Average Error: 0.0 → 0.0
Time: 1.5s
Precision: binary64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}} \cdot \frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}} \cdot \frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}
double code(double x) {
	return (2.0 / ((double) (((double) exp(x)) + ((double) exp(((double) -(x)))))));
}
↓
double code(double x) {
	return ((double) ((((double) sqrt(2.0)) / ((double) sqrt(((double) (((double) exp(x)) + ((double) exp(((double) -(x))))))))) * (((double) sqrt(2.0)) / ((double) sqrt(((double) (((double) exp(x)) + ((double) exp(((double) -(x)))))))))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program Error: 0.0 bits

    \[\frac{2}{e^{x} + e^{-x}}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrtError: 0.8 bits

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

  4. Applied add-sqr-sqrtError: 0.0 bits

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

  5. Applied times-fracError: 0.0 bits

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

  6. Final simplificationError: 0.0 bits

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

Reproduce

herbie shell --seed 2020200 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))