Average Error: 0.0 → 0.0
Time: 2.2s
Precision: binary64
\[\frac{2}{e^{x} + e^{-x}} \]
↓
\[\sqrt{\frac{\frac{2}{e^{x} + e^{-x}}}{\cosh x}} \]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{\frac{\frac{2}{e^{x} + e^{-x}}}{\cosh x}}

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))
↓
(FPCore (x)
 :precision binary64
 (sqrt (/ (/ 2.0 (+ (exp x) (exp (- x)))) (cosh x))))
double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}
↓
double code(double x) {
	return sqrt((2.0 / (exp(x) + exp(-x))) / cosh(x));
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\frac{2}{e^{x} + e^{-x}} \]

  2. Applied add-sqr-sqrt_binary640.8

    \[\leadsto \frac{2}{\color{blue}{\sqrt{e^{x} + e^{-x}} \cdot \sqrt{e^{x} + e^{-x}}}} \]

  3. Applied add-sqr-sqrt_binary640.0

    \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{\sqrt{e^{x} + e^{-x}} \cdot \sqrt{e^{x} + e^{-x}}} \]

  4. Applied times-frac_binary640.0

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}} \cdot \frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}} \]

  5. Applied sqrt-undiv_binary640.0

    \[\leadsto \frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}} \cdot \color{blue}{\sqrt{\frac{2}{e^{x} + e^{-x}}}} \]

  6. Applied sqrt-undiv_binary640.0

    \[\leadsto \color{blue}{\sqrt{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}} \]

  7. Applied sqrt-unprod_binary640.0

    \[\leadsto \color{blue}{\sqrt{\frac{2}{e^{x} + e^{-x}} \cdot \frac{2}{e^{x} + e^{-x}}}} \]

  8. Simplified0.0

    \[\leadsto \sqrt{\color{blue}{\frac{\frac{2}{e^{x} + e^{-x}}}{\cosh x}}} \]

  9. Final simplification0.0

    \[\leadsto \sqrt{\frac{\frac{2}{e^{x} + e^{-x}}}{\cosh x}} \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))