Average Error: 36.0 → 1.2
Time: 15.0s
Precision: binary64
Cost: 27328
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\frac{h \cdot -0.5}{\frac{g}{0.5 \cdot h}}}}{\sqrt[3]{a}} \]
(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 (- (- (* h (* 0.5 (/ h g))) g) g))) (cbrt a))
  (/ (cbrt (/ (* h -0.5) (/ g (* 0.5 h)))) (cbrt a))))
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 * (((h * (0.5 * (h / g))) - g) - g))) / cbrt(a)) + (cbrt(((h * -0.5) / (g / (0.5 * h)))) / cbrt(a));
}

public static double code(double g, double h, double a) {
	return Math.cbrt(((1.0 / (2.0 * a)) * (-g + Math.sqrt(((g * g) - (h * h)))))) + Math.cbrt(((1.0 / (2.0 * a)) * (-g - Math.sqrt(((g * g) - (h * h))))));
}
↓
public static double code(double g, double h, double a) {
	return (Math.cbrt((0.5 * (((h * (0.5 * (h / g))) - g) - g))) / Math.cbrt(a)) + (Math.cbrt(((h * -0.5) / (g / (0.5 * h)))) / Math.cbrt(a));
}

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 * Float64(Float64(Float64(h * Float64(0.5 * Float64(h / g))) - g) - g))) / cbrt(a)) + Float64(cbrt(Float64(Float64(h * -0.5) / Float64(g / Float64(0.5 * h)))) / cbrt(a)))
end

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 * N[(N[(N[(h * N[(0.5 * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - g), $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(h * -0.5), $MachinePrecision] / N[(g / N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

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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. Simplified36.0

    \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
    Proof
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 1 2)) a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 14 points increase in error, 0 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 2 a))) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 14 points increase in error, 0 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) (neg.f64 g))))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 14 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 14 points increase in error, 0 points decrease in error
    (+.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 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 -1)) a)))): 0 points increase in error, 14 points decrease in error
    (+.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 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) -1) a)))): 0 points increase in error, 0 points decrease in error
    (+.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 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (/.f64 1 2) a) -1))))): 0 points increase in error, 0 points decrease in error
    (+.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 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 2 a))) -1)))): 14 points increase in error, 0 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 (/.f64 1 (*.f64 2 a)) -1) (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))): 14 points increase in error, 0 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 1 (*.f64 2 a)) (*.f64 -1 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))))): 0 points increase in error, 14 points decrease in error
    (+.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)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))))): 14 points increase in error, 0 points decrease in error
    (+.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)) (Rewrite<= distribute-neg-out_binary64 (+.f64 (neg.f64 g) (neg.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))))): 0 points increase in error, 0 points decrease in error
    (+.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)) (Rewrite<= sub-neg_binary64 (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))): 0 points increase in error, 14 points decrease in error

  3. Taylor expanded in g around -inf 47.7

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

  4. Simplified47.7

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\color{blue}{\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
    Proof
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (-.f64 (/.f64 (*.f64 1/2 h) (/.f64 g h)) g) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 h (/.f64 g h)))) g) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 6 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (-.f64 (*.f64 1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 h h) g))) g) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 6 points increase in error, 0 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (-.f64 (*.f64 1/2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 h 2)) g)) g) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 6 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 1/2 (/.f64 (pow.f64 h 2) g)) (neg.f64 g))) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 6 points increase in error, 0 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (+.f64 (*.f64 1/2 (/.f64 (pow.f64 h 2) g)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 g))) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 6 points decrease in error

  5. Taylor expanded in g around -inf 19.5

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

  6. Simplified17.3

    \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)} + \sqrt[3]{\left(g + \color{blue}{\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)}\right) \cdot \frac{-0.5}{a}} \]
    Proof
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (-.f64 (/.f64 (*.f64 1/2 h) (/.f64 g h)) g) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 h (/.f64 g h)))) g) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 6 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (-.f64 (*.f64 1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 h h) g))) g) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 6 points increase in error, 0 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (-.f64 (*.f64 1/2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 h 2)) g)) g) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 6 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 1/2 (/.f64 (pow.f64 h 2) g)) (neg.f64 g))) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 6 points increase in error, 0 points decrease in error
    (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (+.f64 (*.f64 1/2 (/.f64 (pow.f64 h 2) g)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 g))) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 6 points decrease in error

