Cubic critical, medium range

Percentage Accurate: 31.9% → 95.3%

Time: 14.5s

Alternatives: 7

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: 31.9% 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: 95.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(c \cdot a\right)}^{4}\\ t_1 := {\left(c \cdot a\right)}^{2} \cdot 2.25\\ \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{t_1}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(a \cdot t_1\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{1.265625 \cdot t_0 + \mathsf{fma}\left(1.5, t_0 \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* c a) 4.0)) (t_1 (* (pow (* c a) 2.0) 2.25)))
   (fma
    -0.5
    (/ c b)
    (*
     -0.16666666666666666
     (+
      (+
       (/ t_1 (* a (pow b 3.0)))
       (/ (fma 1.5 (* c (* a t_1)) (* a (* c 0.0))) (* a (pow b 5.0))))
      (/
       (+ (* 1.265625 t_0) (fma 1.5 (* t_0 3.375) 0.0))
       (* a (pow b 7.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((c * a), 4.0);
	double t_1 = pow((c * a), 2.0) * 2.25;
	return fma(-0.5, (c / b), (-0.16666666666666666 * (((t_1 / (a * pow(b, 3.0))) + (fma(1.5, (c * (a * t_1)), (a * (c * 0.0))) / (a * pow(b, 5.0)))) + (((1.265625 * t_0) + fma(1.5, (t_0 * 3.375), 0.0)) / (a * pow(b, 7.0))))));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * a) ^ 4.0
	t_1 = Float64((Float64(c * a) ^ 2.0) * 2.25)
	return fma(-0.5, Float64(c / b), Float64(-0.16666666666666666 * Float64(Float64(Float64(t_1 / Float64(a * (b ^ 3.0))) + Float64(fma(1.5, Float64(c * Float64(a * t_1)), Float64(a * Float64(c * 0.0))) / Float64(a * (b ^ 5.0)))) + Float64(Float64(Float64(1.265625 * t_0) + fma(1.5, Float64(t_0 * 3.375), 0.0)) / Float64(a * (b ^ 7.0))))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision] * 2.25), $MachinePrecision]}, N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(t$95$1 / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5 * N[(c * N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(a * N[(c * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.265625 * t$95$0), $MachinePrecision] + N[(1.5 * N[(t$95$0 * 3.375), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 33.3%

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

3. Simplified 33.4%

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

5. Applied egg-rr 33.2%

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

6. Taylor expanded in b around inf 96.0%

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

8. Simplified 96.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right), 0\right)}{a \cdot {b}^{7}}\right)\right)} \]

9. Taylor expanded in a around 0 96.0%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \color{blue}{3.375 \cdot \left({a}^{4} \cdot {c}^{4}\right)}, 0\right)}{a \cdot {b}^{7}}\right)\right) \]
10. Step-by-step derivation

    1. *-commutative 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \color{blue}{\left({a}^{4} \cdot {c}^{4}\right) \cdot 3.375}, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    2. metadata-eval 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left({a}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {c}^{4}\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    3. pow-sqr 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left(\color{blue}{\left({a}^{2} \cdot {a}^{2}\right)} \cdot {c}^{4}\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    4. metadata-eval 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left(\left({a}^{2} \cdot {a}^{2}\right) \cdot {c}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    5. pow-sqr 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left(\left({a}^{2} \cdot {a}^{2}\right) \cdot \color{blue}{\left({c}^{2} \cdot {c}^{2}\right)}\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    6. swap-sqr 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    7. unpow2 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    8. unpow2 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left(\left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    9. swap-sqr 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left(\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    10. unpow2 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    11. unpow2 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left({\left(a \cdot c\right)}^{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right)\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    12. unpow2 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left({\left(a \cdot c\right)}^{2} \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    13. swap-sqr 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    14. unpow2 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{{\left(a \cdot c\right)}^{2}}\right) \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    15. pow-sqr 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    16. metadata-eval 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{\color{blue}{4}} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

11. Simplified 96.0%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5, \color{blue}{{\left(a \cdot c\right)}^{4} \cdot 3.375}, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

12. Taylor expanded in a around 0 96.0%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(\color{blue}{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right)} + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]
13. Step-by-step derivation

