Cubic critical, medium range

Percentage Accurate: 30.8% → 95.5%

Time: 17.4s

Alternatives: 9

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 30.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2}}{a \cdot {b}^{3}} + \frac{3.375 \cdot {\left(c \cdot a\right)}^{3}}{a \cdot {b}^{5}}\right) + \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (*
   -0.16666666666666666
   (+
    (+
     (/ (pow (* a (* c -1.5)) 2.0) (* a (pow b 3.0)))
     (/ (* 3.375 (pow (* c a) 3.0)) (* a (pow b 5.0))))
    (/ (pow a 3.0) (/ (pow b 7.0) (* (pow c 4.0) 6.328125)))))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), (-0.16666666666666666 * (((pow((a * (c * -1.5)), 2.0) / (a * pow(b, 3.0))) + ((3.375 * pow((c * a), 3.0)) / (a * pow(b, 5.0)))) + (pow(a, 3.0) / (pow(b, 7.0) / (pow(c, 4.0) * 6.328125))))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(-0.16666666666666666 * Float64(Float64(Float64((Float64(a * Float64(c * -1.5)) ^ 2.0) / Float64(a * (b ^ 3.0))) + Float64(Float64(3.375 * (Float64(c * a) ^ 3.0)) / Float64(a * (b ^ 5.0)))) + Float64((a ^ 3.0) / Float64((b ^ 7.0) / Float64((c ^ 4.0) * 6.328125))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(N[Power[N[(a * N[(c * -1.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(3.375 * N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[Power[c, 4.0], $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip3-- 30.7%

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

   2. div-inv 30.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({\left(b \cdot b\right)}^{3} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}}{3 \cdot a} \]

   3. pow2 30.8%

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

   4. pow-pow 30.9%

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

   5. metadata-eval 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   6. associate-*l* 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   7. unpow-prod-down 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{{3}^{3} \cdot {\left(a \cdot c\right)}^{3}}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   8. metadata-eval 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   9. pow2 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   10. pow2 30.9%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   11. pow-prod-up 30.8%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   12. metadata-eval 30.8%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{\color{blue}{4}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   13. distribute-rgt-out 30.8%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \color{blue}{\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right)}}}}{3 \cdot a} \]

4. Applied egg-rr 30.8%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]

5. Taylor expanded in b around inf 94.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(1.5 \cdot \left(a \cdot \left(c \cdot \left(-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

7. Simplified 94.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + {\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} \cdot -0.5\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right), 0\right)}{a \cdot {b}^{7}}\right)\right)} \]

8. Taylor expanded in a around 0 94.0%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \color{blue}{\frac{{a}^{3} \cdot \left(1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right) \]
9. Step-by-step derivation

   1. associate-/l* 94.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}}}}\right)\right) \]

   2. distribute-rgt-out 94.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{{a}^{3}}{\frac{{b}^{7}}{\color{blue}{{c}^{4} \cdot \left(1.265625 + 5.0625\right)}}}\right)\right) \]

   3. metadata-eval 94.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot \color{blue}{6.328125}}}\right)\right) \]

10. Simplified 94.0%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}}}\right)\right) \]

11. Taylor expanded in c around 0 94.0%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\color{blue}{3.375 \cdot \left({a}^{3} \cdot {c}^{3}\right)}}{a \cdot {b}^{5}}\right) + \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}}\right)\right) \]
12. Step-by-step derivation

    1. cube-prod 94.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{3.375 \cdot \color{blue}{{\left(a \cdot c\right)}^{3}}}{a \cdot {b}^{5}}\right) + \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}}\right)\right) \]

13. Simplified 94.0%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\color{blue}{3.375 \cdot {\left(a \cdot c\right)}^{3}}}{a \cdot {b}^{5}}\right) + \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}}\right)\right) \]

14. Final simplification 94.0%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2}}{a \cdot {b}^{3}} + \frac{3.375 \cdot {\left(c \cdot a\right)}^{3}}{a \cdot {b}^{5}}\right) + \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 6.328125}}\right)\right) \]
15. Add Preprocessing

Alternative 2: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666}{a} \cdot \frac{6.328125 \cdot {\left(c \cdot a\right)}^{4}}{{b}^{7}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+
   (* -0.5 (/ c b))
   (+
    (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
    (*
     (/ -0.16666666666666666 a)
     (/ (* 6.328125 (pow (* c a) 4.0)) (pow b 7.0)))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + ((-0.16666666666666666 / a) * ((6.328125 * pow((c * a), 4.0)) / pow(b, 7.0)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + (((-0.16666666666666666d0) / a) * ((6.328125d0 * ((c * a) ** 4.0d0)) / (b ** 7.0d0)))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + ((-0.16666666666666666 / a) * ((6.328125 * Math.pow((c * a), 4.0)) / Math.pow(b, 7.0)))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + ((-0.16666666666666666 / a) * ((6.328125 * math.pow((c * a), 4.0)) / math.pow(b, 7.0)))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(Float64(-0.16666666666666666 / a) * Float64(Float64(6.328125 * (Float64(c * a) ^ 4.0)) / (b ^ 7.0))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + ((-0.16666666666666666 / a) * ((6.328125 * ((c * a) ^ 4.0)) / (b ^ 7.0)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 / a), $MachinePrecision] * N[(N[(6.328125 * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf 94.0%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

4. Taylor expanded in c around 0 94.0%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
5. Step-by-step derivation

   1. associate-*r/ 94.0%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left({c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)\right)}{a \cdot {b}^{7}}}\right)\right) \]

