Average Error: 12.4 → 4.0
Time: 21.4s
Precision: 64
Internal Precision: 320
$\cos^{-1} \left(\frac{z \cdot y}{\sqrt{x \cdot z} \cdot y}\right)$
$\begin{array}{l} \mathbf{if}\;x \le -2.9785525839490055 \cdot 10^{-210}:\\ \;\;\;\;\cos^{-1} \left(\frac{z}{\sqrt{z \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\frac{z}{\sqrt{z} \cdot \sqrt{x}}\right)\\ \end{array}$

# Try it out

Results

 In Out
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# Derivation

1. Split input into 2 regimes
2. ## if x < -2.9785525839490055e-210

1. Initial program 11.5

$\cos^{-1} \left(\frac{z \cdot y}{\sqrt{x \cdot z} \cdot y}\right)$
2. Initial simplification6.5

$\leadsto \cos^{-1} \left(\frac{z}{\sqrt{x \cdot z}}\right)$

## if -2.9785525839490055e-210 < x

1. Initial program 13.3

$\cos^{-1} \left(\frac{z \cdot y}{\sqrt{x \cdot z} \cdot y}\right)$
2. Initial simplification8.9

$\leadsto \cos^{-1} \left(\frac{z}{\sqrt{x \cdot z}}\right)$
3. Using strategy rm
4. Applied sqrt-prod1.5

$\leadsto \cos^{-1} \left(\frac{z}{\color{blue}{\sqrt{x} \cdot \sqrt{z}}}\right)$
3. Recombined 2 regimes into one program.
4. Final simplification4.0

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.9785525839490055 \cdot 10^{-210}:\\ \;\;\;\;\cos^{-1} \left(\frac{z}{\sqrt{z \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\frac{z}{\sqrt{z} \cdot \sqrt{x}}\right)\\ \end{array}$

# Runtime

Time bar (total: 21.4s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (z y x)
:name "acos(z * y / (sqrt(x * z) * y))"
(acos (/ (* z y) (* (sqrt (* x z)) y))))