expq3 (problem 3.4.2)

Average Error: 60.2 → 0.3

Time: 9.0s

Precision: 64

• could not determine a ground truth for program body

  1. a = 7.321465291114456e-39
  2. b = -5.705042820112819e+286
  3. eps = -2.4185511553388966e-62

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} = -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 0.0087632414871769045\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{e^{a \cdot \varepsilon + \varepsilon \cdot b} + \left(1 - 1 \cdot \left(e^{\varepsilon \cdot b} + e^{a \cdot \varepsilon}\right)\right)}{e^{\left(a + b\right) \cdot \varepsilon} - 1}}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r100040 = eps;
        double r100041 = a;
        double r100042 = b;
        double r100043 = r100041 + r100042;
        double r100044 = r100043 * r100040;
        double r100045 = exp(r100044);
        double r100046 = 1.0;
        double r100047 = r100045 - r100046;
        double r100048 = r100040 * r100047;
        double r100049 = r100041 * r100040;
        double r100050 = exp(r100049);
        double r100051 = r100050 - r100046;
        double r100052 = r100042 * r100040;
        double r100053 = exp(r100052);
        double r100054 = r100053 - r100046;
        double r100055 = r100051 * r100054;
        double r100056 = r100048 / r100055;
        return r100056;
}

↓

double f(double a, double b, double eps) {
        double r100057 = eps;
        double r100058 = a;
        double r100059 = b;
        double r100060 = r100058 + r100059;
        double r100061 = r100060 * r100057;
        double r100062 = exp(r100061);
        double r100063 = 1.0;
        double r100064 = r100062 - r100063;
        double r100065 = r100057 * r100064;
        double r100066 = r100058 * r100057;
        double r100067 = exp(r100066);
        double r100068 = r100067 - r100063;
        double r100069 = r100059 * r100057;
        double r100070 = exp(r100069);
        double r100071 = r100070 - r100063;
        double r100072 = r100068 * r100071;
        double r100073 = r100065 / r100072;
        double r100074 = -inf.0;
        bool r100075 = r100073 <= r100074;
        double r100076 = 0.008763241487176904;
        bool r100077 = r100073 <= r100076;
        double r100078 = !r100077;
        bool r100079 = r100075 || r100078;
        double r100080 = 1.0;
        double r100081 = r100080 / r100059;
        double r100082 = r100080 / r100058;
        double r100083 = r100081 + r100082;
        double r100084 = r100057 * r100059;
        double r100085 = r100066 + r100084;
        double r100086 = exp(r100085);
        double r100087 = exp(r100084);
        double r100088 = r100087 + r100067;
        double r100089 = r100063 * r100088;
        double r100090 = r100063 - r100089;
        double r100091 = r100086 + r100090;
        double r100092 = r100091 / r100064;
        double r100093 = r100057 / r100092;
        double r100094 = r100079 ? r100083 : r100093;
        return r100094;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.2
Target	15.2
Herbie	0.3

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

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 0.008763241487176904 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))

   1. Initial program 64.0

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

   2. Taylor expanded around 0 0.0

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

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 0.008763241487176904

   1. Initial program 4.1

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

   2. Taylor expanded around inf 29.9

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

   3. Simplified 4.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} = -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 0.0087632414871769045\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{e^{a \cdot \varepsilon + \varepsilon \cdot b} + \left(1 - 1 \cdot \left(e^{\varepsilon \cdot b} + e^{a \cdot \varepsilon}\right)\right)}{e^{\left(a + b\right) \cdot \varepsilon} - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))