Average Error: 26.4 → 1.8
Time: 17.4s
Precision: 64
$\left(\left(k \gt 1 \land k \lt 100000.0\right) \land 0 \lt x\right) \land x \lt 1$
$\frac{\sinh \left(k - k \cdot x\right)}{\sinh k}$
$1 - x$
\frac{\sinh \left(k - k \cdot x\right)}{\sinh k}
1 - x
double f(double k, double x) {
double r30639143 = k;
double r30639144 = x;
double r30639145 = r30639143 * r30639144;
double r30639146 = r30639143 - r30639145;
double r30639147 = sinh(r30639146);
double r30639148 = sinh(r30639143);
double r30639149 = r30639147 / r30639148;
return r30639149;
}


double f(double __attribute__((unused)) k, double x) {
double r30639150 = 1.0;
double r30639151 = x;
double r30639152 = r30639150 - r30639151;
return r30639152;
}



# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Initial program 26.4

$\frac{\sinh \left(k - k \cdot x\right)}{\sinh k}$
2. Taylor expanded around 0 2.2

$\leadsto \color{blue}{1 - \left(x + \frac{1}{3} \cdot \left(x \cdot {k}^{2}\right)\right)}$
3. Simplified2.2

$\leadsto \color{blue}{\left(1 - x\right) - \left(x \cdot k\right) \cdot \left(\frac{1}{3} \cdot k\right)}$
4. Taylor expanded around 0 1.8

$\leadsto \color{blue}{1 - x}$
5. Final simplification1.8

$\leadsto 1 - x$

# Reproduce

herbie shell --seed 1
(FPCore (k x)
:name "sinh(k - k * x) / sinh(k)"
:pre (and (and (and (> k 1) (< k 100000.0)) (< 0 x)) (< x 1))
(/ (sinh (- k (* k x))) (sinh k)))