Average Error: 36.0 → 32.4
Time: 10.2s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a}}\\ t_0 \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + t_0 \cdot \sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \end{array} \]
\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}
t_0 := \sqrt[3]{\frac{0.5}{a}}\\
t_0 \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + t_0 \cdot \sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}
\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
 (let* ((t_0 (cbrt (/ 0.5 a))))
   (+
    (* t_0 (cbrt (- (sqrt (- (* g g) (* h h))) g)))
    (* t_0 (cbrt (- (- g) (sqrt (* (+ g h) (- g 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) {
	double t_0 = cbrt(0.5 / a);
	return (t_0 * cbrt(sqrt((g * g) - (h * h)) - g)) + (t_0 * cbrt(-g - sqrt((g + h) * (g - 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 36.0

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

  3. Applied cbrt-prod_binary6434.1

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

  4. Simplified34.1

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

  5. Simplified34.1

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

  7. Applied cbrt-prod_binary6432.4

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

  8. Simplified32.4

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

  10. Applied difference-of-squares_binary6432.4

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

  11. Simplified32.4

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

  12. Final simplification32.4

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

Reproduce

herbie shell --seed 2021196 
(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))))))))