Average Error: 58.2 → 53.3
Time: 3.4m
Precision: 64
Internal Precision: 128
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;M \le -1.3210147326530277 \cdot 10^{-213}:\\ \;\;\;\;\left(\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + M} \cdot \sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} - M}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;M \le -3.1008647698225976 \cdot 10^{-288}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + \sqrt{\left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)}\right) \cdot \left(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + \sqrt{\left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + \sqrt{\left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)}\right)\right)} \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;M \le 1.5661919427365177 \cdot 10^{-262} \lor \neg \left(M \le 3.8425177899430697 \cdot 10^{-196}\right):\\ \;\;\;\;\left(\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + M} \cdot \sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} - M}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} + \sqrt{\left(\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} - M\right) \cdot \left(\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} + M\right)}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if M < -1.3210147326530277e-213 or -3.1008647698225976e-288 < M < 1.5661919427365177e-262 or 3.8425177899430697e-196 < M

    1. Initial program 59.1

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

    2. Initial simplification54.8

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]
    3. Using strategy rm

    4. Applied sqrt-prod55.9

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\color{blue}{\sqrt{M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \sqrt{\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M}} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]
    5. Using strategy rm

    6. Applied associate-*r*55.9

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \sqrt{\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M} + \color{blue}{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}\right)\]
    7. Using strategy rm

    8. Applied associate-*r*55.9

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{M + \color{blue}{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}} \cdot \sqrt{\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M} + \left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}\right)\]
    9. Using strategy rm

    10. Applied associate-*r*54.1

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{M + \left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}} \cdot \sqrt{\color{blue}{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}} - M} + \left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}\right)\]

    if -1.3210147326530277e-213 < M < -3.1008647698225976e-288

    1. Initial program 54.1

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

    2. Initial simplification46.0

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]
    3. Using strategy rm

    4. Applied add-cbrt-cube47.6

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\sqrt[3]{\left(\left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right) \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}}\]

    if 1.5661919427365177e-262 < M < 3.8425177899430697e-196

    1. Initial program 54.3

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

    2. Initial simplification45.5

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]
    3. Using strategy rm

    4. Applied add-log-exp57.8

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \color{blue}{\log \left(e^{\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)}\right)\]

    5. Applied add-log-exp55.8

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\color{blue}{\log \left(e^{\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)}}\right)} + \log \left(e^{\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)\right)\]

    6. Applied sum-log55.4

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\log \left(e^{\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)}} \cdot e^{\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)}\]

    7. Simplified52.0

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \log \color{blue}{\left(e^{\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} + \sqrt{\left(M + \frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}}\right) \cdot \left(\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} - M\right)}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification53.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \le -1.3210147326530277 \cdot 10^{-213}:\\ \;\;\;\;\left(\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + M} \cdot \sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} - M}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;M \le -3.1008647698225976 \cdot 10^{-288}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + \sqrt{\left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)}\right) \cdot \left(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + \sqrt{\left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + \sqrt{\left(M + \left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)}\right)\right)} \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;M \le 1.5661919427365177 \cdot 10^{-262} \lor \neg \left(M \le 3.8425177899430697 \cdot 10^{-196}\right):\\ \;\;\;\;\left(\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + \sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} + M} \cdot \sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} - M}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} + \sqrt{\left(\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} - M\right) \cdot \left(\frac{\frac{c0}{w \cdot h}}{\frac{D}{d} \cdot \frac{D}{d}} + M\right)}}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2018278 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))