Average Error: 58.4 → 27.2
Time: 2.5m
Precision: 64
Internal Precision: 7232
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\frac{c0}{2 \cdot w}\right) \cdot \left(\sqrt{(\left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(\left(-M\right) \cdot M\right))_*}\right) + \left(e^{\log \left(\frac{c0}{w \cdot h}\right) + \log \left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{c0}{2 \cdot w}\right))_* \le 1.6989985471632755 \cdot 10^{+298}:\\ \;\;\;\;(\left(\frac{c0}{2 \cdot w}\right) \cdot \left({\left(\sqrt[3]{(\left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(\left(-M\right) \cdot M\right))_*}\right)}^{\frac{1}{2}} \cdot \left|\sqrt[3]{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right|\right) + \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{d}{D} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d}{D}\right)\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 2 regimes

  2. if (fma (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ c0 (* w h))) (* (* (/ d D) (/ d D)) (/ c0 (* w h))) (* M (- M)))) (* (/ c0 (* w 2)) (exp (+ (log (* (/ d D) (/ d D))) (log (/ c0 (* w h))))))) < 1.6989985471632755e+298

    1. Initial program 46.6

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

    2. Initial simplification22.0

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

    4. Applied associate-*l*21.9

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

    6. Applied pow1/221.9

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

    8. Applied add-cube-cbrt21.9

      \[\leadsto (\left(\frac{c0}{w \cdot 2}\right) \cdot \left({\color{blue}{\left(\left(\sqrt[3]{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(M \cdot \left(-M\right)\right))_*} \cdot \sqrt[3]{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(M \cdot \left(-M\right)\right))_*}\right) \cdot \sqrt[3]{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(M \cdot \left(-M\right)\right))_*}\right)}}^{\frac{1}{2}}\right) + \left(\frac{c0}{w \cdot 2} \cdot \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)\right)\right))_*\]

    9. Applied unpow-prod-down21.9

      \[\leadsto (\left(\frac{c0}{w \cdot 2}\right) \cdot \color{blue}{\left({\left(\sqrt[3]{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(M \cdot \left(-M\right)\right))_*} \cdot \sqrt[3]{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(M \cdot \left(-M\right)\right))_*}\right)}^{\frac{1}{2}} \cdot {\left(\sqrt[3]{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(M \cdot \left(-M\right)\right))_*}\right)}^{\frac{1}{2}}\right)} + \left(\frac{c0}{w \cdot 2} \cdot \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)\right)\right))_*\]

    10. Simplified21.9

      \[\leadsto (\left(\frac{c0}{w \cdot 2}\right) \cdot \left(\color{blue}{\left|\sqrt[3]{(\left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(-M \cdot M\right))_*}\right|} \cdot {\left(\sqrt[3]{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(M \cdot \left(-M\right)\right))_*}\right)}^{\frac{1}{2}}\right) + \left(\frac{c0}{w \cdot 2} \cdot \left(\frac{d}{D} \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot h}\right)\right)\right))_*\]

    if 1.6989985471632755e+298 < (fma (/ c0 (* w 2)) (sqrt (fma (* (* (/ d D) (/ d D)) (/ c0 (* w h))) (* (* (/ d D) (/ d D)) (/ c0 (* w h))) (* M (- M)))) (* (/ c0 (* w 2)) (exp (+ (log (* (/ d D) (/ d D))) (log (/ c0 (* w h)))))))

    1. Initial program 60.6

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

    2. Initial simplification61.6

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

    3. Taylor expanded around inf 28.2

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification27.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;(\left(\frac{c0}{2 \cdot w}\right) \cdot \left(\sqrt{(\left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(\left(-M\right) \cdot M\right))_*}\right) + \left(e^{\log \left(\frac{c0}{w \cdot h}\right) + \log \left(\frac{d}{D} \cdot \frac{d}{D}\right)} \cdot \frac{c0}{2 \cdot w}\right))_* \le 1.6989985471632755 \cdot 10^{+298}:\\ \;\;\;\;(\left(\frac{c0}{2 \cdot w}\right) \cdot \left({\left(\sqrt[3]{(\left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(\left(-M\right) \cdot M\right))_*}\right)}^{\frac{1}{2}} \cdot \left|\sqrt[3]{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{w}}{h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right|\right) + \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{d}{D} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d}{D}\right)\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(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))))))