Average Error: 58.1 → 31.5
Time: 49.5s
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) = -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} + M}, \sqrt{\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} - M}, \frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)}{2} \cdot \frac{c0}{w}\\ \mathbf{elif}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 7.37275217224549 \cdot 10^{+251}:\\ \;\;\;\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) = -\infty:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} + M}, \sqrt{\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} - M}, \frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)}{2} \cdot \frac{c0}{w}\\

\mathbf{elif}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 7.37275217224549 \cdot 10^{+251}:\\
\;\;\;\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double c0, double w, double h, double D, double d, double M) {
        double r1991624 = c0;
        double r1991625 = 2.0;
        double r1991626 = w;
        double r1991627 = r1991625 * r1991626;
        double r1991628 = r1991624 / r1991627;
        double r1991629 = d;
        double r1991630 = r1991629 * r1991629;
        double r1991631 = r1991624 * r1991630;
        double r1991632 = h;
        double r1991633 = r1991626 * r1991632;
        double r1991634 = D;
        double r1991635 = r1991634 * r1991634;
        double r1991636 = r1991633 * r1991635;
        double r1991637 = r1991631 / r1991636;
        double r1991638 = r1991637 * r1991637;
        double r1991639 = M;
        double r1991640 = r1991639 * r1991639;
        double r1991641 = r1991638 - r1991640;
        double r1991642 = sqrt(r1991641);
        double r1991643 = r1991637 + r1991642;
        double r1991644 = r1991628 * r1991643;
        return r1991644;
}
↓
double f(double c0, double w, double h, double D, double d, double M) {
        double r1991645 = c0;
        double r1991646 = w;
        double r1991647 = 2.0;
        double r1991648 = r1991646 * r1991647;
        double r1991649 = r1991645 / r1991648;
        double r1991650 = d;
        double r1991651 = r1991650 * r1991650;
        double r1991652 = r1991645 * r1991651;
        double r1991653 = D;
        double r1991654 = r1991653 * r1991653;
        double r1991655 = h;
        double r1991656 = r1991646 * r1991655;
        double r1991657 = r1991654 * r1991656;
        double r1991658 = r1991652 / r1991657;
        double r1991659 = r1991658 * r1991658;
        double r1991660 = M;
        double r1991661 = r1991660 * r1991660;
        double r1991662 = r1991659 - r1991661;
        double r1991663 = sqrt(r1991662);
        double r1991664 = r1991663 + r1991658;
        double r1991665 = r1991649 * r1991664;
        double r1991666 = -inf.0;
        bool r1991667 = r1991665 <= r1991666;
        double r1991668 = r1991645 / r1991655;
        double r1991669 = r1991650 / r1991653;
        double r1991670 = r1991669 * r1991669;
        double r1991671 = r1991670 / r1991646;
        double r1991672 = r1991668 * r1991671;
        double r1991673 = r1991672 + r1991660;
        double r1991674 = sqrt(r1991673);
        double r1991675 = r1991672 - r1991660;
        double r1991676 = sqrt(r1991675);
        double r1991677 = fma(r1991674, r1991676, r1991672);
        double r1991678 = r1991677 / r1991647;
        double r1991679 = r1991645 / r1991646;
        double r1991680 = r1991678 * r1991679;
        double r1991681 = 7.37275217224549e+251;
        bool r1991682 = r1991665 <= r1991681;
        double r1991683 = 0.0;
        double r1991684 = r1991682 ? r1991665 : r1991683;
        double r1991685 = r1991667 ? r1991680 : r1991684;
        return r1991685;
}

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

    1. Initial program 61.2

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

    2. Simplified56.4

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

    4. Applied difference-of-squares56.4

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

    5. Applied sqrt-prod49.3

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

    6. Applied fma-def49.3

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

    if -inf.0 < (* (/ 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))))) < 7.37275217224549e+251

    1. Initial program 21.9

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

    if 7.37275217224549e+251 < (* (/ 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)))))

    1. Initial program 62.7

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

    2. Simplified56.0

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

    3. Taylor expanded around inf 33.7

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

    4. Taylor expanded around 0 31.7

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

  4. Final simplification31.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) = -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} + M}, \sqrt{\frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w} - M}, \frac{c0}{h} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w}\right)}{2} \cdot \frac{c0}{w}\\ \mathbf{elif}\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right) \le 7.37275217224549 \cdot 10^{+251}:\\ \;\;\;\;\frac{c0}{w \cdot 2} \cdot \left(\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M} + \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +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))))))