Average Error: 43.4 → 0.9
Time: 1.4m
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(\left(\left(im \cdot \frac{-1}{3}\right) \cdot im - 2\right) \cdot im\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(\left(\left(im \cdot \frac{-1}{3}\right) \cdot im - 2\right) \cdot im\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r29400193 = 0.5;
        double r29400194 = re;
        double r29400195 = sin(r29400194);
        double r29400196 = r29400193 * r29400195;
        double r29400197 = im;
        double r29400198 = -r29400197;
        double r29400199 = exp(r29400198);
        double r29400200 = exp(r29400197);
        double r29400201 = r29400199 - r29400200;
        double r29400202 = r29400196 * r29400201;
        return r29400202;
}
↓
double f(double re, double im) {
        double r29400203 = -0.016666666666666666;
        double r29400204 = im;
        double r29400205 = 5.0;
        double r29400206 = pow(r29400204, r29400205);
        double r29400207 = r29400203 * r29400206;
        double r29400208 = 0.5;
        double r29400209 = re;
        double r29400210 = sin(r29400209);
        double r29400211 = r29400208 * r29400210;
        double r29400212 = r29400207 * r29400211;
        double r29400213 = -0.3333333333333333;
        double r29400214 = r29400204 * r29400213;
        double r29400215 = r29400214 * r29400204;
        double r29400216 = 2.0;
        double r29400217 = r29400215 - r29400216;
        double r29400218 = r29400217 * r29400204;
        double r29400219 = r29400218 * r29400211;
        double r29400220 = r29400212 + r29400219;
        return r29400220;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.4
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.9

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

  3. Simplified0.9

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

  5. Applied fma-udef0.9

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

  6. Applied distribute-lft-in0.9

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

  7. Final simplification0.9

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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