  7. Applied egg-rr2.5

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

  8. Simplified2.5

    \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
    Proof
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (cbrt.f64 (*.f64 (+.f64 g (-.f64 (/.f64 (*.f64 1/2 h) (/.f64 g h)) g)) (/.f64 -1/2 a)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 h 1/2) (/.f64 h g))) g) g))) (cbrt.f64 a)) (cbrt.f64 (*.f64 (+.f64 g (-.f64 (/.f64 (*.f64 1/2 h) (/.f64 g h)) g)) (/.f64 -1/2 a)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 h)) (/.f64 h g)) g) g))) (cbrt.f64 a)) (cbrt.f64 (*.f64 (+.f64 g (-.f64 (/.f64 (*.f64 1/2 h) (/.f64 g h)) g)) (/.f64 -1/2 a)))): 4 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (Rewrite=> fma-neg_binary64 (fma.f64 (*.f64 1/2 h) (/.f64 h g) (neg.f64 g))) g))) (cbrt.f64 a)) (cbrt.f64 (*.f64 (+.f64 g (-.f64 (/.f64 (*.f64 1/2 h) (/.f64 g h)) g)) (/.f64 -1/2 a)))): 4 points increase in error, 0 points decrease in error

  9. Applied egg-rr2.3

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

  10. Simplified1.2

    \[\leadsto \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \color{blue}{\frac{\sqrt[3]{\frac{-0.5 \cdot h}{\frac{g}{0.5 \cdot h}}}}{\sqrt[3]{a}}} \]
    Proof
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (/.f64 (cbrt.f64 (/.f64 (*.f64 -1/2 h) (/.f64 g (*.f64 1/2 h)))) (cbrt.f64 a))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (/.f64 (cbrt.f64 (/.f64 (*.f64 -1/2 h) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 g h) 1/2)))) (cbrt.f64 a))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (/.f64 (cbrt.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.f64 h (/.f64 (/.f64 g h) 1/2))))) (cbrt.f64 a))): 11 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (/.f64 (cbrt.f64 (*.f64 -1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 h 1/2) (/.f64 g h))))) (cbrt.f64 a))): 4 points increase in error, 7 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (/.f64 (cbrt.f64 (*.f64 -1/2 (Rewrite<= associate-*r/_binary64 (*.f64 h (/.f64 1/2 (/.f64 g h)))))) (cbrt.f64 a))): 0 points increase in error, 4 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (/.f64 (cbrt.f64 (*.f64 -1/2 (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 h (/.f64 1/2 (/.f64 g h))) 0)))) (cbrt.f64 a))): 0 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (/.f64 (cbrt.f64 (*.f64 -1/2 (+.f64 (*.f64 h (/.f64 1/2 (/.f64 g h))) (Rewrite<= +-inverses_binary64 (-.f64 g g))))) (cbrt.f64 a))): 11 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (/.f64 (cbrt.f64 (*.f64 -1/2 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 h (/.f64 1/2 (/.f64 g h))) g) g)))) (cbrt.f64 a))): 0 points increase in error, 11 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (/.f64 (cbrt.f64 (*.f64 -1/2 (-.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 1/2 (/.f64 g h)) h)) g) g))) (cbrt.f64 a))): 11 points increase in error, 0 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (/.f64 (cbrt.f64 (*.f64 -1/2 (-.f64 (Rewrite<= fma-udef_binary64 (fma.f64 (/.f64 1/2 (/.f64 g h)) h g)) g))) (cbrt.f64 a))): 0 points increase in error, 11 points decrease in error
    (+.f64 (/.f64 (cbrt.f64 (*.f64 1/2 (-.f64 (-.f64 (*.f64 h (*.f64 1/2 (/.f64 h g))) g) g))) (cbrt.f64 a)) (/.f64 (cbrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (fma.f64 (/.f64 1/2 (/.f64 g h)) h g) g) -1/2))) (cbrt.f64 a))): 0 points increase in error, 11 points decrease in error

  11. Final simplification1.2

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

Alternatives

Alternative 1
Error1.2
Cost27328
\[\frac{\sqrt[3]{\frac{h \cdot -0.5}{\frac{g}{0.5 \cdot h}}}}{\sqrt[3]{a}} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g} \]
Alternative 2
Error4.5
Cost21060
\[\begin{array}{l} \mathbf{if}\;h \cdot h \leq 10^{+307}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{\frac{0.25}{\frac{g}{h \cdot h}}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)} + \sqrt[3]{\frac{h}{\frac{a}{h} \cdot \frac{g}{-0.25}}}\\ \end{array} \]
Alternative 3
Error2.9
Cost20928
\[\frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(0.5 \cdot \frac{h}{\frac{g}{h}}\right)} \]
Alternative 4
Error2.4
Cost20928
\[\frac{\sqrt[3]{0.25 \cdot \frac{h}{\frac{g}{h}} - g}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
Alternative 5
Error16.6
Cost14400
\[\sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)} + \sqrt[3]{\frac{h}{\frac{a}{h} \cdot \frac{g}{-0.25}}} \]
Alternative 6
Error17.3
Cost14080
\[\sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-g}{a}} \]
Alternative 7
Error17.6
Cost13568
\[\sqrt[3]{\frac{-g}{a}} + \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} \]
Alternative 8
Error62.1
Cost6848
\[\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} \]

Error

Reproduce

herbie shell --seed 2022340 
(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))))))))