expq3 (problem 3.4.2)

Average Error: 60.4 → 53.8

Time: 33.9s

Precision: 64

• could not determine a ground truth for program body

  1. a = -1.6243967905174018e+136
  2. b = 1.3180249475760962e+302
  3. eps = 1.0692647045277341e-165

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.706776552602809909969013296017709199406 \cdot 10^{142}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)}^{3} \cdot \left(a \cdot {\varepsilon}^{3}\right)\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{elif}\;b \le 1.829472098735899306961801525168728408435 \cdot 10^{68}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left(\left(\left(b \cdot b\right) \cdot {\varepsilon}^{3}\right) \cdot {\left(\sqrt[3]{b}\right)}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\\ \mathbf{elif}\;b \le 3.275590786210768942354859723569296251086 \cdot 10^{179} \lor \neg \left(b \le 2.957657581720416521596292537514259153078 \cdot 10^{293}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)}^{3} \cdot \left(a \cdot {\varepsilon}^{3}\right)\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left({\left(e^{b}\right)}^{\varepsilon} - 1\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r48023 = eps;
        double r48024 = a;
        double r48025 = b;
        double r48026 = r48024 + r48025;
        double r48027 = r48026 * r48023;
        double r48028 = exp(r48027);
        double r48029 = 1.0;
        double r48030 = r48028 - r48029;
        double r48031 = r48023 * r48030;
        double r48032 = r48024 * r48023;
        double r48033 = exp(r48032);
        double r48034 = r48033 - r48029;
        double r48035 = r48025 * r48023;
        double r48036 = exp(r48035);
        double r48037 = r48036 - r48029;
        double r48038 = r48034 * r48037;
        double r48039 = r48031 / r48038;
        return r48039;
}

↓

double f(double a, double b, double eps) {
        double r48040 = b;
        double r48041 = -1.70677655260281e+142;
        bool r48042 = r48040 <= r48041;
        double r48043 = eps;
        double r48044 = a;
        double r48045 = r48044 + r48040;
        double r48046 = r48045 * r48043;
        double r48047 = exp(r48046);
        double r48048 = 1.0;
        double r48049 = r48047 - r48048;
        double r48050 = r48043 * r48049;
        double r48051 = 0.16666666666666666;
        double r48052 = cbrt(r48044);
        double r48053 = r48052 * r48052;
        double r48054 = 3.0;
        double r48055 = pow(r48053, r48054);
        double r48056 = pow(r48043, r48054);
        double r48057 = r48044 * r48056;
        double r48058 = r48055 * r48057;
        double r48059 = r48051 * r48058;
        double r48060 = 0.5;
        double r48061 = 2.0;
        double r48062 = pow(r48044, r48061);
        double r48063 = pow(r48043, r48061);
        double r48064 = r48062 * r48063;
        double r48065 = r48060 * r48064;
        double r48066 = r48044 * r48043;
        double r48067 = r48065 + r48066;
        double r48068 = r48059 + r48067;
        double r48069 = r48040 * r48043;
        double r48070 = exp(r48069);
        double r48071 = r48070 - r48048;
        double r48072 = r48068 * r48071;
        double r48073 = r48050 / r48072;
        double r48074 = 1.8294720987358993e+68;
        bool r48075 = r48040 <= r48074;
        double r48076 = exp(r48066);
        double r48077 = r48076 - r48048;
        double r48078 = r48040 * r48040;
        double r48079 = r48078 * r48056;
        double r48080 = cbrt(r48040);
        double r48081 = pow(r48080, r48054);
        double r48082 = r48079 * r48081;
        double r48083 = r48051 * r48082;
        double r48084 = pow(r48040, r48061);
        double r48085 = r48063 * r48084;
        double r48086 = r48060 * r48085;
        double r48087 = r48043 * r48040;
        double r48088 = r48086 + r48087;
        double r48089 = r48083 + r48088;
        double r48090 = r48077 * r48089;
        double r48091 = r48050 / r48090;
        double r48092 = 3.275590786210769e+179;
        bool r48093 = r48040 <= r48092;
        double r48094 = 2.9576575817204165e+293;
        bool r48095 = r48040 <= r48094;
        double r48096 = !r48095;
        bool r48097 = r48093 || r48096;
        double r48098 = exp(r48040);
        double r48099 = pow(r48098, r48043);
        double r48100 = r48099 - r48048;
        double r48101 = r48077 * r48100;
        double r48102 = r48050 / r48101;
        double r48103 = r48097 ? r48073 : r48102;
        double r48104 = r48075 ? r48091 : r48103;
        double r48105 = r48042 ? r48073 : r48104;
        return r48105;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	14.7
Herbie	53.8

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

Derivation

1. Split input into 3 regimes

2. if b < -1.70677655260281e+142 or 1.8294720987358993e+68 < b < 3.275590786210769e+179 or 2.9576575817204165e+293 < b

   1. Initial program 53.6

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

   2. Taylor expanded around 0 46.5

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

   4. Applied add-cube-cbrt 46.5

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

   5. Applied unpow-prod-down 46.5

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

   6. Applied associate-*l* 46.1

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

   7. Simplified 46.1

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

   if -1.70677655260281e+142 < b < 1.8294720987358993e+68

   1. Initial program 63.2

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

   2. Taylor expanded around 0 56.5

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

   4. Applied add-cube-cbrt 56.5

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

   5. Applied unpow-prod-down 56.5

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

   6. Applied associate-*r* 56.3

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

   7. Simplified 56.3

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

   if 3.275590786210769e+179 < b < 2.9576575817204165e+293

   1. Initial program 50.4

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

   2. Taylor expanded around inf 50.4

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

   3. Simplified 50.5

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

4. Final simplification 53.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.706776552602809909969013296017709199406 \cdot 10^{142}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)}^{3} \cdot \left(a \cdot {\varepsilon}^{3}\right)\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{elif}\;b \le 1.829472098735899306961801525168728408435 \cdot 10^{68}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left(\left(\left(b \cdot b\right) \cdot {\varepsilon}^{3}\right) \cdot {\left(\sqrt[3]{b}\right)}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\\ \mathbf{elif}\;b \le 3.275590786210768942354859723569296251086 \cdot 10^{179} \lor \neg \left(b \le 2.957657581720416521596292537514259153078 \cdot 10^{293}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)}^{3} \cdot \left(a \cdot {\varepsilon}^{3}\right)\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left({\left(e^{b}\right)}^{\varepsilon} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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))))