Average Error: 0.1 → 0.1
Time: 2.4s
Precision: binary64
\[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
↓
\[1 - \left(\sqrt{\sqrt{0.12}} \cdot \left(x \cdot \left(x \cdot {\left(\sqrt{0.12}\right)}^{1.5}\right)\right) + x \cdot 0.253\right) \]
1 - x \cdot \left(0.253 + x \cdot 0.12\right)
↓
1 - \left(\sqrt{\sqrt{0.12}} \cdot \left(x \cdot \left(x \cdot {\left(\sqrt{0.12}\right)}^{1.5}\right)\right) + x \cdot 0.253\right)

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))
↓
(FPCore (x)
 :precision binary64
 (-
  1.0
  (+ (* (sqrt (sqrt 0.12)) (* x (* x (pow (sqrt 0.12) 1.5)))) (* x 0.253))))
double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}
↓
double code(double x) {
	return 1.0 - ((sqrt(sqrt(0.12)) * (x * (x * pow(sqrt(0.12), 1.5)))) + (x * 0.253));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]

  3. Taylor expanded in x around 0 0.1

    \[\leadsto \color{blue}{1 - \left(0.12 \cdot {x}^{2} + 0.253 \cdot x\right)} \]

  4. Applied add-sqr-sqrt_binary640.1

    \[\leadsto 1 - \left(\color{blue}{\left(\sqrt{0.12} \cdot \sqrt{0.12}\right)} \cdot {x}^{2} + 0.253 \cdot x\right) \]

  5. Applied associate-*l*_binary640.1

    \[\leadsto 1 - \left(\color{blue}{\sqrt{0.12} \cdot \left(\sqrt{0.12} \cdot {x}^{2}\right)} + 0.253 \cdot x\right) \]

  6. Applied add-sqr-sqrt_binary640.3

    \[\leadsto 1 - \left(\color{blue}{\left(\sqrt{\sqrt{0.12}} \cdot \sqrt{\sqrt{0.12}}\right)} \cdot \left(\sqrt{0.12} \cdot {x}^{2}\right) + 0.253 \cdot x\right) \]

  7. Applied associate-*l*_binary640.3

    \[\leadsto 1 - \left(\color{blue}{\sqrt{\sqrt{0.12}} \cdot \left(\sqrt{\sqrt{0.12}} \cdot \left(\sqrt{0.12} \cdot {x}^{2}\right)\right)} + 0.253 \cdot x\right) \]

  8. Simplified0.1

    \[\leadsto 1 - \left(\sqrt{\sqrt{0.12}} \cdot \color{blue}{\left(x \cdot \left(x \cdot {\left(\sqrt{0.12}\right)}^{1.5}\right)\right)} + 0.253 \cdot x\right) \]

  9. Final simplification0.1

    \[\leadsto 1 - \left(\sqrt{\sqrt{0.12}} \cdot \left(x \cdot \left(x \cdot {\left(\sqrt{0.12}\right)}^{1.5}\right)\right) + x \cdot 0.253\right) \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))