expq3 (problem 3.4.2)

Average Error: 60.2 → 0.3

Time: 9.4s

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

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

↓

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

C

double f(double a, double b, double eps) {
        double r106535 = eps;
        double r106536 = a;
        double r106537 = b;
        double r106538 = r106536 + r106537;
        double r106539 = r106538 * r106535;
        double r106540 = exp(r106539);
        double r106541 = 1.0;
        double r106542 = r106540 - r106541;
        double r106543 = r106535 * r106542;
        double r106544 = r106536 * r106535;
        double r106545 = exp(r106544);
        double r106546 = r106545 - r106541;
        double r106547 = r106537 * r106535;
        double r106548 = exp(r106547);
        double r106549 = r106548 - r106541;
        double r106550 = r106546 * r106549;
        double r106551 = r106543 / r106550;
        return r106551;
}

↓

double f(double a, double b, double eps) {
        double r106552 = eps;
        double r106553 = a;
        double r106554 = b;
        double r106555 = r106553 + r106554;
        double r106556 = r106555 * r106552;
        double r106557 = exp(r106556);
        double r106558 = 1.0;
        double r106559 = r106557 - r106558;
        double r106560 = r106552 * r106559;
        double r106561 = r106553 * r106552;
        double r106562 = exp(r106561);
        double r106563 = r106562 - r106558;
        double r106564 = r106554 * r106552;
        double r106565 = exp(r106564);
        double r106566 = r106565 - r106558;
        double r106567 = r106563 * r106566;
        double r106568 = r106560 / r106567;
        double r106569 = -inf.0;
        bool r106570 = r106568 <= r106569;
        double r106571 = 0.008763241487176904;
        bool r106572 = r106568 <= r106571;
        double r106573 = !r106572;
        bool r106574 = r106570 || r106573;
        double r106575 = 1.0;
        double r106576 = r106575 / r106554;
        double r106577 = r106575 / r106553;
        double r106578 = r106576 + r106577;
        double r106579 = r106552 * r106554;
        double r106580 = fma(r106553, r106552, r106579);
        double r106581 = exp(r106580);
        double r106582 = exp(r106579);
        double r106583 = r106558 * r106562;
        double r106584 = fma(r106558, r106582, r106583);
        double r106585 = r106558 - r106584;
        double r106586 = r106581 + r106585;
        double r106587 = r106586 / r106559;
        double r106588 = r106552 / r106587;
        double r106589 = r106574 ? r106578 : r106588;
        return r106589;
}

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

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

4. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020056 +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))))