\[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
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: 29.6 s
Input Error: 22.7
Output Error: 15.5
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left(\sqrt{\left(\left({\left(\sqrt{(e^{d} - 1)^* \cdot c}\right)}^2\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2 & \text{when } d \le 6.45665f-40 \\ \left(\left(\left(-c\right) \cdot (e^{-d} - 1)^*\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-d}\right)\right) & \text{otherwise} \end{cases}\)

    if d < 6.45665f-40

    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
      \[\color{red}{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)} \leadsto \color{blue}{{\left(\sqrt{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2}\]
      7.7
    4. Using strategy rm
      7.7
    5. Applied add-sqr-sqrt to get
      \[{\left(\sqrt{\left(\color{red}{\left((e^{d} - 1)^* \cdot c\right)} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2 \leadsto {\left(\sqrt{\left(\color{blue}{\left({\left(\sqrt{(e^{d} - 1)^* \cdot c}\right)}^2\right)} \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2\]
      7.8

    if 6.45665f-40 < d

    1. Started with
      \[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      30.2
    2. Using strategy rm
      30.2
    3. Applied add-sqr-sqrt to get
      \[\color{red}{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)} \leadsto \color{blue}{{\left(\sqrt{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2}\]
      30.2
    4. Applied taylor to get
      \[{\left(\sqrt{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2 \leadsto {\left(\sqrt{\left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)}\right)}^2\]
      29.6
    5. Taylor expanded around -inf to get
      \[{\left(\sqrt{\color{red}{\left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)}}\right)}^2 \leadsto {\left(\sqrt{\color{blue}{\left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)}}\right)}^2\]
      29.6
    6. Applied simplify to get
      \[\color{red}{{\left(\sqrt{\left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)}\right)}^2} \leadsto \color{blue}{\left(\left((e^{\frac{-1}{d}} - 1)^* \cdot \frac{-1}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)}\]
      29.9
    7. Applied taylor to get
      \[\left(\left((e^{\frac{-1}{d}} - 1)^* \cdot \frac{-1}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right) \leadsto \left(\left(-1 \cdot \left(c \cdot (e^{-1 \cdot d} - 1)^*\right)\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-1 \cdot d}\right)\right)\]
      19.2
    8. Taylor expanded around inf to get
      \[\color{red}{\left(\left(-1 \cdot \left(c \cdot (e^{-1 \cdot d} - 1)^*\right)\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-1 \cdot d}\right)\right)} \leadsto \color{blue}{\left(\left(-1 \cdot \left(c \cdot (e^{-1 \cdot d} - 1)^*\right)\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-1 \cdot d}\right)\right)}\]
      19.2
    9. Applied simplify to get
      \[\left(\left(-1 \cdot \left(c \cdot (e^{-1 \cdot d} - 1)^*\right)\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-1 \cdot d}\right)\right) \leadsto \left(\left(\left(-c\right) \cdot (e^{-d} - 1)^*\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{-d}\right)\right)\]
      19.2
    10. 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)))