Henrywood and Agarwal, Equation (13)

Average Error: 59.1 → 46.2

Time: 54.3s

Precision: 64

• could not determine a ground truth for program body

  1. c0 = 4.211238809870961e+95
  2. w = 1.1412903097154827e+35
  3. h = -2.4212175173215518e-214
  4. D = -1.5312717063394058e-305
  5. d = 2.5188017914938334e+295
  6. M = 1.2523134913899472e-164

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \le 3.34750491238746575446583831459992275621 \cdot 10^{-281}:\\ \;\;\;\;\frac{\left(\frac{d}{\sqrt[3]{w}} \cdot \left(\frac{0}{h \cdot D} \cdot \frac{2}{D}\right)\right) \cdot \frac{c0}{w}}{2}\\ \mathbf{elif}\;M \le 1.03379078061023875438654714236971941009 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{c0} \cdot \sqrt[3]{c0}\right) \cdot \left(\left(\left(\frac{\frac{c0}{\sqrt[3]{w}}}{D} \cdot \frac{\frac{d}{\sqrt[3]{w}}}{h}\right) \cdot \frac{d}{\sqrt[3]{w}}\right) \cdot \frac{2}{D}\right)\right) \cdot \frac{\sqrt[3]{c0}}{w}}{2}\\ \mathbf{elif}\;M \le 2.289096923733291240007388617918193953003 \cdot 10^{-126}:\\ \;\;\;\;\frac{\left(\frac{d}{\sqrt[3]{w}} \cdot \left(\frac{0}{h \cdot D} \cdot \frac{2}{D}\right)\right) \cdot \frac{c0}{w}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\left(\frac{2}{D} \cdot \frac{\left(\sqrt[3]{c0} \cdot \sqrt[3]{c0}\right) \cdot \left(\frac{\sqrt[3]{c0}}{\sqrt[3]{w}} \cdot \frac{d}{\sqrt[3]{w}}\right)}{h \cdot D}\right) \cdot \frac{d}{\sqrt[3]{w}}\right)}{2}\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r7023475 = c0;
        double r7023476 = 2.0;
        double r7023477 = w;
        double r7023478 = r7023476 * r7023477;
        double r7023479 = r7023475 / r7023478;
        double r7023480 = d;
        double r7023481 = r7023480 * r7023480;
        double r7023482 = r7023475 * r7023481;
        double r7023483 = h;
        double r7023484 = r7023477 * r7023483;
        double r7023485 = D;
        double r7023486 = r7023485 * r7023485;
        double r7023487 = r7023484 * r7023486;
        double r7023488 = r7023482 / r7023487;
        double r7023489 = r7023488 * r7023488;
        double r7023490 = M;
        double r7023491 = r7023490 * r7023490;
        double r7023492 = r7023489 - r7023491;
        double r7023493 = sqrt(r7023492);
        double r7023494 = r7023488 + r7023493;
        double r7023495 = r7023479 * r7023494;
        return r7023495;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r7023496 = M;
        double r7023497 = 3.347504912387466e-281;
        bool r7023498 = r7023496 <= r7023497;
        double r7023499 = d;
        double r7023500 = w;
        double r7023501 = cbrt(r7023500);
        double r7023502 = r7023499 / r7023501;
        double r7023503 = 0.0;
        double r7023504 = h;
        double r7023505 = D;
        double r7023506 = r7023504 * r7023505;
        double r7023507 = r7023503 / r7023506;
        double r7023508 = 2.0;
        double r7023509 = r7023508 / r7023505;
        double r7023510 = r7023507 * r7023509;
        double r7023511 = r7023502 * r7023510;
        double r7023512 = c0;
        double r7023513 = r7023512 / r7023500;
        double r7023514 = r7023511 * r7023513;
        double r7023515 = 2.0;
        double r7023516 = r7023514 / r7023515;
        double r7023517 = 1.0337907806102388e-184;
        bool r7023518 = r7023496 <= r7023517;
        double r7023519 = cbrt(r7023512);
        double r7023520 = r7023519 * r7023519;
        double r7023521 = r7023512 / r7023501;
        double r7023522 = r7023521 / r7023505;
        double r7023523 = r7023502 / r7023504;
        double r7023524 = r7023522 * r7023523;
        double r7023525 = r7023524 * r7023502;
        double r7023526 = r7023525 * r7023509;
        double r7023527 = r7023520 * r7023526;
        double r7023528 = r7023519 / r7023500;
        double r7023529 = r7023527 * r7023528;
        double r7023530 = r7023529 / r7023515;
        double r7023531 = 2.2890969237332912e-126;
        bool r7023532 = r7023496 <= r7023531;
        double r7023533 = r7023519 / r7023501;
        double r7023534 = r7023533 * r7023502;
        double r7023535 = r7023520 * r7023534;
        double r7023536 = r7023535 / r7023506;
        double r7023537 = r7023509 * r7023536;
        double r7023538 = r7023537 * r7023502;
        double r7023539 = r7023513 * r7023538;
        double r7023540 = r7023539 / r7023515;
        double r7023541 = r7023532 ? r7023516 : r7023540;
        double r7023542 = r7023518 ? r7023530 : r7023541;
        double r7023543 = r7023498 ? r7023516 : r7023542;
        return r7023543;
}

