Average Error: 58.2 → 28.2
Time: 5.8m
Precision: 64
Internal Precision: 6976
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot M}{\sqrt{\left(\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d} \cdot h} + M\right) \cdot \left(\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d} \cdot h} - M\right)} - \frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d} \cdot h}} \cdot \frac{-c0}{w \cdot 2} = -\infty:\\ \;\;\;\;\frac{\frac{c0}{w \cdot 2}}{h \cdot w} \cdot \frac{2 \cdot c0}{\frac{D}{d} \cdot \frac{D}{d}}\\ \mathbf{if}\;\frac{M \cdot M}{\sqrt{\left(\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d} \cdot h} + M\right) \cdot \left(\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d} \cdot h} - M\right)} - \frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d} \cdot h}} \cdot \frac{-c0}{w \cdot 2} \le +\infty:\\ \;\;\;\;\frac{M \cdot M}{\sqrt{\left(\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d} \cdot h} + M\right) \cdot \left(\frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d} \cdot h} - M\right)} - \frac{\frac{\frac{c0}{w}}{\frac{D}{d}}}{\frac{D}{d} \cdot h}} \cdot \frac{-c0}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 (* 2 w)) (* 1 (/ (- (* M M)) (- (sqrt (* (+ (/ (/ (/ c0 w) (/ D d)) (* h (/ D d))) M) (- (/ (/ (/ c0 w) (/ D d)) (* h (/ D d))) M))) (/ (/ (/ c0 w) (/ D d)) (* h (/ D d))))))) < -inf.0

    1. Initial program 44.5

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

    3. Applied flip-+61.5

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

    4. Applied simplify61.5

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

    6. Applied *-un-lft-identity61.5

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

    7. Applied *-un-lft-identity61.5

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

    8. Applied times-frac61.5

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

    9. Applied simplify61.5

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

    10. Applied simplify61.5

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

    11. Taylor expanded around 0 44.6

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

    12. Applied simplify25.7

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

    if -inf.0 < (* (/ c0 (* 2 w)) (* 1 (/ (- (* M M)) (- (sqrt (* (+ (/ (/ (/ c0 w) (/ D d)) (* h (/ D d))) M) (- (/ (/ (/ c0 w) (/ D d)) (* h (/ D d))) M))) (/ (/ (/ c0 w) (/ D d)) (* h (/ D d))))))) < +inf.0

    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. Using strategy rm

    3. Applied flip-+60.5

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

    4. Applied simplify32.4

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

    6. Applied *-un-lft-identity32.4

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

    7. Applied *-un-lft-identity32.4

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

    8. Applied times-frac32.4

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

    9. Applied simplify32.4

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

    10. Applied simplify25.1

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

    12. Applied frac-2neg25.1

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

    13. Applied simplify19.3

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

    if +inf.0 < (* (/ c0 (* 2 w)) (* 1 (/ (- (* M M)) (- (sqrt (* (+ (/ (/ (/ c0 w) (/ D d)) (* h (/ D d))) M) (- (/ (/ (/ c0 w) (/ D d)) (* h (/ D d))) M))) (/ (/ (/ c0 w) (/ D d)) (* h (/ D d)))))))

    1. Initial program 58.2

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

    2. Taylor expanded around inf 49.9

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

    3. Applied simplify40.4

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

  4. Applied simplify28.2

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

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(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))))))