\[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
Test:
(fmod (* (expm1 d) c) (atan2 8.61318337292339e-131 d))
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Bits error versus d
Time: 18.1 s
Input Error: 23.1
Output Error: 2.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\left({\left({\left(\sqrt{\sqrt{(e^{d} - 1)^* \cdot c}}\right)}^2\right)}^2\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) & \text{when } d \le -4.5869585f-38 \\ \left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right) & \text{otherwise} \end{cases}\)

    if d < -4.5869585f-38

    1. Started with
      \[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      7.4
    2. Using strategy rm
      7.4
    3. Applied add-sqr-sqrt to get
      \[\left(\color{red}{\left((e^{d} - 1)^* \cdot c\right)} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) \leadsto \left(\color{blue}{\left({\left(\sqrt{(e^{d} - 1)^* \cdot c}\right)}^2\right)} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      7.7
    4. Using strategy rm
      7.7
    5. Applied add-sqr-sqrt to get
      \[\left(\left({\color{red}{\left(\sqrt{(e^{d} - 1)^* \cdot c}\right)}}^2\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) \leadsto \left(\left({\color{blue}{\left({\left(\sqrt{\sqrt{(e^{d} - 1)^* \cdot c}}\right)}^2\right)}}^2\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      7.9

    if -4.5869585f-38 < d

    1. Started with
      \[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      30.8
    2. Using strategy rm
      30.8
    3. Applied add-sqr-sqrt to get
      \[\left(\color{red}{\left((e^{d} - 1)^* \cdot c\right)} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) \leadsto \left(\color{blue}{\left({\left(\sqrt{(e^{d} - 1)^* \cdot c}\right)}^2\right)} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      30.9
    4. Using strategy rm
      30.9
    5. Applied add-sqr-sqrt to get
      \[\left(\left({\color{red}{\left(\sqrt{(e^{d} - 1)^* \cdot c}\right)}}^2\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) \leadsto \left(\left({\color{blue}{\left({\left(\sqrt{\sqrt{(e^{d} - 1)^* \cdot c}}\right)}^2\right)}}^2\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      30.9
    6. Applied taylor to get
      \[\left(\left({\left({\left(\sqrt{\sqrt{(e^{d} - 1)^* \cdot c}}\right)}^2\right)}^2\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) \leadsto \left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      30.7
    7. Taylor expanded around inf to get
      \[\left(\color{red}{0} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) \leadsto \left(\color{blue}{0} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      30.7
    8. Applied taylor to get
      \[\left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) \leadsto \left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)\]
      0.3
    9. Taylor expanded around -inf to get
      \[\color{red}{\left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)} \leadsto \color{blue}{\left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)}\]
      0.3
    10. Applied simplify to get
      \[\left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right) \leadsto \left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)\]
      0.3
    11. Applied final simplification
  1. Removed slow pow expressions

Original test:


(lambda ((a default) (b default) (c default) (d default))
  #:name "(fmod (* (expm1 d) c) (atan2 8.61318337292339e-131 d))"
  (fmod (* (expm1 d) c) (atan2 8.61318337292339e-131 d)))