   2. distribute-rgt-in 94.0%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666 \cdot \color{blue}{\left(\left(1.265625 \cdot {a}^{4}\right) \cdot {c}^{4} + \left(5.0625 \cdot {a}^{4}\right) \cdot {c}^{4}\right)}}{a \cdot {b}^{7}}\right)\right) \]

   3. associate-*r* 94.0%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666 \cdot \left(\color{blue}{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right)} + \left(5.0625 \cdot {a}^{4}\right) \cdot {c}^{4}\right)}{a \cdot {b}^{7}}\right)\right) \]

   4. associate-*r* 94.0%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666 \cdot \left(1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}\right)}{a \cdot {b}^{7}}\right)\right) \]

   5. times-frac 94.0%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666}{a} \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right) \]

6. Simplified 94.0%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666}{a} \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7}}}\right)\right) \]

7. Final simplification 94.0%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666}{a} \cdot \frac{6.328125 \cdot {\left(c \cdot a\right)}^{4}}{{b}^{7}}\right)\right) \]
8. Add Preprocessing

Alternative 3: 94.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (+
   (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
   (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip3-- 30.7%

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

   2. div-inv 30.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({\left(b \cdot b\right)}^{3} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}}{3 \cdot a} \]

   3. pow2 30.8%

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

   4. pow-pow 30.9%

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

   5. metadata-eval 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   6. associate-*l* 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   7. unpow-prod-down 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{{3}^{3} \cdot {\left(a \cdot c\right)}^{3}}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   8. metadata-eval 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   9. pow2 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   10. pow2 30.9%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   11. pow-prod-up 30.8%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   12. metadata-eval 30.8%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{\color{blue}{4}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   13. distribute-rgt-out 30.8%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \color{blue}{\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right)}}}}{3 \cdot a} \]

4. Applied egg-rr 30.8%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]

5. Taylor expanded in b around inf 94.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(1.5 \cdot \left(a \cdot \left(c \cdot \left(-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

7. Simplified 94.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + {\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} \cdot -0.5\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot \left(c \cdot -1.5\right)\right)}^{2} + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right), 0\right)}{a \cdot {b}^{7}}\right)\right)} \]

8. Taylor expanded in a around 0 92.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}}\right) \]

9. Final simplification 92.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]
10. Add Preprocessing

Alternative 4: 94.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf 92.1%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

4. Final simplification 92.1%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]
5. Add Preprocessing

Alternative 5: 83.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4e-11)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4e-11) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4e-11)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4e-11], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.99999999999999976e-11

   1. Initial program 66.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. +-commutative 66.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}}{3 \cdot a} \]

      2. sqr-neg 66.2%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}{3 \cdot a} \]

      3. unsub-neg 66.2%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a} \]

      4. div-sub 65.2%

         \[\leadsto \color{blue}{\frac{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} - \frac{b}{3 \cdot a}} \]

      5. --rgt-identity 65.2%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

      6. div-sub 66.2%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0\right) - b}{3 \cdot a}} \]

   3. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}} \]
   4. Add Preprocessing

   if -3.99999999999999976e-11 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 8.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 97.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative 97.2%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

      2. associate-*l/ 97.2%

         \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]

   5. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -4e-11) t_0 (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -4e-11) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-4d-11)) then
        tmp = t_0
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -4e-11) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -4e-11:
		tmp = t_0
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -4e-11)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -4e-11)
		tmp = t_0;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-11], t$95$0, N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.99999999999999976e-11

   1. Initial program 66.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   if -3.99999999999999976e-11 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 8.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 97.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative 97.2%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

      2. associate-*l/ 97.2%

         \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]

   5. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf 89.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

4. Final simplification 89.2%

   \[\leadsto -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]
5. Add Preprocessing

Alternative 8: 81.7% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (* -0.5 c) b))

C

double code(double a, double b, double c) {
	return (-0.5 * c) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * c) / b
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * c) / b;
}

Python

def code(a, b, c):
	return (-0.5 * c) / b

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * c) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 30.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf 81.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. *-commutative 81.4%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   2. associate-*l/ 81.4%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]

5. Simplified 81.4%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]

6. Final simplification 81.4%

   \[\leadsto \frac{-0.5 \cdot c}{b} \]
7. Add Preprocessing

Alternative 9: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 30.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip3-- 30.7%

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

   2. div-inv 30.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({\left(b \cdot b\right)}^{3} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}}{3 \cdot a} \]

   3. pow2 30.8%

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

   4. pow-pow 30.9%

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

   5. metadata-eval 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   6. associate-*l* 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   7. unpow-prod-down 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{{3}^{3} \cdot {\left(a \cdot c\right)}^{3}}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   8. metadata-eval 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   9. pow2 30.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   10. pow2 30.9%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   11. pow-prod-up 30.8%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   12. metadata-eval 30.8%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{\color{blue}{4}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   13. distribute-rgt-out 30.8%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \color{blue}{\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right)}}}}{3 \cdot a} \]

4. Applied egg-rr 30.8%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. div-inv 30.8%

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{\left({b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}\right) \cdot \frac{1}{3 \cdot a}} \]

6. Applied egg-rr 30.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}}{\mathsf{fma}\left(\left(a \cdot 3\right) \cdot c, \mathsf{fma}\left(b, b, \left(a \cdot 3\right) \cdot c\right), {b}^{4}\right)}}\right) \cdot \frac{1}{a \cdot 3}} \]

7. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(b + -1 \cdot b\right)}{a}} \]

   2. distribute-rgt1-in 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

9. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

10. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024036 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))