Cubic critical, narrow range

Percentage Accurate: 54.9% → 90.6%

Time: 14.8s

Alternatives: 14

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 54.9% 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: 90.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(c \cdot a\right)}^{4}\\ \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{t_0 \cdot 1.265625 + t_0 \cdot 5.0625}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* c a) 4.0)))
   (fma
    -0.5625
    (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
    (fma
     -0.16666666666666666
     (/ (+ (* t_0 1.265625) (* t_0 5.0625)) (* a (pow b 7.0)))
     (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((c * a), 4.0);
	return fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.16666666666666666, (((t_0 * 1.265625) + (t_0 * 5.0625)) / (a * pow(b, 7.0))), fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0)))));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * a) ^ 4.0
	return fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.16666666666666666, Float64(Float64(Float64(t_0 * 1.265625) + Float64(t_0 * 5.0625)) / Float64(a * (b ^ 7.0))), fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]}, N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(t$95$0 * 1.265625), $MachinePrecision] + N[(t$95$0 * 5.0625), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.3%

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

   1. /-rgt-identity 54.3%

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

   2. metadata-eval 54.3%

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

   3. associate-/r/ 54.3%

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

   4. metadata-eval 54.3%

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

   5. metadata-eval 54.3%

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

   6. times-frac 54.3%

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

   7. *-commutative 54.3%

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

   8. times-frac 54.3%

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

   9. *-commutative 54.3%

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

   10. associate-/r* 54.3%

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

   11. associate-*l/ 54.3%

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

3. Simplified 54.4%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

4. Taylor expanded in b around inf 92.2%

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

   1. fma-def 92.2%

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

   2. associate-/l* 92.2%

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

   3. unpow2 92.2%

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

   4. fma-def 92.2%

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

6. Simplified 92.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 92.2%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({c}^{4} \cdot {a}^{4}\right)\right)}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]

   2. expm1-udef 91.6%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({c}^{4} \cdot {a}^{4}\right)} - 1\right)}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]

   3. *-commutative 91.6%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{a}^{4} \cdot {c}^{4}}\right)} - 1\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]

   4. pow-prod-down 91.6%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(a \cdot c\right)}^{4}}\right)} - 1\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]

8. Applied egg-rr 91.6%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{4}\right)} - 1\right)}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]
9. Step-by-step derivation

   1. expm1-def 92.2%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot c\right)}^{4}\right)\right)}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]

   2. expm1-log1p 92.2%

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

   3. *-commutative 92.2%

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

10. Simplified 92.2%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \color{blue}{{\left(c \cdot a\right)}^{4}}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right) \]
11. Step-by-step derivation

    1. unpow-prod-down 92.2%

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

    2. unpow-prod-down 92.2%

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

    3. pow-prod-down 92.2%

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

    4. pow-prod-up 92.2%

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

    5. metadata-eval 92.2%

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

    6. pow-prod-down 92.2%

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

    7. pow-prod-up 92.2%

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

    8. metadata-eval 92.2%

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

    9. *-commutative 92.2%

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

    10. pow-prod-down 92.2%

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

    11. metadata-eval 92.2%

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

12. Applied egg-rr 92.2%

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

13. Final simplification 92.2%

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

Alternative 2: 89.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{0.3333333333333333}{a}}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.55:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(t_0 \cdot {t_0}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (cbrt (/ 0.3333333333333333 a))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.55)
     (*
      (- (sqrt (fma b b (* a (* c -3.0)))) b)
      (exp (log (* t_0 (pow t_0 2.0)))))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = cbrt((0.3333333333333333 / a));
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.55) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * exp(log((t_0 * pow(t_0, 2.0))));
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = cbrt(Float64(0.3333333333333333 / a))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.55)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * exp(log(Float64(t_0 * (t_0 ^ 2.0)))));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(0.3333333333333333 / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.55], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[Exp[N[Log[N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.55000000000000004

   1. Initial program 82.4%

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

      1. neg-sub0 82.4%

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

      2. associate-+l- 82.4%

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

      3. sub0-neg 82.4%

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

      4. neg-mul-1 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. metadata-eval 82.4%

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

      8. metadata-eval 82.4%

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

      9. times-frac 82.4%

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

      10. *-commutative 82.4%

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

      11. times-frac 82.4%

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

   3. Simplified 82.5%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. add-exp-log 82.5%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{e^{\log \left(\frac{0.3333333333333333}{a}\right)}} \]

   5. Applied egg-rr 82.5%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{e^{\log \left(\frac{0.3333333333333333}{a}\right)}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 82.6%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \color{blue}{\left(\left(\sqrt[3]{\frac{0.3333333333333333}{a}} \cdot \sqrt[3]{\frac{0.3333333333333333}{a}}\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a}}\right)}} \]

      2. pow2 82.6%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\color{blue}{{\left(\sqrt[3]{\frac{0.3333333333333333}{a}}\right)}^{2}} \cdot \sqrt[3]{\frac{0.3333333333333333}{a}}\right)} \]

