Average Error: 58.3 → 29.4
Time: 5.9m
Precision: 64
Internal Precision: 6784
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le -1.956189728316902 \cdot 10^{-252}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}}\\ \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le 1.0501759608785294 \cdot 10^{-150}:\\ \;\;\;\;\left(\frac{\left|M\right|}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - \sqrt{\left(\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}\right) - M \cdot M}} \cdot \frac{\left|M\right|}{1}\right) \cdot \frac{c0}{2 \cdot w}\\ \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le +\infty:\\ \;\;\;\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \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

Derivation

  1. Split input into 3 regimes

  2. if (* (/ (/ c0 w) (/ (+ 0 (* M M)) (/ (* M M) 2))) (+ (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (sqrt (- (* (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (/ (* (/ d D) (/ d D)) (/ (* w h) c0))) (* M M))))) < -1.956189728316902e-252 or 1.0501759608785294e-150 < (* (/ (/ c0 w) (/ (+ 0 (* M M)) (/ (* M M) 2))) (+ (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (sqrt (- (* (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (/ (* (/ d D) (/ d D)) (/ (* w h) c0))) (* M M))))) < +inf.0

    1. Initial program 55.2

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

    3. Applied flip-+61.9

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

    4. Applied simplify58.3

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{0 + M \cdot M}}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\]

    5. Taylor expanded around 0 58.3

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \color{blue}{\frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(h \cdot w\right)}} - M \cdot M}}\]

    6. Applied simplify55.8

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

    8. Applied flip--60.9

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

    9. Applied associate-/r/60.9

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

    10. Applied simplify43.3

      \[\leadsto \color{blue}{\frac{\frac{c0}{w}}{\frac{0 + M \cdot M}{\frac{M \cdot M}{2}}}} \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} + \sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M}\right)\]

    if -1.956189728316902e-252 < (* (/ (/ c0 w) (/ (+ 0 (* M M)) (/ (* M M) 2))) (+ (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (sqrt (- (* (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (/ (* (/ d D) (/ d D)) (/ (* w h) c0))) (* M M))))) < 1.0501759608785294e-150

    1. Initial program 54.6

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

    3. Applied flip-+56.5

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

    4. Applied simplify40.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{0 + M \cdot M}}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity40.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{0 + M \cdot M}{\color{blue}{1 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}}\]

    7. Applied add-sqr-sqrt40.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{\color{blue}{\sqrt{0 + M \cdot M} \cdot \sqrt{0 + M \cdot M}}}{1 \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)}\]

    8. Applied times-frac40.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{\sqrt{0 + M \cdot M}}{1} \cdot \frac{\sqrt{0 + M \cdot M}}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\right)}\]

    9. Applied simplify40.4

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{\left|M\right|}{1}} \cdot \frac{\sqrt{0 + M \cdot M}}{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}}\right)\]

    10. Applied simplify16.7

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{\left|M\right|}{1} \cdot \color{blue}{\frac{\left|M\right|}{\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - \sqrt{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) - M \cdot M}}}\right)\]

    if +inf.0 < (* (/ (/ c0 w) (/ (+ 0 (* M M)) (/ (* M M) 2))) (+ (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (sqrt (- (* (/ (* (/ d D) (/ d D)) (/ (* w h) c0)) (/ (* (/ d D) (/ d D)) (/ (* w h) c0))) (* M M)))))

    1. Initial program 58.9

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

    2. Taylor expanded around inf 32.9

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0}\]

    3. Applied simplify28.1

      \[\leadsto \color{blue}{0}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify29.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le -1.956189728316902 \cdot 10^{-252}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}}\\ \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le 1.0501759608785294 \cdot 10^{-150}:\\ \;\;\;\;\left(\frac{\left|M\right|}{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - \sqrt{\left(\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}\right) - M \cdot M}} \cdot \frac{\left|M\right|}{1}\right) \cdot \frac{c0}{2 \cdot w}\\ \mathbf{if}\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}} \le +\infty:\\ \;\;\;\;\left(\sqrt{\frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}} - M \cdot M} + \frac{\frac{d}{D} \cdot \frac{d}{D}}{\frac{w \cdot h}{c0}}\right) \cdot \frac{\frac{c0}{w}}{\frac{M \cdot M}{\frac{M \cdot M}{2}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}}\]

Runtime

Time bar (total: 5.9m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' 
(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))))))