Average Error: 60.0 → 1.3
Time: 13.3s
Precision: binary64
\[-1 < \varepsilon \land \varepsilon < 1\]
\[[a, b]=\mathsf{sort}([a, b])\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
↓
\[\begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ t_2 := \frac{1}{a} + \frac{1}{b}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, a \cdot \left(\varepsilon \cdot \varepsilon\right), t_2\right)\\ \mathbf{elif}\;t_1 \leq 2.8347809522572864 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon \cdot \mathsf{expm1}\left(t_0\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
\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}
t_0 := \varepsilon \cdot \left(a + b\right)\\
t_1 := \frac{\varepsilon \cdot \left(e^{t_0} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\
t_2 := \frac{1}{a} + \frac{1}{b}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;\mathsf{fma}\left(0.08333333333333333, a \cdot \left(\varepsilon \cdot \varepsilon\right), t_2\right)\\

\mathbf{elif}\;t_1 \leq 2.8347809522572864 \cdot 10^{-70}:\\
\;\;\;\;\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon \cdot \mathsf{expm1}\left(t_0\right)}}\\

\mathbf{else}:\\
\;\;\;\;t_2\\


\end{array}

(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* (+ a b) eps)) 1.0))
  (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))
↓
(FPCore (a b eps)
 :precision binary64
 (let* ((t_0 (* eps (+ a b)))
        (t_1
         (/
          (* eps (- (exp t_0) 1.0))
          (* (- (exp (* eps a)) 1.0) (- (exp (* eps b)) 1.0))))
        (t_2 (+ (/ 1.0 a) (/ 1.0 b))))
   (if (<= t_1 (- INFINITY))
     (fma 0.08333333333333333 (* a (* eps eps)) t_2)
     (if (<= t_1 2.8347809522572864e-70)
       (/ 1.0 (/ (* (expm1 (* eps a)) (expm1 (* eps b))) (* eps (expm1 t_0))))
       t_2))))
double code(double a, double b, double eps) {
	return (eps * (exp((a + b) * eps) - 1.0)) / ((exp(a * eps) - 1.0) * (exp(b * eps) - 1.0));
}
↓
double code(double a, double b, double eps) {
	double t_0 = eps * (a + b);
	double t_1 = (eps * (exp(t_0) - 1.0)) / ((exp(eps * a) - 1.0) * (exp(eps * b) - 1.0));
	double t_2 = (1.0 / a) + (1.0 / b);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(0.08333333333333333, (a * (eps * eps)), t_2);
	} else if (t_1 <= 2.8347809522572864e-70) {
		tmp = 1.0 / ((expm1(eps * a) * expm1(eps * b)) / (eps * expm1(t_0)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original60.0
Target15.1
Herbie1.3
\[\frac{a + b}{a \cdot b} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < -inf.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. Simplified17.6

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

    3. Taylor expanded in eps around 0 12.4

      \[\leadsto \color{blue}{0.08333333333333333 \cdot \left({\varepsilon}^{2} \cdot a\right) + \left(\frac{1}{b} + \left(\frac{1}{a} + 0.08333333333333333 \cdot \left({\varepsilon}^{2} \cdot b\right)\right)\right)} \]

    4. Simplified12.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, a \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{1}{b} + \mathsf{fma}\left(0.08333333333333333, b \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{1}{a}\right)\right)} \]

    5. Taylor expanded in b around 0 6.0

      \[\leadsto \mathsf{fma}\left(0.08333333333333333, a \cdot \left(\varepsilon \cdot \varepsilon\right), \color{blue}{\frac{1}{b} + \frac{1}{a}}\right) \]

    6. Simplified6.0

      \[\leadsto \mathsf{fma}\left(0.08333333333333333, a \cdot \left(\varepsilon \cdot \varepsilon\right), \color{blue}{\frac{1}{a} + \frac{1}{b}}\right) \]

    if -inf.0 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < 2.834780952257286e-70

    1. Initial program 2.7

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

    3. Applied clear-num_binary640.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}}} \]

    if 2.834780952257286e-70 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1)))

    1. Initial program 63.3

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

    2. Simplified47.6

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]

    3. Taylor expanded in eps around 0 14.8

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

    4. Taylor expanded in a around 0 0.7

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, a \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{1}{a} + \frac{1}{b}\right)\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 2.8347809522572864 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2021344 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :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))))