   7. Applied egg-rr 82.6%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \color{blue}{\left({\left(\sqrt[3]{\frac{0.3333333333333333}{a}}\right)}^{2} \cdot \sqrt[3]{\frac{0.3333333333333333}{a}}\right)}} \]

   if -0.55000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 48.6%

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

      1. /-rgt-identity 48.6%

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

      2. metadata-eval 48.6%

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

      3. associate-/r/ 48.6%

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

      4. metadata-eval 48.6%

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

      5. metadata-eval 48.6%

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

      6. times-frac 48.6%

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

      7. *-commutative 48.6%

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

      8. times-frac 48.6%

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

      9. *-commutative 48.6%

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

      10. associate-/r* 48.6%

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

      11. associate-*l/ 48.6%

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

   3. Simplified 48.8%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

   4. Taylor expanded in b around inf 93.0%

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

      1. fma-def 93.0%

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

      2. associate-/l* 93.0%

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

      3. unpow2 93.0%

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

      4. fma-def 93.0%

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

      5. associate-*r/ 93.0%

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

      6. *-commutative 93.0%

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

      7. unpow2 93.0%

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

   6. Simplified 93.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.55:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\sqrt[3]{\frac{0.3333333333333333}{a}} \cdot {\left(\sqrt[3]{\frac{0.3333333333333333}{a}}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 3: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.55:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left({\left(\sqrt[3]{\frac{0.3333333333333333}{a}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.55)
   (*
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (exp (log (pow (cbrt (/ 0.3333333333333333 a)) 3.0))))
   (fma
    -0.5625
    (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
    (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.55) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * exp(log(pow(cbrt((0.3333333333333333 / a)), 3.0)));
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.55)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * exp(log((cbrt(Float64(0.3333333333333333 / a)) ^ 3.0))));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.55], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[Exp[N[Log[N[Power[N[Power[N[(0.3333333333333333 / a), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.55000000000000004

   1. Initial program 82.4%

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

      1. neg-sub0 82.4%

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

      2. associate-+l- 82.4%

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

      3. sub0-neg 82.4%

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

      4. neg-mul-1 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. metadata-eval 82.4%

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

      8. metadata-eval 82.4%

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

      9. times-frac 82.4%

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

      10. *-commutative 82.4%

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

      11. times-frac 82.4%

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

   3. Simplified 82.5%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. add-exp-log 82.5%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{e^{\log \left(\frac{0.3333333333333333}{a}\right)}} \]

   5. Applied egg-rr 82.5%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{e^{\log \left(\frac{0.3333333333333333}{a}\right)}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 82.6%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \color{blue}{\left(\left(\sqrt[3]{\frac{0.3333333333333333}{a}} \cdot \sqrt[3]{\frac{0.3333333333333333}{a}}\right) \cdot \sqrt[3]{\frac{0.3333333333333333}{a}}\right)}} \]

      2. pow3 82.6%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \color{blue}{\left({\left(\sqrt[3]{\frac{0.3333333333333333}{a}}\right)}^{3}\right)}} \]

   7. Applied egg-rr 82.6%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \color{blue}{\left({\left(\sqrt[3]{\frac{0.3333333333333333}{a}}\right)}^{3}\right)}} \]

   if -0.55000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 48.6%

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

      1. /-rgt-identity 48.6%

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

      2. metadata-eval 48.6%

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

      3. associate-/r/ 48.6%

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

      4. metadata-eval 48.6%

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

      5. metadata-eval 48.6%

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

      6. times-frac 48.6%

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

      7. *-commutative 48.6%

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

      8. times-frac 48.6%

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

      9. *-commutative 48.6%

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

      10. associate-/r* 48.6%

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

      11. associate-*l/ 48.6%

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

   3. Simplified 48.8%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

   4. Taylor expanded in b around inf 93.0%

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

      1. fma-def 93.0%

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

      2. associate-/l* 93.0%

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

      3. unpow2 93.0%

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

      4. fma-def 93.0%

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

      5. associate-*r/ 93.0%

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

      6. *-commutative 93.0%

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

      7. unpow2 93.0%

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

   6. Simplified 93.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.55:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left({\left(\sqrt[3]{\frac{0.3333333333333333}{a}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 4: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.55:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\sqrt{\frac{0.1111111111111111}{a \cdot a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.55)
   (*
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (exp (log (sqrt (/ 0.1111111111111111 (* a a))))))
   (fma
    -0.5625
    (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
    (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.55) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * exp(log(sqrt((0.1111111111111111 / (a * a)))));
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.55)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * exp(log(sqrt(Float64(0.1111111111111111 / Float64(a * a))))));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.55], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[Exp[N[Log[N[Sqrt[N[(0.1111111111111111 / N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.55000000000000004

   1. Initial program 82.4%

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

      1. neg-sub0 82.4%

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

      2. associate-+l- 82.4%

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

      3. sub0-neg 82.4%

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

      4. neg-mul-1 82.4%

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

      5. associate-*r/ 82.4%

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

      6. *-commutative 82.4%

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

      7. metadata-eval 82.4%

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

      8. metadata-eval 82.4%

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

      9. times-frac 82.4%

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

      10. *-commutative 82.4%

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

      11. times-frac 82.4%

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

   3. Simplified 82.5%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. add-exp-log 82.5%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{e^{\log \left(\frac{0.3333333333333333}{a}\right)}} \]

