Average Error: 44.0 → 0.7
Time: 36.3s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left({im}^{3} \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(im \cdot -2 + {im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left({im}^{3} \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(im \cdot -2 + {im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r176370 = 0.5;
        double r176371 = re;
        double r176372 = sin(r176371);
        double r176373 = r176370 * r176372;
        double r176374 = im;
        double r176375 = -r176374;
        double r176376 = exp(r176375);
        double r176377 = exp(r176374);
        double r176378 = r176376 - r176377;
        double r176379 = r176373 * r176378;
        return r176379;
}
↓
double f(double re, double im) {
        double r176380 = im;
        double r176381 = 3.0;
        double r176382 = pow(r176380, r176381);
        double r176383 = -0.3333333333333333;
        double r176384 = r176382 * r176383;
        double r176385 = 0.5;
        double r176386 = re;
        double r176387 = sin(r176386);
        double r176388 = r176385 * r176387;
        double r176389 = r176384 * r176388;
        double r176390 = -2.0;
        double r176391 = r176380 * r176390;
        double r176392 = 5.0;
        double r176393 = pow(r176380, r176392);
        double r176394 = -0.016666666666666666;
        double r176395 = r176393 * r176394;
        double r176396 = r176391 + r176395;
        double r176397 = r176396 * r176388;
        double r176398 = r176389 + r176397;
        return r176398;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original44.0
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 44.0

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

  2. Taylor expanded around 0 0.7

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

  3. Simplified0.7

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left({im}^{3} \cdot \frac{-1}{3} - \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)}\]
  4. Using strategy rm

  5. Applied sub-neg0.7

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left({im}^{3} \cdot \frac{-1}{3} + \left(-\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

  6. Applied distribute-lft-in0.7

    \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left({im}^{3} \cdot \frac{-1}{3}\right) + \left(0.5 \cdot \sin re\right) \cdot \left(-\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)}\]

  7. Simplified0.7

    \[\leadsto \color{blue}{\left({im}^{3} \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \sin re\right)} + \left(0.5 \cdot \sin re\right) \cdot \left(-\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\]

  8. Simplified0.7

    \[\leadsto \left({im}^{3} \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \color{blue}{\left(im \cdot -2 + {im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(0.5 \cdot \sin re\right)}\]

  9. Final simplification0.7

    \[\leadsto \left({im}^{3} \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(im \cdot -2 + {im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(0.5 \cdot \sin re\right)\]

Reproduce

herbie shell --seed 2019199 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))