Average Error: 36.0 → 31.7

Time: 10.7s

Precision: binary64

Cost: 33665

\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;g \leq -3.6722565929769024 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{g - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]

\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}

↓

\begin{array}{l}
\mathbf{if}\;g \leq -3.6722565929769024 \cdot 10^{-188}:\\
\;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{g - g}}{\sqrt[3]{a \cdot 2}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}}\\

\end{array}

(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h))))))
  (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))

↓

(FPCore (g h a)
 :precision binary64
 (if (<= g -3.6722565929769024e-188)
   (+
    (/ (cbrt (- (sqrt (- (* g g) (* h h))) g)) (cbrt (* a 2.0)))
    (/ (cbrt (- g g)) (cbrt (* a 2.0))))
   (+
    (cbrt (* (- (sqrt (- (* g g) (* h h))) g) (/ 1.0 (* a 2.0))))
    (/ (cbrt (- (- g) g)) (cbrt (* a 2.0))))))

double code(double g, double h, double a) {
	return cbrt((1.0 / (2.0 * a)) * (-g + sqrt((g * g) - (h * h)))) + cbrt((1.0 / (2.0 * a)) * (-g - sqrt((g * g) - (h * h))));
}

↓

double code(double g, double h, double a) {
	double tmp;
	if (g <= -3.6722565929769024e-188) {
		tmp = (cbrt(sqrt((g * g) - (h * h)) - g) / cbrt(a * 2.0)) + (cbrt(g - g) / cbrt(a * 2.0));
	} else {
		tmp = cbrt((sqrt((g * g) - (h * h)) - g) * (1.0 / (a * 2.0))) + (cbrt(-g - g) / cbrt(a * 2.0));
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `g`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	32.2
Cost	40192

\[\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}}\]

Alternative 2
Error	33.6
Cost	27457

\[\begin{array}{l} \mathbf{if}\;g \leq -3.6722565929769024 \cdot 10^{-188}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\left(g - g\right) \cdot \frac{1}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]

Alternative 3
Error	34.3
Cost	27457

\[\begin{array}{l} \mathbf{if}\;g \leq -8.018616055635176 \cdot 10^{-176}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\left(g - g\right) \cdot \frac{1}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\left(g - g\right) \cdot \frac{1}{a \cdot 2}}\\ \end{array}\]

Alternative 4
Error	36.2
Cost	21121

\[\begin{array}{l} \mathbf{if}\;g \leq 1.562051483471885 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\left(g - g\right) \cdot \frac{1}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\left(g + g\right) \cdot \frac{-1}{a \cdot 2}}\\ \end{array}\]

Alternative 5
Error	36.2
Cost	21121

\[\begin{array}{l} \mathbf{if}\;g \leq -1.5608046214930196 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\left(g - g\right) \cdot \frac{1}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(g - g\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}\\ \end{array}\]

Alternative 6
Error	49.4
Cost	20800

\[\sqrt[3]{\left(g - g\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\frac{-1}{a \cdot 2} \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}\]

Alternative 7
Error	61.1
Cost	64

\[-1\]

Alternative 8
Error	61.2
Cost	64

\[1\]

Derivation

Split input into 2 regimes
if g < -3.67225659297690243e-188
1. Initial program 35.5
  \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
2. Using strategy rm
3. Applied associate-*l/_binary64_547735.5
  \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}{2 \cdot a}}}\]
4. Applied cbrt-div_binary64_556635.4
  \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}}\]
5. Simplified35.4
  \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\color{blue}{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}}{\sqrt[3]{2 \cdot a}}\]
6. Using strategy rm
7. Applied associate-*l/_binary64_547735.4
  \[\leadsto \sqrt[3]{\color{blue}{\frac{1 \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}{2 \cdot a}}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{2 \cdot a}}\]
8. Applied cbrt-div_binary64_556631.5
  \[\leadsto \color{blue}{\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{2 \cdot a}}\]
9. Simplified31.5
  \[\leadsto \frac{\color{blue}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{2 \cdot a}}\]
10. Taylor expanded around -inf 31.8
  \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(-g\right) - \color{blue}{-1 \cdot g}}}{\sqrt[3]{2 \cdot a}}\]
11. Simplified31.8
  \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(-g\right) - \color{blue}{\left(-g\right)}}}{\sqrt[3]{2 \cdot a}}\]
12. Simplified31.8
  \[\leadsto \color{blue}{\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{g - g}}{\sqrt[3]{a \cdot 2}}}\]
if -3.67225659297690243e-188 < g
1. Initial program 36.5
  \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
2. Using strategy rm
3. Applied associate-*l/_binary64_547736.5
  \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}{2 \cdot a}}}\]
4. Applied cbrt-div_binary64_556632.8
  \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}}\]
5. Simplified32.8
  \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\color{blue}{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}}{\sqrt[3]{2 \cdot a}}\]
6. Taylor expanded around inf 31.7
  \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\left(-g\right) - \color{blue}{g}}}{\sqrt[3]{2 \cdot a}}\]
7. Simplified31.7
  \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}}}\]
Recombined 2 regimes into one program.
Final simplification31.7
\[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -3.6722565929769024 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{g - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))

Error

Try it out

Alternatives

Error

Derivation

`if g < -3.67225659297690243e-188`

`if -3.67225659297690243e-188 < g`

Reproduce