expq3 (problem 3.4.2)

Average Error: 60.3 → 0.3

Time: 19.0s

Precision: binary64

\[-1 < \varepsilon \land \varepsilon < 1\]

\[[a, b]=\mathsf{sort}([a, b])\]

Math

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

↓

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, a \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{1}{a} + \frac{1}{b}\right)\\ \mathbf{elif}\;t_1 \leq 1.7185873133137824 \cdot 10^{-18}:\\ \;\;\;\;\left(\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \mathsf{expm1}\left(t_0\right)\right) \cdot \frac{1}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \mathsf{fma}\left(b \cdot \left(\varepsilon \cdot \varepsilon\right), 0.08333333333333333, \frac{1}{a}\right)\\ \end{array} \]

FPCore

(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)))))
   (if (<= t_1 (- INFINITY))
     (fma 0.08333333333333333 (* a (* eps eps)) (+ (/ 1.0 a) (/ 1.0 b)))
     (if (<= t_1 1.7185873133137824e-18)
       (* (* (/ eps (expm1 (* eps a))) (expm1 t_0)) (/ 1.0 (expm1 (* eps b))))
       (+ (/ 1.0 b) (fma (* b (* eps eps)) 0.08333333333333333 (/ 1.0 a)))))))

C

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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(0.08333333333333333, (a * (eps * eps)), ((1.0 / a) + (1.0 / b)));
	} else if (t_1 <= 1.7185873133137824e-18) {
		tmp = ((eps / expm1(eps * a)) * expm1(t_0)) * (1.0 / expm1(eps * b));
	} else {
		tmp = (1.0 / b) + fma((b * (eps * eps)), 0.08333333333333333, (1.0 / a));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	15.1
Herbie	0.3

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

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. Simplified 17.4

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

   3. Taylor expanded in eps around 0 9.2

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

   4. Simplified 9.2

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

   5. Applied add-sqr-sqrt_binary64 35.9

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

   6. Taylor expanded in b around 0 0

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

   7. Simplified 0

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

   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))) < 1.7185873133137824e-18

   1. Initial program 3.8

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

   2. Simplified 0.1

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

   3. Applied associate-/r*_binary64 0.1

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

   4. Simplified 0.1

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

   5. Applied div-inv_binary64 0.1

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

   6. Simplified 0.1

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

   if 1.7185873133137824e-18 < (/.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 63.7

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

   2. Simplified 47.5

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

   3. Taylor expanded in eps around 0 2.1

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

   4. Simplified 2.1

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

   5. Taylor expanded in a around 0 0.4

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

   6. Simplified 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot b, 0.08333333333333333, \frac{1}{a}\right) + \frac{1}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, a \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{1}{a} + \frac{1}{b}\right)\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 1.7185873133137824 \cdot 10^{-18}:\\ \;\;\;\;\left(\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)\right) \cdot \frac{1}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \mathsf{fma}\left(b \cdot \left(\varepsilon \cdot \varepsilon\right), 0.08333333333333333, \frac{1}{a}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022005 
(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))))