\[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
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: 22.1 s
Input Error: 24.6
Output Error: 19.3
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left(\sqrt{{\left(\sqrt[3]{{\left(\sqrt{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2}\right)}^3}\right)}^2 & \text{when } d \le -2.287137f-38 \\ \left(\left(\frac{(e^{-d} - 1)^*}{\frac{c}{-1}}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right) & \text{otherwise} \end{cases}\)

    if d < -2.287137f-38

    1. Started with
      \[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)\]
      19.3
    2. Using strategy rm
      19.3
    3. Applied add-sqr-sqrt to get
      \[\color{red}{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)} \leadsto \color{blue}{{\left(\sqrt{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2}\]
      19.0
    4. Using strategy rm
      19.0
    5. Applied add-cube-cbrt to get
      \[{\left(\sqrt{\color{red}{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}}\right)}^2 \leadsto {\left(\sqrt{\color{blue}{{\left(\sqrt[3]{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^3}}\right)}^2\]
      19.1
    6. Using strategy rm
      19.1
    7. Applied add-sqr-sqrt to get
      \[{\left(\sqrt{{\left(\sqrt[3]{\color{red}{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}}\right)}^3}\right)}^2 \leadsto {\left(\sqrt{{\left(\sqrt[3]{\color{blue}{{\left(\sqrt{\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right)}\right)}^2}}\right)}^3}\right)}^2\]
      19.1

    if -2.287137f-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
      \[\left(\left((e^{d} - 1)^* \cdot c\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{d}\right)\right) \leadsto \left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)\]
      28.6
    3. Taylor expanded around -inf to get
      \[\color{red}{\left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)} \leadsto \color{blue}{\left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)}\]
      28.6
    4. Applied taylor to get
      \[\left(\left(-1 \cdot \frac{(e^{\frac{-1}{d}} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right) \leadsto \left(\left(-1 \cdot \frac{(e^{-1 \cdot d} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)\]
      19.5
    5. Taylor expanded around inf to get
      \[\left(\left(-1 \cdot \color{red}{\frac{(e^{-1 \cdot d} - 1)^*}{c}}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right) \leadsto \left(\left(-1 \cdot \color{blue}{\frac{(e^{-1 \cdot d} - 1)^*}{c}}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)\]
      19.5
    6. Applied simplify to get
      \[\left(\left(-1 \cdot \frac{(e^{-1 \cdot d} - 1)^*}{c}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right) \leadsto \left(\left(\frac{(e^{-d} - 1)^*}{\frac{c}{-1}}\right) \bmod \left(\tan^{-1}_* \frac{8.61318337292339 \cdot 10^{-131}}{\frac{-1}{d}}\right)\right)\]
      19.5
    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)))