(+ 0.5641895835477563 (/ (* (/ 1 (pow x 2)) (+ (* (+ (* (+ (* (+ (* (+ (* (- 0.0163153871373021) (/ 1 (pow x 2))) (- 0.3053266349612323)) (/ 1 (pow x 2))) (- 0.3603448999498044)) (/ 1 (pow x 2))) (- 0.1257817261112292)) (/ 1 (pow x 2))) (- 0.01608378514874228)) (/ 1 (pow x 2))) (- 0.0006587491615298378))) (+ (* (+ (* (+ (* (+ (* (+ (/ 1 (pow x 2)) 2.568520192289822) (/ 1 (pow x 2))) 1.87295284992346) (/ 1 (pow x 2))) 0.5279051029514285) (/ 1 (pow x 2))) 0.06051834131244132) (/ 1 (pow x 2))) 0.002335204976268692)))

Average Error: 17.6 → 17.6

Time: 4.4s

Precision: binary64

Math

\[0.564189583547756279 + \frac{\frac{1}{{x}^{2}} \cdot \left(\left(\left(\left(\left(\left(-0.016315387137302101\right) \cdot \frac{1}{{x}^{2}} + \left(-0.305326634961232302\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.360344899949804398\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.125781726111229203\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.016083785148742279\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-6.5874916152983782 \cdot 10^{-4}\right)\right)}{\left(\left(\left(\left(\frac{1}{{x}^{2}} + 2.5685201922898222\right) \cdot \frac{1}{{x}^{2}} + 1.87295284992346001\right) \cdot \frac{1}{{x}^{2}} + 0.527905102951428451\right) \cdot \frac{1}{{x}^{2}} + 0.0605183413124413191\right) \cdot \frac{1}{{x}^{2}} + 0.0023352049762686922}\]

↓

\[0.564189583547756279 + \frac{\frac{1}{{x}^{2}} \cdot \left(\left(\left(\left(\left(\left(-0.016315387137302101\right) \cdot \frac{1}{{x}^{2}} + \left(-0.305326634961232302\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.360344899949804398\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.125781726111229203\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.016083785148742279\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-6.5874916152983782 \cdot 10^{-4}\right)\right)}{\left(\left(\left(\left(\frac{1}{{x}^{2}} + 2.5685201922898222\right) \cdot \frac{1}{{x}^{2}} + 1.87295284992346001\right) \cdot \frac{1}{{x}^{2}} + 0.527905102951428451\right) \cdot \frac{1}{{x}^{2}} + 0.0605183413124413191\right) \cdot \frac{1}{{x}^{2}} + 0.0023352049762686922}\]

Error

Bits error versus x

Derivation

1. Initial program 17.6

   \[0.564189583547756279 + \frac{\frac{1}{{x}^{2}} \cdot \left(\left(\left(\left(\left(\left(-0.016315387137302101\right) \cdot \frac{1}{{x}^{2}} + \left(-0.305326634961232302\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.360344899949804398\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.125781726111229203\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.016083785148742279\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-6.5874916152983782 \cdot 10^{-4}\right)\right)}{\left(\left(\left(\left(\frac{1}{{x}^{2}} + 2.5685201922898222\right) \cdot \frac{1}{{x}^{2}} + 1.87295284992346001\right) \cdot \frac{1}{{x}^{2}} + 0.527905102951428451\right) \cdot \frac{1}{{x}^{2}} + 0.0605183413124413191\right) \cdot \frac{1}{{x}^{2}} + 0.0023352049762686922}\]

2. Final simplification 17.6

   \[\leadsto 0.564189583547756279 + \frac{\frac{1}{{x}^{2}} \cdot \left(\left(\left(\left(\left(\left(-0.016315387137302101\right) \cdot \frac{1}{{x}^{2}} + \left(-0.305326634961232302\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.360344899949804398\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.125781726111229203\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-0.016083785148742279\right)\right) \cdot \frac{1}{{x}^{2}} + \left(-6.5874916152983782 \cdot 10^{-4}\right)\right)}{\left(\left(\left(\left(\frac{1}{{x}^{2}} + 2.5685201922898222\right) \cdot \frac{1}{{x}^{2}} + 1.87295284992346001\right) \cdot \frac{1}{{x}^{2}} + 0.527905102951428451\right) \cdot \frac{1}{{x}^{2}} + 0.0605183413124413191\right) \cdot \frac{1}{{x}^{2}} + 0.0023352049762686922}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x)
  :name "(+ 0.5641895835477563 (/ (* (/ 1 (pow x 2)) (+ (* (+ (* (+ (* (+ (* (+ (* (- 0.0163153871373021) (/ 1 (pow x 2))) (- 0.3053266349612323)) (/ 1 (pow x 2))) (- 0.3603448999498044)) (/ 1 (pow x 2))) (- 0.1257817261112292)) (/ 1 (pow x 2))) (- 0.01608378514874228)) (/ 1 (pow x 2))) (- 0.0006587491615298378))) (+ (* (+ (* (+ (* (+ (* (+ (/ 1 (pow x 2)) 2.568520192289822) (/ 1 (pow x 2))) 1.87295284992346) (/ 1 (pow x 2))) 0.5279051029514285) (/ 1 (pow x 2))) 0.06051834131244132) (/ 1 (pow x 2))) 0.002335204976268692)))"
  :precision binary64
  (+ 0.5641895835477563 (/ (* (/ 1.0 (pow x 2.0)) (+ (* (+ (* (+ (* (+ (* (+ (* (neg 0.0163153871373021) (/ 1.0 (pow x 2.0))) (neg 0.3053266349612323)) (/ 1.0 (pow x 2.0))) (neg 0.3603448999498044)) (/ 1.0 (pow x 2.0))) (neg 0.1257817261112292)) (/ 1.0 (pow x 2.0))) (neg 0.01608378514874228)) (/ 1.0 (pow x 2.0))) (neg 0.0006587491615298378))) (+ (* (+ (* (+ (* (+ (* (+ (/ 1.0 (pow x 2.0)) 2.568520192289822) (/ 1.0 (pow x 2.0))) 1.87295284992346) (/ 1.0 (pow x 2.0))) 0.5279051029514285) (/ 1.0 (pow x 2.0))) 0.06051834131244132) (/ 1.0 (pow x 2.0))) 0.002335204976268692))))