\[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
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: 11.2 s
Input Error: 22.7
Output Error: 15.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\left({\left((e^{\log_* (1 + \sqrt{(e^{d} - 1)^* \cdot c})} - 1)^*\right)}^2\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) & \text{when } d \le -4.0174092f-38 \\ \left(\left(\left(-c\right) \cdot (e^{-d} - 1)^*\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-d}\right)\right) & \text{otherwise} \end{cases}\)

    if d < -4.0174092f-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 expm1-log1p-u to get
      \[\left(\left({\color{red}{\left(\sqrt{(e^{d} - 1)^* \cdot c}\right)}}^2\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) \leadsto \left(\left({\color{blue}{\left((e^{\log_* (1 + \sqrt{(e^{d} - 1)^* \cdot c})} - 1)^*\right)}}^2\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      7.9

    if -4.0174092f-38 < d

    1. Started with
      \[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      30.1
    2. Applied taylor to get
      \[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) \leadsto \left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)\]
      29.9
    3. Taylor expanded around -inf to get
      \[\color{red}{\left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)} \leadsto \color{blue}{\left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)}\]
      29.9
    4. Applied taylor to get
      \[\left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right) \leadsto \left(\left(-1 \cdot \left(c \cdot (e^{-1 \cdot d} - 1)^*\right)\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-1 \cdot d}\right)\right)\]
      19.6
    5. Taylor expanded around inf to get
      \[\color{red}{\left(\left(-1 \cdot \left(c \cdot (e^{-1 \cdot d} - 1)^*\right)\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-1 \cdot d}\right)\right)} \leadsto \color{blue}{\left(\left(-1 \cdot \left(c \cdot (e^{-1 \cdot d} - 1)^*\right)\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-1 \cdot d}\right)\right)}\]
      19.6
    6. Applied simplify to get
      \[\left(\left(-1 \cdot \left(c \cdot (e^{-1 \cdot d} - 1)^*\right)\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-1 \cdot d}\right)\right) \leadsto \left(\left(\left(-c\right) \cdot (e^{-d} - 1)^*\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-d}\right)\right)\]
      19.6
    7. 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)))