expq3 (problem 3.4.2)

Average Error: 60.2 → 49.7

Time: 16.3s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{\varepsilon \cdot b} - 1\\ t_1 := \varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)\\ t_2 := e^{\varepsilon \cdot a} - 1\\ t_3 := t_2 \cdot t_0\\ t_4 := \frac{t_1}{t_3}\\ \mathbf{if}\;t_4 \leq -0.321337824977454:\\ \;\;\;\;\frac{t_1}{\left(\varepsilon \cdot a\right) \cdot t_0}\\ \mathbf{elif}\;t_4 \leq 3.4699057758429277 \cdot 10^{-47}:\\ \;\;\;\;\frac{t_1}{\log \left(e^{t_3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_5 := \sqrt[3]{t_2}\\ \frac{t_1}{\left(t_5 \cdot t_5\right) \cdot \left(t_5 \cdot \left(b \cdot \left(\varepsilon + b \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(b \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)\right)\right)} \end{array}\\ \end{array} \]

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 (- (exp (* eps b)) 1.0))
        (t_1 (* eps (- (exp (* eps (+ a b))) 1.0)))
        (t_2 (- (exp (* eps a)) 1.0))
        (t_3 (* t_2 t_0))
        (t_4 (/ t_1 t_3)))
   (if (<= t_4 -0.321337824977454)
     (/ t_1 (* (* eps a) t_0))
     (if (<= t_4 3.4699057758429277e-47)
       (/ t_1 (log (exp t_3)))
       (let* ((t_5 (cbrt t_2)))
         (/
          t_1
          (*
           (* t_5 t_5)
           (*
            t_5
            (*
             b
             (+
              eps
              (*
               b
               (*
                (* eps eps)
                (+ (* eps (* b 0.16666666666666666)) 0.5)))))))))))))

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 = exp(eps * b) - 1.0;
	double t_1 = eps * (exp(eps * (a + b)) - 1.0);
	double t_2 = exp(eps * a) - 1.0;
	double t_3 = t_2 * t_0;
	double t_4 = t_1 / t_3;
	double tmp;
	if (t_4 <= -0.321337824977454) {
		tmp = t_1 / ((eps * a) * t_0);
	} else if (t_4 <= 3.4699057758429277e-47) {
		tmp = t_1 / log(exp(t_3));
	} else {
		double t_5 = cbrt(t_2);
		tmp = t_1 / ((t_5 * t_5) * (t_5 * (b * (eps + (b * ((eps * eps) * ((eps * (b * 0.16666666666666666)) + 0.5)))))));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.2
Target	14.7
Herbie	49.7

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

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))) < -0.321337824977454

   1. Initial program 63.9

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

   2. Taylor expanded around 0 20.6

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

   if -0.321337824977454 < (/.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))) < 3.46990577584292773e-47

   1. Initial program 4.3

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Using strategy rm

   3. Applied add-log-exp_binary64 5.0

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

   if 3.46990577584292773e-47 < (/.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 63.6

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

   2. Taylor expanded around 0 57.7

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(0.5 \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)\right)}} \]

   3. Simplified 57.4

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(b \cdot b\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot b\right)\right)}} \]
   4. Using strategy rm

   5. Applied add-cube-cbrt_binary64 57.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\left(\sqrt[3]{e^{a \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{a \cdot \varepsilon} - 1}\right) \cdot \sqrt[3]{e^{a \cdot \varepsilon} - 1}\right)} \cdot \left(\varepsilon \cdot b + \left(b \cdot b\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot b\right)\right)} \]

   6. Applied associate-*l*_binary64 57.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\sqrt[3]{e^{a \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{a \cdot \varepsilon} - 1}\right) \cdot \left(\sqrt[3]{e^{a \cdot \varepsilon} - 1} \cdot \left(\varepsilon \cdot b + \left(b \cdot b\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot b\right)\right)\right)}} \]

   7. Simplified 57.3

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt[3]{e^{a \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{a \cdot \varepsilon} - 1}\right) \cdot \color{blue}{\left(\left(b \cdot \left(b \cdot \left(0.16666666666666666 \cdot \left(b \cdot {\varepsilon}^{3}\right) + 0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon\right)\right) \cdot \sqrt[3]{e^{\varepsilon \cdot a} - 1}\right)}} \]
   8. Using strategy rm

   9. Applied *-un-lft-identity_binary64 57.3

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt[3]{e^{a \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{a \cdot \varepsilon} - 1}\right) \cdot \left(\left(b \cdot \left(b \cdot \left(0.16666666666666666 \cdot \left(b \cdot {\varepsilon}^{3}\right) + 0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon\right)\right) \cdot \sqrt[3]{\color{blue}{1 \cdot \left(e^{\varepsilon \cdot a} - 1\right)}}\right)} \]

   10. Applied cbrt-prod_binary64 57.3

       \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt[3]{e^{a \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{a \cdot \varepsilon} - 1}\right) \cdot \left(\left(b \cdot \left(b \cdot \left(0.16666666666666666 \cdot \left(b \cdot {\varepsilon}^{3}\right) + 0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon\right)\right) \cdot \color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{e^{\varepsilon \cdot a} - 1}\right)}\right)} \]

   11. Applied associate-*r*_binary64 57.3

       \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt[3]{e^{a \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{a \cdot \varepsilon} - 1}\right) \cdot \color{blue}{\left(\left(\left(b \cdot \left(b \cdot \left(0.16666666666666666 \cdot \left(b \cdot {\varepsilon}^{3}\right) + 0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon\right)\right) \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{e^{\varepsilon \cdot a} - 1}\right)}} \]

   12. Simplified 57.3

       \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt[3]{e^{a \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{a \cdot \varepsilon} - 1}\right) \cdot \left(\color{blue}{\left(b \cdot \left(\varepsilon + b \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(b \cdot 0.16666666666666666\right) \cdot \varepsilon + 0.5\right)\right)\right)\right)} \cdot \sqrt[3]{e^{\varepsilon \cdot a} - 1}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq -0.321337824977454:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(\varepsilon \cdot a\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \leq 3.4699057758429277 \cdot 10^{-47}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\log \left(e^{\left(e^{\varepsilon \cdot a} - 1\right) \cdot \left(e^{\varepsilon \cdot b} - 1\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} - 1\right)}{\left(\sqrt[3]{e^{\varepsilon \cdot a} - 1} \cdot \sqrt[3]{e^{\varepsilon \cdot a} - 1}\right) \cdot \left(\sqrt[3]{e^{\varepsilon \cdot a} - 1} \cdot \left(b \cdot \left(\varepsilon + b \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(b \cdot 0.16666666666666666\right) + 0.5\right)\right)\right)\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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))))