Average Error: 29.5 → 1.6
Time: 13.5s
Precision: 64
Internal Precision: 1344
$e^{\sin x} - 1$
$\begin{array}{l} \mathbf{if}\;\sin x \le -0.12014515666984021:\\ \;\;\;\;\frac{1 + e^{\sin x}}{\frac{1 + e^{\sin x}}{e^{\sin x} - 1}}\\ \mathbf{elif}\;\sin x \le 1.198484680558077 \cdot 10^{-06}:\\ \;\;\;\;\left(x + \frac{1}{2} \cdot {x}^{2}\right) - \frac{1}{8} \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\sin x + \sin x} - 1}{1 + e^{\sin x}}\\ \end{array}$

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# Derivation

1. Split input into 3 regimes
2. ## if (sin x) < -0.12014515666984021

1. Initial program 0.4

$e^{\sin x} - 1$
2. Initial simplification0.4

$\leadsto e^{\sin x} - 1$
3. Using strategy rm
4. Applied flip--0.5

$\leadsto \color{blue}{\frac{e^{\sin x} \cdot e^{\sin x} - 1 \cdot 1}{e^{\sin x} + 1}}$
5. Using strategy rm
6. Applied difference-of-squares0.4

$\leadsto \frac{\color{blue}{\left(e^{\sin x} + 1\right) \cdot \left(e^{\sin x} - 1\right)}}{e^{\sin x} + 1}$
7. Applied associate-/l*0.4

$\leadsto \color{blue}{\frac{e^{\sin x} + 1}{\frac{e^{\sin x} + 1}{e^{\sin x} - 1}}}$

## if -0.12014515666984021 < (sin x) < 1.198484680558077e-06

1. Initial program 56.8

$e^{\sin x} - 1$
2. Initial simplification56.8

$\leadsto e^{\sin x} - 1$
3. Using strategy rm
4. Applied flip--56.8

$\leadsto \color{blue}{\frac{e^{\sin x} \cdot e^{\sin x} - 1 \cdot 1}{e^{\sin x} + 1}}$
5. Taylor expanded around 0 2.5

$\leadsto \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + x\right) - \frac{1}{8} \cdot {x}^{4}}$

## if 1.198484680558077e-06 < (sin x)

1. Initial program 1.0

$e^{\sin x} - 1$
2. Initial simplification1.0

$\leadsto e^{\sin x} - 1$
3. Using strategy rm
4. Applied flip--1.1

$\leadsto \color{blue}{\frac{e^{\sin x} \cdot e^{\sin x} - 1 \cdot 1}{e^{\sin x} + 1}}$
5. Using strategy rm
6. Applied prod-exp0.8

$\leadsto \frac{\color{blue}{e^{\sin x + \sin x}} - 1 \cdot 1}{e^{\sin x} + 1}$
3. Recombined 3 regimes into one program.
4. Final simplification1.6

$\leadsto \begin{array}{l} \mathbf{if}\;\sin x \le -0.12014515666984021:\\ \;\;\;\;\frac{1 + e^{\sin x}}{\frac{1 + e^{\sin x}}{e^{\sin x} - 1}}\\ \mathbf{elif}\;\sin x \le 1.198484680558077 \cdot 10^{-06}:\\ \;\;\;\;\left(x + \frac{1}{2} \cdot {x}^{2}\right) - \frac{1}{8} \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\sin x + \sin x} - 1}{1 + e^{\sin x}}\\ \end{array}$

# Runtime

Time bar (total: 13.5s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x)
:name "exp(sin(x))-1"
(- (exp (sin x)) 1))