expq3 (problem 3.4.2)

Average Error: 60.4 → 54.6

Time: 35.3s

Precision: 64

• could not determine a ground truth for program body

  1. a = 1.771691118882549e+275
  2. b = 2.9315320292795686e-164
  3. eps = 1.414912756308087e-16

\[-1 \lt \varepsilon \land \varepsilon \lt 1\]

Math

\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.295050564857762529537510118555827115194 \cdot 10^{166}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\left(\sqrt{e^{b \cdot \varepsilon}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{b \cdot \varepsilon}} - \sqrt{1}\right)\right)}\\ \mathbf{elif}\;a \le 1.038142947290451481468833360456566540417 \cdot 10^{117}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot {\left(\varepsilon \cdot a\right)}^{3} + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot {\left(b \cdot \varepsilon\right)}^{3} + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r80937 = eps;
        double r80938 = a;
        double r80939 = b;
        double r80940 = r80938 + r80939;
        double r80941 = r80940 * r80937;
        double r80942 = exp(r80941);
        double r80943 = 1.0;
        double r80944 = r80942 - r80943;
        double r80945 = r80937 * r80944;
        double r80946 = r80938 * r80937;
        double r80947 = exp(r80946);
        double r80948 = r80947 - r80943;
        double r80949 = r80939 * r80937;
        double r80950 = exp(r80949);
        double r80951 = r80950 - r80943;
        double r80952 = r80948 * r80951;
        double r80953 = r80945 / r80952;
        return r80953;
}

↓

double f(double a, double b, double eps) {
        double r80954 = a;
        double r80955 = -1.2950505648577625e+166;
        bool r80956 = r80954 <= r80955;
        double r80957 = eps;
        double r80958 = b;
        double r80959 = r80954 + r80958;
        double r80960 = r80959 * r80957;
        double r80961 = exp(r80960);
        double r80962 = 1.0;
        double r80963 = r80961 - r80962;
        double r80964 = r80957 * r80963;
        double r80965 = r80954 * r80957;
        double r80966 = exp(r80965);
        double r80967 = r80966 - r80962;
        double r80968 = r80958 * r80957;
        double r80969 = exp(r80968);
        double r80970 = sqrt(r80969);
        double r80971 = sqrt(r80962);
        double r80972 = r80970 + r80971;
        double r80973 = r80970 - r80971;
        double r80974 = r80972 * r80973;
        double r80975 = r80967 * r80974;
        double r80976 = r80964 / r80975;
        double r80977 = 1.0381429472904515e+117;
        bool r80978 = r80954 <= r80977;
        double r80979 = 0.16666666666666666;
        double r80980 = r80957 * r80954;
        double r80981 = 3.0;
        double r80982 = pow(r80980, r80981);
        double r80983 = r80979 * r80982;
        double r80984 = 0.5;
        double r80985 = 2.0;
        double r80986 = pow(r80954, r80985);
        double r80987 = pow(r80957, r80985);
        double r80988 = r80986 * r80987;
        double r80989 = r80984 * r80988;
        double r80990 = r80989 + r80965;
        double r80991 = r80983 + r80990;
        double r80992 = r80969 - r80962;
        double r80993 = r80991 * r80992;
        double r80994 = r80964 / r80993;
        double r80995 = pow(r80968, r80981);
        double r80996 = r80979 * r80995;
        double r80997 = pow(r80958, r80985);
        double r80998 = r80987 * r80997;
        double r80999 = r80984 * r80998;
        double r81000 = r80957 * r80958;
        double r81001 = r80999 + r81000;
        double r81002 = r80996 + r81001;
        double r81003 = r80967 * r81002;
        double r81004 = r80964 / r81003;
        double r81005 = r80978 ? r80994 : r81004;
        double r81006 = r80956 ? r80976 : r81005;
        return r81006;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	14.7
Herbie	54.6

\[\frac{a + b}{a \cdot b}\]

Derivation

1. Split input into 3 regimes

2. if a < -1.2950505648577625e+166

   1. Initial program 49.8

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 49.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}\right)}\]

   4. Applied add-sqr-sqrt 49.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\color{blue}{\sqrt{e^{b \cdot \varepsilon}} \cdot \sqrt{e^{b \cdot \varepsilon}}} - \sqrt{1} \cdot \sqrt{1}\right)}\]

   5. Applied difference-of-squares 49.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\left(\sqrt{e^{b \cdot \varepsilon}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{b \cdot \varepsilon}} - \sqrt{1}\right)\right)}}\]

   if -1.2950505648577625e+166 < a < 1.0381429472904515e+117

   1. Initial program 62.8

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   2. Taylor expanded around 0 57.0

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
   3. Using strategy rm

   4. Applied pow-prod-down 56.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \color{blue}{{\left(a \cdot \varepsilon\right)}^{3}} + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   5. Simplified 56.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot {\color{blue}{\left(\varepsilon \cdot a\right)}}^{3} + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   if 1.0381429472904515e+117 < a

   1. Initial program 52.6

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   2. Taylor expanded around 0 45.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}}\]
   3. Using strategy rm

   4. Applied pow-prod-down 44.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{{\left(\varepsilon \cdot b\right)}^{3}} + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\]

   5. Simplified 44.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot {\color{blue}{\left(b \cdot \varepsilon\right)}}^{3} + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 54.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.295050564857762529537510118555827115194 \cdot 10^{166}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\left(\sqrt{e^{b \cdot \varepsilon}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{b \cdot \varepsilon}} - \sqrt{1}\right)\right)}\\ \mathbf{elif}\;a \le 1.038142947290451481468833360456566540417 \cdot 10^{117}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot {\left(\varepsilon \cdot a\right)}^{3} + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot {\left(b \cdot \varepsilon\right)}^{3} + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1 eps) (< eps 1))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))