expq3 (problem 3.4.2)

Average Error: 60.3 → 53.8

Time: 13.2s

Precision: 64

• could not determine a ground truth for program body

  1. a = -4.7556502650462125e+294
  2. b = 2.729864436188278e+247
  3. eps = -3.4107494527315723e-29

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -6.317274345536705554494934198648981426267 \cdot 10^{196}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\frac{\mathsf{fma}\left(-1, 1, {\left(e^{a}\right)}^{\left(2 \cdot \varepsilon\right)}\right)}{e^{a \cdot \varepsilon} + 1} \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\\ \mathbf{elif}\;a \le -5.631940663491904405136254441698421041501 \cdot 10^{81}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\log \left(e^{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)}\\ \mathbf{elif}\;a \le 38308603704447418368:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r97087 = eps;
        double r97088 = a;
        double r97089 = b;
        double r97090 = r97088 + r97089;
        double r97091 = r97090 * r97087;
        double r97092 = exp(r97091);
        double r97093 = 1.0;
        double r97094 = r97092 - r97093;
        double r97095 = r97087 * r97094;
        double r97096 = r97088 * r97087;
        double r97097 = exp(r97096);
        double r97098 = r97097 - r97093;
        double r97099 = r97089 * r97087;
        double r97100 = exp(r97099);
        double r97101 = r97100 - r97093;
        double r97102 = r97098 * r97101;
        double r97103 = r97095 / r97102;
        return r97103;
}

↓

double f(double a, double b, double eps) {
        double r97104 = a;
        double r97105 = -6.3172743455367056e+196;
        bool r97106 = r97104 <= r97105;
        double r97107 = eps;
        double r97108 = b;
        double r97109 = r97104 + r97108;
        double r97110 = r97109 * r97107;
        double r97111 = exp(r97110);
        double r97112 = 1.0;
        double r97113 = r97111 - r97112;
        double r97114 = r97107 * r97113;
        double r97115 = -r97112;
        double r97116 = exp(r97104);
        double r97117 = 2.0;
        double r97118 = r97117 * r97107;
        double r97119 = pow(r97116, r97118);
        double r97120 = fma(r97115, r97112, r97119);
        double r97121 = r97104 * r97107;
        double r97122 = exp(r97121);
        double r97123 = r97122 + r97112;
        double r97124 = r97120 / r97123;
        double r97125 = 0.16666666666666666;
        double r97126 = 3.0;
        double r97127 = pow(r97107, r97126);
        double r97128 = pow(r97108, r97126);
        double r97129 = r97127 * r97128;
        double r97130 = 0.5;
        double r97131 = pow(r97107, r97117);
        double r97132 = pow(r97108, r97117);
        double r97133 = r97131 * r97132;
        double r97134 = r97107 * r97108;
        double r97135 = fma(r97130, r97133, r97134);
        double r97136 = fma(r97125, r97129, r97135);
        double r97137 = r97124 * r97136;
        double r97138 = r97114 / r97137;
        double r97139 = -5.631940663491904e+81;
        bool r97140 = r97104 <= r97139;
        double r97141 = r97122 - r97112;
        double r97142 = r97108 * r97107;
        double r97143 = exp(r97142);
        double r97144 = r97143 - r97112;
        double r97145 = r97141 * r97144;
        double r97146 = exp(r97145);
        double r97147 = log(r97146);
        double r97148 = r97114 / r97147;
        double r97149 = 3.830860370444742e+19;
        bool r97150 = r97104 <= r97149;
        double r97151 = pow(r97104, r97126);
        double r97152 = r97151 * r97127;
        double r97153 = pow(r97104, r97117);
        double r97154 = r97153 * r97131;
        double r97155 = fma(r97130, r97154, r97121);
        double r97156 = fma(r97125, r97152, r97155);
        double r97157 = r97156 * r97144;
        double r97158 = r97114 / r97157;
        double r97159 = r97141 * r97136;
        double r97160 = r97114 / r97159;
        double r97161 = r97150 ? r97158 : r97160;
        double r97162 = r97140 ? r97148 : r97161;
        double r97163 = r97106 ? r97138 : r97162;
        return r97163;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	14.3
Herbie	53.8

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

Derivation

1. Split input into 4 regimes

2. if a < -6.3172743455367056e+196

   1. Initial program 52.0

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

   2. Taylor expanded around 0 40.0

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

   3. Simplified 40.0

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

   5. Applied flip-- 40.2

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

   6. Simplified 41.7

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

   if -6.3172743455367056e+196 < a < -5.631940663491904e+81

   1. Initial program 55.6

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

   3. Applied add-log-exp 55.9

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

   if -5.631940663491904e+81 < a < 3.830860370444742e+19

   1. Initial program 63.7

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

   2. Taylor expanded around 0 56.1

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

   3. Simplified 56.1

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

   if 3.830860370444742e+19 < a

   1. Initial program 55.0

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

   2. Taylor expanded around 0 49.8

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

   3. Simplified 49.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 53.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.317274345536705554494934198648981426267 \cdot 10^{196}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\frac{\mathsf{fma}\left(-1, 1, {\left(e^{a}\right)}^{\left(2 \cdot \varepsilon\right)}\right)}{e^{a \cdot \varepsilon} + 1} \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\\ \mathbf{elif}\;a \le -5.631940663491904405136254441698421041501 \cdot 10^{81}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\log \left(e^{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)}\\ \mathbf{elif}\;a \le 38308603704447418368:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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))))