Henrywood and Agarwal, Equation (12)

Average Error: 26.2 → 19.8

Time: 24.0s

Precision: 64

Math

\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -1.16224426293113532 \cdot 10^{227}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \le -3.27880730867629335 \cdot 10^{-10}:\\ \;\;\;\;\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \le 3.48850499663846863 \cdot 10^{-81}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{1}{\frac{\ell}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}}\right)\\ \mathbf{elif}\;\ell \le 3.846423325806426 \cdot 10^{236}:\\ \;\;\;\;1 \cdot \left({\left({d}^{1}\right)}^{1} \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r266948 = d;
        double r266949 = h;
        double r266950 = r266948 / r266949;
        double r266951 = 1.0;
        double r266952 = 2.0;
        double r266953 = r266951 / r266952;
        double r266954 = pow(r266950, r266953);
        double r266955 = l;
        double r266956 = r266948 / r266955;
        double r266957 = pow(r266956, r266953);
        double r266958 = r266954 * r266957;
        double r266959 = M;
        double r266960 = D;
        double r266961 = r266959 * r266960;
        double r266962 = r266952 * r266948;
        double r266963 = r266961 / r266962;
        double r266964 = pow(r266963, r266952);
        double r266965 = r266953 * r266964;
        double r266966 = r266949 / r266955;
        double r266967 = r266965 * r266966;
        double r266968 = r266951 - r266967;
        double r266969 = r266958 * r266968;
        return r266969;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r266970 = l;
        double r266971 = -1.1622442629311353e+227;
        bool r266972 = r266970 <= r266971;
        double r266973 = d;
        double r266974 = h;
        double r266975 = r266973 / r266974;
        double r266976 = 1.0;
        double r266977 = 2.0;
        double r266978 = r266976 / r266977;
        double r266979 = pow(r266975, r266978);
        double r266980 = 1.0;
        double r266981 = cbrt(r266970);
        double r266982 = r266981 * r266981;
        double r266983 = r266980 / r266982;
        double r266984 = pow(r266983, r266978);
        double r266985 = r266973 / r266981;
        double r266986 = pow(r266985, r266978);
        double r266987 = r266984 * r266986;
        double r266988 = r266979 * r266987;
        double r266989 = M;
        double r266990 = D;
        double r266991 = r266989 * r266990;
        double r266992 = r266977 * r266973;
        double r266993 = r266991 / r266992;
        double r266994 = pow(r266993, r266977);
        double r266995 = r266978 * r266994;
        double r266996 = r266974 / r266970;
        double r266997 = r266995 * r266996;
        double r266998 = r266976 - r266997;
        double r266999 = r266988 * r266998;
        double r267000 = -3.2788073086762934e-10;
        bool r267001 = r266970 <= r267000;
        double r267002 = cbrt(r266974);
        double r267003 = r267002 * r267002;
        double r267004 = r266980 / r267003;
        double r267005 = pow(r267004, r266978);
        double r267006 = r266973 / r267002;
        double r267007 = pow(r267006, r266978);
        double r267008 = r267005 * r267007;
        double r267009 = r266973 / r266970;
        double r267010 = pow(r267009, r266978);
        double r267011 = r267008 * r267010;
        double r267012 = r267011 * r266998;
        double r267013 = 3.4885049966384686e-81;
        bool r267014 = r266970 <= r267013;
        double r267015 = cbrt(r266973);
        double r267016 = r267015 * r267015;
        double r267017 = r267016 / r266982;
        double r267018 = pow(r267017, r266978);
        double r267019 = r267015 / r266981;
        double r267020 = pow(r267019, r266978);
        double r267021 = r267018 * r267020;
        double r267022 = r266979 * r267021;
        double r267023 = r266995 * r266974;
        double r267024 = r266970 / r267023;
        double r267025 = r266980 / r267024;
        double r267026 = r266976 - r267025;
        double r267027 = r267022 * r267026;
        double r267028 = 3.846423325806426e+236;
        bool r267029 = r266970 <= r267028;
        double r267030 = pow(r266973, r266976);
        double r267031 = pow(r267030, r266976);
        double r267032 = pow(r266974, r266976);
        double r267033 = pow(r266970, r266976);
        double r267034 = r267032 * r267033;
        double r267035 = r266980 / r267034;
        double r267036 = 0.5;
        double r267037 = pow(r267035, r267036);
        double r267038 = r267031 * r267037;
        double r267039 = r266976 * r267038;
        double r267040 = r267029 ? r267039 : r266999;
        double r267041 = r267014 ? r267027 : r267040;
        double r267042 = r267001 ? r267012 : r267041;
        double r267043 = r266972 ? r266999 : r267042;
        return r267043;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 4 regimes

2. if l < -1.1622442629311353e+227 or 3.846423325806426e+236 < l

   1. Initial program 30.7

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 30.9

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

   4. Applied *-un-lft-identity 30.9

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

   5. Applied times-frac 30.9

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

   6. Applied unpow-prod-down 24.8

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

   if -1.1622442629311353e+227 < l < -3.2788073086762934e-10

   1. Initial program 22.5

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 22.8

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

   4. Applied *-un-lft-identity 22.8

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

   5. Applied times-frac 22.8

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

   6. Applied unpow-prod-down 17.0

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

   if -3.2788073086762934e-10 < l < 3.4885049966384686e-81

   1. Initial program 29.0

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 29.3

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

   4. Applied add-cube-cbrt 29.4

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

   5. Applied times-frac 29.4

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

   6. Applied unpow-prod-down 24.2

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   7. Using strategy rm

   8. Applied associate-*r/ 14.3

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \color{blue}{\frac{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{\ell}}\right)\]
   9. Using strategy rm

   10. Applied clear-num 14.3

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

   if 3.4885049966384686e-81 < l < 3.846423325806426e+236

   1. Initial program 23.9

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   2. Taylor expanded around 0 24.4

      \[\leadsto \color{blue}{1 \cdot \left({\left({d}^{1}\right)}^{1} \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 19.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.16224426293113532 \cdot 10^{227}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \le -3.27880730867629335 \cdot 10^{-10}:\\ \;\;\;\;\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \le 3.48850499663846863 \cdot 10^{-81}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{1}{\frac{\ell}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}}\right)\\ \mathbf{elif}\;\ell \le 3.846423325806426 \cdot 10^{236}:\\ \;\;\;\;1 \cdot \left({\left({d}^{1}\right)}^{1} \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))