expq3 (problem 3.4.2)

Average Error: 60.4 → 0.4

Time: 35.5s

Precision: 64

• could not determine a ground truth for program body

  1. a = 7.706155067127394e+81
  2. b = 6.88703791615021e+280
  3. eps = 8.102269269768392e-109

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 1.570835326132957750197283763877702962205 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r6000017 = eps;
        double r6000018 = a;
        double r6000019 = b;
        double r6000020 = r6000018 + r6000019;
        double r6000021 = r6000020 * r6000017;
        double r6000022 = exp(r6000021);
        double r6000023 = 1.0;
        double r6000024 = r6000022 - r6000023;
        double r6000025 = r6000017 * r6000024;
        double r6000026 = r6000018 * r6000017;
        double r6000027 = exp(r6000026);
        double r6000028 = r6000027 - r6000023;
        double r6000029 = r6000019 * r6000017;
        double r6000030 = exp(r6000029);
        double r6000031 = r6000030 - r6000023;
        double r6000032 = r6000028 * r6000031;
        double r6000033 = r6000025 / r6000032;
        return r6000033;
}

↓

double f(double a, double b, double eps) {
        double r6000034 = a;
        double r6000035 = b;
        double r6000036 = r6000034 + r6000035;
        double r6000037 = eps;
        double r6000038 = r6000036 * r6000037;
        double r6000039 = exp(r6000038);
        double r6000040 = 1.0;
        double r6000041 = r6000039 - r6000040;
        double r6000042 = r6000041 * r6000037;
        double r6000043 = r6000037 * r6000035;
        double r6000044 = exp(r6000043);
        double r6000045 = r6000044 - r6000040;
        double r6000046 = r6000037 * r6000034;
        double r6000047 = exp(r6000046);
        double r6000048 = r6000047 - r6000040;
        double r6000049 = r6000045 * r6000048;
        double r6000050 = r6000042 / r6000049;
        double r6000051 = -inf.0;
        bool r6000052 = r6000050 <= r6000051;
        double r6000053 = 1.0;
        double r6000054 = r6000053 / r6000034;
        double r6000055 = r6000053 / r6000035;
        double r6000056 = r6000054 + r6000055;
        double r6000057 = 1.5708353261329578e-35;
        bool r6000058 = r6000050 <= r6000057;
        double r6000059 = r6000058 ? r6000050 : r6000056;
        double r6000060 = r6000052 ? r6000056 : r6000059;
        return r6000060;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	14.6
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < -inf.0 or 1.5708353261329578e-35 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))

   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 58.1

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

   3. Simplified 57.4

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

   4. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 1.5708353261329578e-35

   1. Initial program 3.1

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 1.570835326132957750197283763877702962205 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1.0 eps) (< eps 1.0))

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

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))