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 M < 3.347504912387466e-281 or 1.0337907806102388e-184 < M < 2.2890969237332912e-126

   1. Initial program 58.6

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

   2. Simplified 52.9

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

   3. Taylor expanded around 0 59.2

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

   4. Simplified 56.7

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

   6. Applied add-cube-cbrt 56.7

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

   7. Applied times-frac 55.0

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

   8. Applied associate-*r* 54.0

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

   9. Simplified 50.8

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

   10. Taylor expanded around 0 42.6

       \[\leadsto \frac{\left(\left(\frac{2}{D} \cdot \frac{\color{blue}{0}}{D \cdot h}\right) \cdot \frac{d}{\sqrt[3]{w}}\right) \cdot \frac{c0}{w}}{2}\]

   if 3.347504912387466e-281 < M < 1.0337907806102388e-184

   1. Initial program 55.0

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

   2. Simplified 44.7

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

   3. Taylor expanded around 0 57.3

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

   4. Simplified 53.5

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

   6. Applied add-cube-cbrt 53.5

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

   7. Applied times-frac 52.3

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

   8. Applied associate-*r* 50.8

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

   9. Simplified 45.5

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

   11. Applied *-un-lft-identity 45.5

       \[\leadsto \frac{\left(\left(\frac{2}{D} \cdot \frac{\frac{c0}{\sqrt[3]{w}} \cdot \frac{d}{\sqrt[3]{w}}}{D \cdot h}\right) \cdot \frac{d}{\sqrt[3]{w}}\right) \cdot \frac{c0}{\color{blue}{1 \cdot w}}}{2}\]

   12. Applied add-cube-cbrt 45.5

       \[\leadsto \frac{\left(\left(\frac{2}{D} \cdot \frac{\frac{c0}{\sqrt[3]{w}} \cdot \frac{d}{\sqrt[3]{w}}}{D \cdot h}\right) \cdot \frac{d}{\sqrt[3]{w}}\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{c0} \cdot \sqrt[3]{c0}\right) \cdot \sqrt[3]{c0}}}{1 \cdot w}}{2}\]

   13. Applied times-frac 45.5

       \[\leadsto \frac{\left(\left(\frac{2}{D} \cdot \frac{\frac{c0}{\sqrt[3]{w}} \cdot \frac{d}{\sqrt[3]{w}}}{D \cdot h}\right) \cdot \frac{d}{\sqrt[3]{w}}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{1} \cdot \frac{\sqrt[3]{c0}}{w}\right)}}{2}\]

   14. Applied associate-*r* 45.8

       \[\leadsto \frac{\color{blue}{\left(\left(\left(\frac{2}{D} \cdot \frac{\frac{c0}{\sqrt[3]{w}} \cdot \frac{d}{\sqrt[3]{w}}}{D \cdot h}\right) \cdot \frac{d}{\sqrt[3]{w}}\right) \cdot \frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{1}\right) \cdot \frac{\sqrt[3]{c0}}{w}}}{2}\]

