Average Error: 35.9 → 31.7
Time: 12.4s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\]
\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}
↓
\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}
(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
 (+
  (/ (cbrt (* 0.5 (- (sqrt (- (* g g) (* h h))) g))) (cbrt a))
  (* (cbrt (/ 0.5 a)) (cbrt (- (- g) (sqrt (- (* g g) (* h h))))))))
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) {
	return (cbrt(0.5 * (sqrt((g * g) - (h * h)) - g)) / cbrt(a)) + (cbrt(0.5 / a) * cbrt(-g - sqrt((g * g) - (h * h))));
}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 35.9

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

  3. Applied cbrt-prod_binary64_54133.7

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

  4. Simplified33.7

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

  6. Applied add-cbrt-cube_binary64_54633.7

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

  7. Simplified33.7

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

  9. Applied associate-*l/_binary64_63633.7

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

  10. Applied cbrt-div_binary64_54233.7

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

  11. Applied cube-div_binary64_54833.7

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

  12. Applied cbrt-div_binary64_54231.7

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

  13. Simplified31.7

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

  14. Simplified31.7

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

  15. Final simplification31.7

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

Reproduce

herbie shell --seed 2020231 
(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))))))))