expfmod

Average Error: 59.6 → 24.3

Time: 11.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\begin{array}{l} t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t_0\\ t_1 \leq 0 \lor \neg \left(t_1 \leq 1\right) \end{array}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \sqrt[3]{\cos x}\\ e^{\log \left(\left(e^{x}\right) \bmod \left(\left|t_2\right| \cdot \sqrt{t_2}\right)\right) - x} \end{array}\\ \end{array} \]

FPCore

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

↓

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

C

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

↓

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x))) * t_0;
	double tmp;
	if ((t_1 <= 0.0) || !(t_1 <= 1.0)) {
		tmp = t_0;
	} else {
		double t_2 = cbrt(cos(x));
		tmp = exp(log(fmod(exp(x), (fabs(t_2) * sqrt(t_2)))) - x);
	}
	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 61.7

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

   2. Simplified 61.7

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

   3. Applied add-cube-cbrt_binary64 61.7

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}}}\right)\right)}{e^{x}} \]

   4. Applied sqrt-prod_binary64 61.7

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt{\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}} \cdot \sqrt{\sqrt[3]{\cos x}}\right)}\right)}{e^{x}} \]

   5. Simplified 61.7

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\left|\sqrt[3]{\cos x}\right|} \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}{e^{x}} \]

   6. Applied add-exp-log_binary64 61.7

      \[\leadsto \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}}}{e^{x}} \]

   7. Applied div-exp_binary64 61.7

      \[\leadsto \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right) - x}} \]

   8. Taylor expanded in x around inf 24.8

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

   9. Simplified 24.8

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

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

   1. Initial program 13.4

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

   2. Simplified 13.3

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

   3. Applied add-cube-cbrt_binary64 13.6

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}}}\right)\right)}{e^{x}} \]

   4. Applied sqrt-prod_binary64 13.6

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt{\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}} \cdot \sqrt{\sqrt[3]{\cos x}}\right)}\right)}{e^{x}} \]

   5. Simplified 13.6

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\left|\sqrt[3]{\cos x}\right|} \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}{e^{x}} \]

   6. Applied add-exp-log_binary64 13.5

      \[\leadsto \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right)}}}{e^{x}} \]

   7. Applied div-exp_binary64 13.5

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

4. Final simplification 24.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0 \lor \neg \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 1\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left(e^{x}\right) \bmod \left(\left|\sqrt[3]{\cos x}\right| \cdot \sqrt{\sqrt[3]{\cos x}}\right)\right) - x}\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(FPCore (x)
  :name "expfmod"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))