   15. Simplified 44.6

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

   if 2.2890969237332912e-126 < M

   1. Initial program 61.8

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

   2. Simplified 59.1

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

   3. Taylor expanded around 0 61.3

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

   4. Simplified 59.0

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

   6. Applied add-cube-cbrt 59.1

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

   7. Applied times-frac 58.1

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

   8. Applied associate-*r* 57.2

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

   9. Simplified 54.3

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

   11. Applied *-un-lft-identity 54.3

       \[\leadsto \frac{\left(\left(\frac{2}{D} \cdot \frac{\frac{c0}{\color{blue}{1 \cdot \sqrt[3]{w}}} \cdot \frac{d}{\sqrt[3]{w}}}{D \cdot h}\right) \cdot \frac{d}{\sqrt[3]{w}}\right) \cdot \frac{c0}{w}}{2}\]

   12. Applied add-cube-cbrt 54.4

       \[\leadsto \frac{\left(\left(\frac{2}{D} \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{c0} \cdot \sqrt[3]{c0}\right) \cdot \sqrt[3]{c0}}}{1 \cdot \sqrt[3]{w}} \cdot \frac{d}{\sqrt[3]{w}}}{D \cdot h}\right) \cdot \frac{d}{\sqrt[3]{w}}\right) \cdot \frac{c0}{w}}{2}\]

   13. Applied times-frac 54.4

       \[\leadsto \frac{\left(\left(\frac{2}{D} \cdot \frac{\color{blue}{\left(\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{1} \cdot \frac{\sqrt[3]{c0}}{\sqrt[3]{w}}\right)} \cdot \frac{d}{\sqrt[3]{w}}}{D \cdot h}\right) \cdot \frac{d}{\sqrt[3]{w}}\right) \cdot \frac{c0}{w}}{2}\]

   14. Applied associate-*l* 54.6

       \[\leadsto \frac{\left(\left(\frac{2}{D} \cdot \frac{\color{blue}{\frac{\sqrt[3]{c0} \cdot \sqrt[3]{c0}}{1} \cdot \left(\frac{\sqrt[3]{c0}}{\sqrt[3]{w}} \cdot \frac{d}{\sqrt[3]{w}}\right)}}{D \cdot h}\right) \cdot \frac{d}{\sqrt[3]{w}}\right) \cdot \frac{c0}{w}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 46.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \le 3.34750491238746575446583831459992275621 \cdot 10^{-281}:\\ \;\;\;\;\frac{\left(\frac{d}{\sqrt[3]{w}} \cdot \left(\frac{0}{h \cdot D} \cdot \frac{2}{D}\right)\right) \cdot \frac{c0}{w}}{2}\\ \mathbf{elif}\;M \le 1.03379078061023875438654714236971941009 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{c0} \cdot \sqrt[3]{c0}\right) \cdot \left(\left(\left(\frac{\frac{c0}{\sqrt[3]{w}}}{D} \cdot \frac{\frac{d}{\sqrt[3]{w}}}{h}\right) \cdot \frac{d}{\sqrt[3]{w}}\right) \cdot \frac{2}{D}\right)\right) \cdot \frac{\sqrt[3]{c0}}{w}}{2}\\ \mathbf{elif}\;M \le 2.289096923733291240007388617918193953003 \cdot 10^{-126}:\\ \;\;\;\;\frac{\left(\frac{d}{\sqrt[3]{w}} \cdot \left(\frac{0}{h \cdot D} \cdot \frac{2}{D}\right)\right) \cdot \frac{c0}{w}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\left(\frac{2}{D} \cdot \frac{\left(\sqrt[3]{c0} \cdot \sqrt[3]{c0}\right) \cdot \left(\frac{\sqrt[3]{c0}}{\sqrt[3]{w}} \cdot \frac{d}{\sqrt[3]{w}}\right)}{h \cdot D}\right) \cdot \frac{d}{\sqrt[3]{w}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2.0 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))))))