2-ancestry mixing, positive discriminant

Average Error: 36.6 → 1.3

Time: 17.2s

Precision: binary64

Cost: 33600

Math

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

↓

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

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
 (+
  (* (cbrt (/ 0.5 a)) (cbrt (* -0.5 (/ h (/ g h)))))
  (/ (cbrt (* -0.5 (+ g (fma (/ h g) (* -0.5 h) g)))) (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) {
	return (cbrt((0.5 / a)) * cbrt((-0.5 * (h / (g / h))))) + (cbrt((-0.5 * (g + fma((h / g), (-0.5 * h), g)))) / cbrt(a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(-g) + sqrt(Float64(Float64(g * g) - Float64(h * h)))))) + cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(-g) - sqrt(Float64(Float64(g * g) - Float64(h * h)))))))
end

↓

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(0.5 / a)) * cbrt(Float64(-0.5 * Float64(h / Float64(g / h))))) + Float64(cbrt(Float64(-0.5 * Float64(g + fma(Float64(h / g), Float64(-0.5 * h), g)))) / cbrt(a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[((-g) + N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[((-g) - N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

↓

code[g_, h_, a_] := N[(N[(N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(-0.5 * N[(h / N[(g / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(g + N[(N[(h / g), $MachinePrecision] * N[(-0.5 * h), $MachinePrecision] + g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6

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

2. Simplified 36.5

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

   [Start] 36.6	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   +-commutative [=>] 36.6	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   associate-/r* [=>] 36.5	\[ \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   metadata-eval [=>] 36.5	\[ \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   unsub-neg [=>] 36.5	\[ \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   sub-neg [=>] 36.5	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-g\right) + \left(-\sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   distribute-neg-out [=>] 36.5	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-\left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   neg-mul-1 [=>] 36.5	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-1 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   associate-*r* [=>] 36.5	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(\frac{1}{2 \cdot a} \cdot -1\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}} \]
   *-commutative [<=] 36.5	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \left(\frac{1}{2 \cdot a} \cdot -1\right)}} \]
   associate-*l/ [=>] 36.5	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \color{blue}{\frac{1 \cdot -1}{2 \cdot a}}} \]

3. Applied egg-rr 34.2

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

4. Taylor expanded in g around inf 48.6

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

5. Simplified 48.5

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

   [Start] 48.6	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{{h}^{2}}{g}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   unpow2 [=>] 48.6	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{\color{blue}{h \cdot h}}{g}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   associate-/l* [=>] 48.5	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \color{blue}{\frac{h}{\frac{g}{h}}}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

6. Taylor expanded in g around inf 19.5

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

7. Simplified 17.2

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

   [Start] 19.5	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{h}{\frac{g}{h}}} + \sqrt[3]{\left(g + \left(-0.5 \cdot \frac{{h}^{2}}{g} + g\right)\right) \cdot \frac{-0.5}{a}} \]
   fma-def [=>] 19.5	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{h}{\frac{g}{h}}} + \sqrt[3]{\left(g + \color{blue}{\mathsf{fma}\left(-0.5, \frac{{h}^{2}}{g}, g\right)}\right) \cdot \frac{-0.5}{a}} \]
   unpow2 [=>] 19.5	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{h}{\frac{g}{h}}} + \sqrt[3]{\left(g + \mathsf{fma}\left(-0.5, \frac{\color{blue}{h \cdot h}}{g}, g\right)\right) \cdot \frac{-0.5}{a}} \]
   associate-/l* [=>] 17.2	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{h}{\frac{g}{h}}} + \sqrt[3]{\left(g + \mathsf{fma}\left(-0.5, \color{blue}{\frac{h}{\frac{g}{h}}}, g\right)\right) \cdot \frac{-0.5}{a}} \]

8. Applied egg-rr 1.3

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

9. Simplified 1.3

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

   [Start] 1.3	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{h}{\frac{g}{h}}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \mathsf{fma}\left(-0.5, h \cdot \frac{h}{g}, g\right)\right)}}{\sqrt[3]{a}} \]
   fma-udef [=>] 1.3	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{h}{\frac{g}{h}}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \color{blue}{\left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right) + g\right)}\right)}}{\sqrt[3]{a}} \]
   associate-*r* [=>] 1.3	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{h}{\frac{g}{h}}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \left(\color{blue}{\left(-0.5 \cdot h\right) \cdot \frac{h}{g}} + g\right)\right)}}{\sqrt[3]{a}} \]
   *-commutative [=>] 1.3	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{h}{\frac{g}{h}}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \left(\color{blue}{\frac{h}{g} \cdot \left(-0.5 \cdot h\right)} + g\right)\right)}}{\sqrt[3]{a}} \]
   fma-def [=>] 1.3	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{h}{\frac{g}{h}}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \color{blue}{\mathsf{fma}\left(\frac{h}{g}, -0.5 \cdot h, g\right)}\right)}}{\sqrt[3]{a}} \]

10. Final simplification 1.3

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

Alternatives

Alternative 1
Error	1.9
Cost	48324

\[\begin{array}{l} t_0 := \frac{\sqrt[3]{\frac{h \cdot 0.25}{\frac{g}{h}} - g}}{\sqrt[3]{a}}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(t_1 - g\right)} + \sqrt[3]{\left(g + t_1\right) \cdot \frac{-1}{a \cdot 2}} \leq \infty:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{-0.5}{a} \cdot \left(0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g - g\right)}\\ \end{array} \]

Alternative 2
Error	1.3
Cost	33600

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

Alternative 3
Error	17.3
Cost	20416

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

Alternative 4
Error	2.7
Cost	20416

\[\frac{\sqrt[3]{\frac{h \cdot 0.25}{\frac{g}{h}} - g}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g - g\right)} \]

Alternative 5
Error	18.1
Cost	14528

\[\sqrt[3]{\frac{\frac{h \cdot 0.25}{\frac{g}{h}} - g}{a}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + \left(0.5 \cdot \frac{h}{\frac{g}{h}} - g\right)\right)} \]

Alternative 6
Error	18.4
Cost	13568

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

Alternative 7
Error	62.1
Cost	6848

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

Reproduce

herbie shell --seed 2023059 
(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))))))))