expfmod

Average Error: 59.5 → 23.7

Time: 14.7s

Precision: binary64

Math

\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

↓

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;\begin{array}{l} t_2 := t_0 \cdot t_1\\ t_2 \leq 0 \lor \neg \left(t_2 \leq 1\right) \end{array}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \log \log \left(e^{t_0}\right)\\ e^{\frac{t_3 \cdot t_3 - x \cdot x}{x + t_3}} \end{array}\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))) (t_1 (exp (- x))))
   (if (let* ((t_2 (* t_0 t_1))) (or (<= t_2 0.0) (not (<= t_2 1.0))))
     t_1
     (let* ((t_3 (log (log (exp t_0)))))
       (exp (/ (- (* t_3 t_3) (* x x)) (+ x t_3)))))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

↓

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = exp(-x);
	double t_2 = t_0 * t_1;
	double tmp;
	if ((t_2 <= 0.0) || !(t_2 <= 1.0)) {
		tmp = t_1;
	} else {
		double t_3 = log(log(exp(t_0)));
		tmp = exp(((t_3 * t_3) - (x * x)) / (x + t_3));
	}
	return tmp;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0 or 1 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 61.8

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Simplified 61.8

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]

   3. Applied add-exp-log_binary64 61.8

      \[\leadsto \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}{e^{x}} \]

   4. Applied div-exp_binary64 61.8

      \[\leadsto \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}} \]

   5. Taylor expanded in x around inf 24.2

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]

   6. Simplified 24.2

      \[\leadsto e^{\color{blue}{-x}} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 1

   1. Initial program 11.4

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Simplified 11.2

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]

   3. Applied add-exp-log_binary64 11.2

      \[\leadsto \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}{e^{x}} \]

   4. Applied div-exp_binary64 11.2

      \[\leadsto \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}} \]

   5. Applied add-log-exp_binary64 13.0

      \[\leadsto e^{\log \color{blue}{\log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)} - x} \]

   6. Applied flip--_binary64 13.1

      \[\leadsto e^{\color{blue}{\frac{\log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot \log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) - x \cdot x}{\log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0 \lor \neg \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 1\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) \cdot \log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right) - x \cdot x}{x + \log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021344 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))