Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Average Error: 14.7 → 0.2

Time: 11.3s

Precision: binary64

Math

\[x \cdot \log \left(\frac{x}{y}\right) - z \]

↓

\[\mathsf{fma}\left(x, 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right), x \cdot \log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}{\sqrt[3]{y}}\right)\right) - z \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (log (/ x y))) z))

↓

(FPCore (x y z)
 :precision binary64
 (-
  (fma
   x
   (* 2.0 (log (/ (cbrt x) (cbrt y))))
   (* x (log (/ (cbrt (* (cbrt x) (* (cbrt x) (cbrt x)))) (cbrt y)))))
  z))

C

double code(double x, double y, double z) {
	return (x * log(x / y)) - z;
}

↓

double code(double x, double y, double z) {
	return fma(x, (2.0 * log(cbrt(x) / cbrt(y))), (x * log(cbrt(cbrt(x) * (cbrt(x) * cbrt(x))) / cbrt(y)))) - z;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	14.7
Target	7.5
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y < 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Derivation

1. Initial program 14.7

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]

2. Applied add-cube-cbrt_binary64 14.7

   \[\leadsto x \cdot \log \left(\frac{x}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right) - z \]

3. Applied add-cube-cbrt_binary64 14.7

   \[\leadsto x \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right) - z \]

4. Applied times-frac_binary64 14.7

   \[\leadsto x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} - z \]

5. Applied log-prod_binary64 3.7

   \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)} - z \]

6. Applied distribute-rgt-in_binary64 3.7

   \[\leadsto \color{blue}{\left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot x + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x\right)} - z \]

7. Simplified 0.2

   \[\leadsto \left(\color{blue}{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x\right) - z \]

8. Simplified 0.2

   \[\leadsto \left(x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + \color{blue}{x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)}\right) - z \]

9. Applied fma-def_binary64 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right), x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)} - z \]

10. Applied add-cbrt-cube_binary64 0.2

    \[\leadsto \mathsf{fma}\left(x, 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right), x \cdot \log \left(\frac{\color{blue}{\sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{\sqrt[3]{y}}\right)\right) - z \]

11. Final simplification 0.2

    \[\leadsto \mathsf{fma}\left(x, 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right), x \cdot \log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}{\sqrt[3]{y}}\right)\right) - z \]

Reproduce

herbie shell --seed 2021313 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))