\[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
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: 13.8 s
Input Error: 46.0
Output Error: 36.0
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left({\left(\sqrt{\sqrt{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}}\right)}^2\right)}^2 & \text{when } d \le 1.8818443968586205 \\ \left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) & \text{otherwise} \end{cases}\)

    if d < 1.8818443968586205

    1. Started with
      \[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      40.3
    2. Using strategy rm
      40.3
    3. Applied add-sqr-sqrt to get
      \[\color{red}{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)} \leadsto \color{blue}{{\left(\sqrt{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2}\]
      40.1
    4. Using strategy rm
      40.1
    5. Applied add-sqr-sqrt to get
      \[{\color{red}{\left(\sqrt{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}}^2 \leadsto {\color{blue}{\left({\left(\sqrt{\sqrt{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}}\right)}^2\right)}}^2\]
      40.2

    if 1.8818443968586205 < d

    1. Started with
      \[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      63.0
    2. Using strategy rm
      63.0
    3. Applied add-sqr-sqrt to get
      \[\color{red}{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)} \leadsto \color{blue}{{\left(\sqrt{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2}\]
      63.0
    4. Using strategy rm
      63.0
    5. Applied add-cbrt-cube to get
      \[{\left(\sqrt{\left(\color{red}{\left((e^{d} - 1)^* \cdot c\right)} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2 \leadsto {\left(\sqrt{\left(\color{blue}{\left(\sqrt[3]{{\left((e^{d} - 1)^* \cdot c\right)}^3}\right)} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2\]
      62.7
    6. Applied taylor to get
      \[{\left(\sqrt{\left(\left(\sqrt[3]{{\left((e^{d} - 1)^* \cdot c\right)}^3}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2 \leadsto {\left(\sqrt{\left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2\]
      23.6
    7. Taylor expanded around inf to get
      \[{\left(\sqrt{\left(\color{red}{0} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2 \leadsto {\left(\sqrt{\left(\color{blue}{0} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2\]
      23.6
    8. Applied simplify to get
      \[{\left(\sqrt{\left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2 \leadsto \left(0 \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      23.6
    9. 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)))