}{\sqrt{\sqrt[3]{e^{im}} \cdot \sqrt[3]{e^{im}}}}}{\sqrt{\sqrt{e^{im}}}} \cdot \frac{\frac{\sin re}{\sqrt{\sqrt[3]{e^{im}}}}}{\sqrt{\sqrt{e^{im}}}}}\]

  13. Simplified0.1

    \[\leadsto e^{im} \cdot \left(0.5 \cdot \sin re\right) + \color{blue}{\frac{\frac{0.5}{\sqrt{\sqrt{e^{im}}}}}{\left|\sqrt[3]{e^{im}}\right|}} \cdot \frac{\frac{\sin re}{\sqrt{\sqrt[3]{e^{im}}}}}{\sqrt{\sqrt{e^{im}}}}\]

  14. Final simplification0.1

    \[\leadsto \left(\sin re \cdot 0.5\right) \cdot e^{im} + \frac{\frac{\sin re}{\sqrt{\sqrt[3]{e^{im}}}}}{\sqrt{\sqrt{e^{im}}}} \cdot \frac{\frac{0.5}{\sqrt{\sqrt{e^{im}}}}}{\left|\sqrt[3]{e^{im}}\right|}\]

Reproduce

herbie shell --seed 2019102 
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))