2-ancestry mixing, positive discriminant

Average Error: 36.1 → 31.3

Time: 15.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq \infty:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \left(\sqrt[3]{g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\frac{-1}{a}}\right) \cdot \sqrt[3]{0.5}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

FPCore

(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 (<=
      (+
       (cbrt (* (/ 1.0 (* 2.0 a)) (- (sqrt (- (* g g) (* h h))) g)))
       (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ -1.0 (* 2.0 a)))))
      INFINITY)
   (+
    (/ (cbrt (- (sqrt (- (* g g) (* h h))) g)) (cbrt (* 2.0 a)))
    (*
     (* (cbrt (+ g (sqrt (* (+ g h) (- g h))))) (cbrt (/ -1.0 a)))
     (cbrt 0.5)))
   -1.0))

C

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 ((cbrt((1.0 / (2.0 * a)) * (sqrt((g * g) - (h * h)) - g)) + cbrt((g + sqrt((g * g) - (h * h))) * (-1.0 / (2.0 * a)))) <= ((double) INFINITY)) {
		tmp = (cbrt(sqrt((g * g) - (h * h)) - g) / cbrt(2.0 * a)) + ((cbrt(g + sqrt((g + h) * (g - h))) * cbrt(-1.0 / a)) * cbrt(0.5));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Split input into 2 regimes

2. if (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < +inf.0

   1. Initial program 14.0

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

   3. Applied associate-*l/_binary64_2749 14.0

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

   4. Applied cbrt-div_binary64_2838 10.2

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

   5. Simplified 10.2

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

   6. Taylor expanded around -inf 37.4

      \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \color{blue}{e^{0.3333333333333333 \cdot \left(\log \left(\frac{-1}{a}\right) + \log \left(\sqrt{{g}^{2} - {h}^{2}} + g\right)\right)} \cdot \sqrt[3]{0.5}}\]

   7. Simplified 7.4

      \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \color{blue}{\left(\sqrt[3]{g + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{-1}{a}}\right) \cdot \sqrt[3]{0.5}}\]
   8. Using strategy rm

   9. Applied difference-of-squares_binary64_2775 7.4

      \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \left(\sqrt[3]{g + \sqrt{\color{blue}{\left(g + h\right) \cdot \left(g - h\right)}}} \cdot \sqrt[3]{\frac{-1}{a}}\right) \cdot \sqrt[3]{0.5}\]

   10. Simplified 7.4

       \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \left(\sqrt[3]{g + \sqrt{\color{blue}{\left(h + g\right)} \cdot \left(g - h\right)}} \cdot \sqrt[3]{\frac{-1}{a}}\right) \cdot \sqrt[3]{0.5}\]

   if +inf.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

   1. Initial program 61.3

      \[-1\]
3. Recombined 2 regimes into one program.

4. Final simplification 31.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq \infty:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \left(\sqrt[3]{g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\frac{-1}{a}}\right) \cdot \sqrt[3]{0.5}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021102 
(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))))))))