2-ancestry mixing, positive discriminant

Average Error: 36.0 → 31.6

Time: 12.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{2 \cdot a}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \frac{\sqrt[3]{\left(-g\right) - t_1}}{t_0}\\ t_3 := t_1 - g\\ t_4 := \sqrt[3]{\frac{1}{2 \cdot a} \cdot t_3} + \sqrt[3]{\left(g + t_1\right) \cdot \frac{-1}{2 \cdot a}}\\ t_5 := \sqrt[3]{t_3}\\ \mathbf{if}\;t_4 \leq -2.6688122044117306 \cdot 10^{-103}:\\ \;\;\;\;\frac{t_5}{t_0} + t_2\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;\sqrt[3]{\frac{t_3}{a}} \cdot \sqrt[3]{0.5} + \frac{\sqrt[3]{\left(-g\right) - g}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2 + \sqrt[3]{0.5} \cdot \frac{t_5}{\sqrt[3]{a}}\\ \end{array} \]

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
 (let* ((t_0 (cbrt (* 2.0 a)))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2 (/ (cbrt (- (- g) t_1)) t_0))
        (t_3 (- t_1 g))
        (t_4
         (+
          (cbrt (* (/ 1.0 (* 2.0 a)) t_3))
          (cbrt (* (+ g t_1) (/ -1.0 (* 2.0 a))))))
        (t_5 (cbrt t_3)))
   (if (<= t_4 -2.6688122044117306e-103)
     (+ (/ t_5 t_0) t_2)
     (if (<= t_4 0.0)
       (+ (* (cbrt (/ t_3 a)) (cbrt 0.5)) (/ (cbrt (- (- g) g)) t_0))
       (+ t_2 (* (cbrt 0.5) (/ t_5 (cbrt a))))))))

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 t_0 = cbrt(2.0 * a);
	double t_1 = sqrt((g * g) - (h * h));
	double t_2 = cbrt(-g - t_1) / t_0;
	double t_3 = t_1 - g;
	double t_4 = cbrt((1.0 / (2.0 * a)) * t_3) + cbrt((g + t_1) * (-1.0 / (2.0 * a)));
	double t_5 = cbrt(t_3);
	double tmp;
	if (t_4 <= -2.6688122044117306e-103) {
		tmp = (t_5 / t_0) + t_2;
	} else if (t_4 <= 0.0) {
		tmp = (cbrt(t_3 / a) * cbrt(0.5)) + (cbrt(-g - g) / t_0);
	} else {
		tmp = t_2 + (cbrt(0.5) * (t_5 / cbrt(a)));
	}
	return tmp;
}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Split input into 3 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))))))) < -2.6688122044117306e-103

   1. Initial program 10.2

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

   3. Applied associate-*l/_binary64 10.2

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

   4. Applied cbrt-div_binary64 7.9

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

   5. Simplified 7.9

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

   7. Applied associate-*l/_binary64 7.9

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

   8. Applied cbrt-div_binary64 5.7

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

   9. Simplified 5.7

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

   if -2.6688122044117306e-103 < (+.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 58.1

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

   3. Applied associate-*l/_binary64 58.1

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

   4. Applied cbrt-div_binary64 37.6

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

   5. Simplified 37.6

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

   6. Taylor expanded around 0 44.7

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

   7. Simplified 37.5

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

   8. Taylor expanded around inf 9.1

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

   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)))))))

   1. Initial program 44.3

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

   3. Applied associate-*l/_binary64 44.3

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

   4. Applied cbrt-div_binary64 43.3

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

   5. Simplified 43.3

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

   6. Taylor expanded around 0 53.9

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

   7. Simplified 43.3

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

   9. Applied cbrt-div_binary64 42.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{a}}} \cdot \sqrt[3]{0.5} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{2 \cdot a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -2.6688122044117306 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 0:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{a}} \cdot \sqrt[3]{0.5} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{0.5} \cdot \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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))))))))