Average Error: 15.0 → 10.8
Time: 29.4s
Precision: 64
Internal Precision: 2368
$\cos^{-1} \left(x1 \cdot y1 + \frac{x2 \cdot y2}{\sqrt{x1^2 + x2^2}^* \cdot \sqrt{y1^2 + y2^2}^*}\right)$
$\cos^{-1} \left(\frac{x2}{\sqrt{x1^2 + x2^2}^*} \cdot \frac{y2}{\sqrt{y1^2 + y2^2}^*} + x1 \cdot y1\right)$

# Try it out

Results

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# Derivation

1. Initial program 15.0

$\cos^{-1} \left(x1 \cdot y1 + \frac{x2 \cdot y2}{\sqrt{x1^2 + x2^2}^* \cdot \sqrt{y1^2 + y2^2}^*}\right)$
2. Using strategy rm
3. Applied times-frac10.8

$\leadsto \cos^{-1} \left(x1 \cdot y1 + \color{blue}{\frac{x2}{\sqrt{x1^2 + x2^2}^*} \cdot \frac{y2}{\sqrt{y1^2 + y2^2}^*}}\right)$
4. Final simplification10.8

$\leadsto \cos^{-1} \left(\frac{x2}{\sqrt{x1^2 + x2^2}^*} \cdot \frac{y2}{\sqrt{y1^2 + y2^2}^*} + x1 \cdot y1\right)$

# Runtime

Time bar (total: 29.4s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x1 y1 x2 y2)
:name "acos(x1 * y1 + x2 * y2 / (hypot(x1, x2) * hypot(y1, y2)))"
(acos (+ (* x1 y1) (/ (* x2 y2) (* (hypot x1 x2) (hypot y1 y2))))))