2-ancestry mixing, positive discriminant

Average Error: 36.0 → 30.5

Time: 15.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -4.012135255503389 \cdot 10^{-107}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 0:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq \infty:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

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)))))
      -4.012135255503389e-107)
   (+
    (* (cbrt (/ 0.5 a)) (cbrt (- (sqrt (- (* g g) (* h h))) g)))
    (/ (cbrt (- (- g) (sqrt (- (* g g) (* h h))))) (cbrt (* 2.0 a))))
   (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)))))
        0.0)
     (+
      (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ -1.0 (* 2.0 a))))
      (* (cbrt (/ 0.5 a)) (cbrt (- (- g) g))))
     (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 (/ 0.5 a)) (cbrt (- (sqrt (- (* g g) (* h h))) g)))
        (* (cbrt (/ 0.5 a)) (cbrt (- (- g) (sqrt (- (* g g) (* h h)))))))
       -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)))) <= -4.012135255503389e-107) {
		tmp = (cbrt(0.5 / a) * cbrt(sqrt((g * g) - (h * h)) - g)) + (cbrt(-g - sqrt((g * g) - (h * h))) / cbrt(2.0 * a));
	} else 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)))) <= 0.0) {
		tmp = cbrt((g + sqrt((g * g) - (h * h))) * (-1.0 / (2.0 * a))) + (cbrt(0.5 / a) * cbrt(-g - g));
	} else 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(0.5 / a) * cbrt(sqrt((g * g) - (h * h)) - g)) + (cbrt(0.5 / a) * cbrt(-g - sqrt((g * g) - (h * h))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Split input into 4 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))))))) < -4.01213525550338899e-107

   1. Initial program 11.0

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

   3. Applied cbrt-prod_binary64_6588 8.7

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

   4. Simplified 8.7

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

   5. Simplified 8.7

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

   7. Applied associate-*l/_binary64_6500 8.7

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

   8. Applied cbrt-div_binary64_6589 6.1

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

   if -4.01213525550338899e-107 < (+.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))))))) < 0.0

   1. Initial program 60.6

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

   3. Applied cbrt-prod_binary64_6588 41.7

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

   4. Simplified 41.5

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

   5. Simplified 41.5

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

   6. Taylor expanded around -inf 7.5

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

   if 0.0 < (+.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))))))) < +inf.0

   1. Initial program 10.5

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

   3. Applied cbrt-prod_binary64_6588 7.9

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

   4. Simplified 7.9

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

   5. Simplified 7.9

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

   7. Applied cbrt-prod_binary64_6588 6.0

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

   8. Simplified 6.0

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

   10. Applied *-un-lft-identity_binary64_6557 6.0

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

   11. Applied sqrt-prod_binary64_6573 6.0

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

   12. Simplified 6.0

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

   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 4 regimes into one program.

4. Final simplification 30.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -4.012135255503389 \cdot 10^{-107}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 0:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq \infty:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Reproduce

herbie shell --seed 2021084 
(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))))))))