    1. metadata-eval 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left({a}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {c}^{4}\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    2. pow-sqr 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left(\color{blue}{\left({a}^{2} \cdot {a}^{2}\right)} \cdot {c}^{4}\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    3. metadata-eval 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left(\left({a}^{2} \cdot {a}^{2}\right) \cdot {c}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    4. pow-sqr 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left(\left({a}^{2} \cdot {a}^{2}\right) \cdot \color{blue}{\left({c}^{2} \cdot {c}^{2}\right)}\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    5. swap-sqr 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)} + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    6. unpow2 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    7. unpow2 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left(\left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    8. swap-sqr 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left(\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    9. unpow2 96.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    10. unpow2 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right)\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    11. unpow2 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    12. swap-sqr 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    13. unpow2 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{{\left(a \cdot c\right)}^{2}}\right) + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    14. pow-sqr 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}} + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

    15. metadata-eval 96.0%

        \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(1.265625 \cdot {\left(a \cdot c\right)}^{\color{blue}{4}} + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

14. Simplified 96.0%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(\left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right) \cdot a\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(\color{blue}{1.265625 \cdot {\left(a \cdot c\right)}^{4}} + 0\right) + \mathsf{fma}\left(1.5, {\left(a \cdot c\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

15. Final simplification 96.0%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(c \cdot a\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, c \cdot \left(a \cdot \left({\left(c \cdot a\right)}^{2} \cdot 2.25\right)\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{1.265625 \cdot {\left(c \cdot a\right)}^{4} + \mathsf{fma}\left(1.5, {\left(c \cdot a\right)}^{4} \cdot 3.375, 0\right)}{a \cdot {b}^{7}}\right)\right) \]

Alternative 2: 95.3% accurate, 0.2× speedup

Math

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

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
     (* (/ (pow (* c a) 4.0) a) (/ 6.328125 (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 * ((pow((c * a), 4.0) / a) * (6.328125 / 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) * ((((c * a) ** 4.0d0) / a) * (6.328125d0 / (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 * ((Math.pow((c * a), 4.0) / a) * (6.328125 / 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 * ((math.pow((c * a), 4.0) / a) * (6.328125 / 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(-0.16666666666666666 * Float64(Float64((Float64(c * a) ^ 4.0) / a) * Float64(6.328125 / (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 * ((((c * a) ^ 4.0) / a) * (6.328125 / (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[(-0.16666666666666666 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.3%

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

2. Taylor expanded in b around inf 96.0%

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

3. Taylor expanded in c around 0 96.0%

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

   1. distribute-rgt-out 96.0%

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

   2. associate-*l* 96.0%

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

   3. *-commutative 96.0%

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

   4. times-frac 96.0%

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

5. Simplified 96.0%

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

6. Final simplification 96.0%

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

Alternative 3: 93.7% 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 33.3%

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

2. Taylor expanded in b around inf 94.1%

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

3. Final simplification 94.1%

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

Alternative 4: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -2e-5)
   (/ (- (sqrt (fma b b (* c (* a -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)) <= -2e-5) {
		tmp = (sqrt(fma(b, b, (c * (a * -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)) <= -2e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -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], -2e-5], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -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)) < -2.00000000000000016e-5

   1. Initial program 70.2%

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

   3. Simplified 70.4%

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

   if -2.00000000000000016e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 18.9%

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

   2. Taylor expanded in b around inf 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*l/ 90.6%

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

   4. Simplified 90.6%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 5: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -2e-5) 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 <= -2e-5) {
		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 <= (-2d-5)) 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 <= -2e-5) {
		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 <= -2e-5:
		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 <= -2e-5)
		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 <= -2e-5)
		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, -2e-5], 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)) < -2.00000000000000016e-5

   1. Initial program 70.2%

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

   if -2.00000000000000016e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 18.9%

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

   2. Taylor expanded in b around inf 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*l/ 90.6%

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

   4. Simplified 90.6%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 6: 90.4% 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 33.3%

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

2. Taylor expanded in b around inf 90.4%

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

3. Final simplification 90.4%

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

Alternative 7: 80.8% 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 33.3%

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

2. Taylor expanded in b around inf 80.0%

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

   1. *-commutative 80.0%

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

   2. associate-*l/ 80.0%

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

4. Simplified 80.0%

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

5. Final simplification 80.0%

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

Reproduce

herbie shell --seed 2023310 
(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)))