Average Error: 4.3 → 0.1
Time: 21.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{x} + 1}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{e^{x} + 1}
double f(double x) {
        double r503122 = 2.0;
        double r503123 = x;
        double r503124 = r503122 * r503123;
        double r503125 = exp(r503124);
        double r503126 = 1.0;
        double r503127 = r503125 - r503126;
        double r503128 = exp(r503123);
        double r503129 = r503128 - r503126;
        double r503130 = r503127 / r503129;
        double r503131 = sqrt(r503130);
        return r503131;
}
↓
double f(double x) {
        double r503132 = x;
        double r503133 = exp(r503132);
        double r503134 = 1.0;
        double r503135 = r503133 + r503134;
        double r503136 = sqrt(r503135);
        return r503136;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 4.3

    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\sqrt{e^{x} + 1}}\]

  3. Final simplification0.1

    \[\leadsto \sqrt{e^{x} + 1}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))