Average Error: 37.1 → 0.4
Time: 35.5s
Precision: 64
Internal Precision: 2368
$\sin \left(x + e\right) - \sin x$
$\begin{array}{l} \mathbf{if}\;e \le -8.235254616753446 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos e + \cos x \cdot \sin e\right) - \sin x\\ \mathbf{if}\;e \le 3.219014701150901 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{x + \left(e + x\right)}{2}\right) \cdot \sin \left(\frac{e}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos e + \left(\cos x \cdot \sin e - \sin x\right)\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 3 regimes
2. ## if e < -8.235254616753446e-09

1. Initial program 29.6

$\sin \left(x + e\right) - \sin x$
2. Using strategy rm
3. Applied sin-sum0.6

$\leadsto \color{blue}{\left(\sin x \cdot \cos e + \cos x \cdot \sin e\right)} - \sin x$

## if -8.235254616753446e-09 < e < 3.219014701150901e-11

1. Initial program 44.8

$\sin \left(x + e\right) - \sin x$
2. Using strategy rm
3. Applied diff-sin44.8

$\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + e\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + e\right) + x}{2}\right)\right)}$
4. Applied simplify0.3

$\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{x + \left(e + x\right)}{2}\right) \cdot \sin \left(\frac{e}{2}\right)\right)}$

## if 3.219014701150901e-11 < e

1. Initial program 29.7

$\sin \left(x + e\right) - \sin x$
2. Using strategy rm
3. Applied sin-sum0.6

$\leadsto \color{blue}{\left(\sin x \cdot \cos e + \cos x \cdot \sin e\right)} - \sin x$
4. Applied associate--l+0.6

$\leadsto \color{blue}{\sin x \cdot \cos e + \left(\cos x \cdot \sin e - \sin x\right)}$
3. Recombined 3 regimes into one program.

# Runtime

Time bar (total: 35.5s)Debug log

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