   5. Applied egg-rr 82.5%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{e^{\log \left(\frac{0.3333333333333333}{a}\right)}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 82.6%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \color{blue}{\left(\sqrt{\frac{0.3333333333333333}{a}} \cdot \sqrt{\frac{0.3333333333333333}{a}}\right)}} \]

      2. sqrt-unprod 82.5%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \color{blue}{\left(\sqrt{\frac{0.3333333333333333}{a} \cdot \frac{0.3333333333333333}{a}}\right)}} \]

      3. frac-times 82.6%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\sqrt{\color{blue}{\frac{0.3333333333333333 \cdot 0.3333333333333333}{a \cdot a}}}\right)} \]

      4. metadata-eval 82.6%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\sqrt{\frac{\color{blue}{0.1111111111111111}}{a \cdot a}}\right)} \]

   7. Applied egg-rr 82.6%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \color{blue}{\left(\sqrt{\frac{0.1111111111111111}{a \cdot a}}\right)}} \]

   if -0.55000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 48.6%

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

      1. /-rgt-identity 48.6%

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

      2. metadata-eval 48.6%

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

      3. associate-/r/ 48.6%

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

      4. metadata-eval 48.6%

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

      5. metadata-eval 48.6%

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

      6. times-frac 48.6%

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

      7. *-commutative 48.6%

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

      8. times-frac 48.6%

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

      9. *-commutative 48.6%

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

      10. associate-/r* 48.6%

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

      11. associate-*l/ 48.6%

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

   3. Simplified 48.8%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

   4. Taylor expanded in b around inf 93.0%

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

      1. fma-def 93.0%

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

      2. associate-/l* 93.0%

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

      3. unpow2 93.0%

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

      4. fma-def 93.0%

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

      5. associate-*r/ 93.0%

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

      6. *-commutative 93.0%

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

      7. unpow2 93.0%

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

   6. Simplified 93.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.55:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\sqrt{\frac{0.1111111111111111}{a \cdot a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\\ \end{array} \]

Alternative 5: 85.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\sqrt{\frac{0.1111111111111111}{a \cdot a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(\left(c \cdot c\right) \cdot -0.3333333333333333\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{b \cdot b}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.004)
   (*
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (exp (log (sqrt (/ 0.1111111111111111 (* a a))))))
   (fma
    -0.5
    (/ c b)
    (*
     (* (* c c) -0.3333333333333333)
     (fma
      0.5
      (- (* 2.25 (/ a (pow b 3.0))) (/ 2.25 (* b b)))
      (/ 1.125 (* b b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * exp(log(sqrt((0.1111111111111111 / (a * a)))));
	} else {
		tmp = fma(-0.5, (c / b), (((c * c) * -0.3333333333333333) * fma(0.5, ((2.25 * (a / pow(b, 3.0))) - (2.25 / (b * b))), (1.125 / (b * b)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * exp(log(sqrt(Float64(0.1111111111111111 / Float64(a * a))))));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(Float64(c * c) * -0.3333333333333333) * fma(0.5, Float64(Float64(2.25 * Float64(a / (b ^ 3.0))) - Float64(2.25 / Float64(b * b))), Float64(1.125 / Float64(b * b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[Exp[N[Log[N[Sqrt[N[(0.1111111111111111 / N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(0.5 * N[(N[(2.25 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.25 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.125 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0040000000000000001

   1. Initial program 78.4%

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

      1. neg-sub0 78.4%

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

      2. associate-+l- 78.4%

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

      3. sub0-neg 78.4%

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

      4. neg-mul-1 78.4%

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

      5. associate-*r/ 78.4%

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

      6. *-commutative 78.4%

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

      7. metadata-eval 78.4%

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

      8. metadata-eval 78.4%

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

      9. times-frac 78.4%

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

      10. *-commutative 78.4%

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

      11. times-frac 78.4%

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

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. add-exp-log 78.5%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{e^{\log \left(\frac{0.3333333333333333}{a}\right)}} \]

   5. Applied egg-rr 78.5%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{e^{\log \left(\frac{0.3333333333333333}{a}\right)}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 78.5%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \color{blue}{\left(\sqrt{\frac{0.3333333333333333}{a}} \cdot \sqrt{\frac{0.3333333333333333}{a}}\right)}} \]

      2. sqrt-unprod 78.5%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \color{blue}{\left(\sqrt{\frac{0.3333333333333333}{a} \cdot \frac{0.3333333333333333}{a}}\right)}} \]

      3. frac-times 78.5%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\sqrt{\color{blue}{\frac{0.3333333333333333 \cdot 0.3333333333333333}{a \cdot a}}}\right)} \]

      4. metadata-eval 78.5%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\sqrt{\frac{\color{blue}{0.1111111111111111}}{a \cdot a}}\right)} \]

   7. Applied egg-rr 78.5%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \color{blue}{\left(\sqrt{\frac{0.1111111111111111}{a \cdot a}}\right)}} \]

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. metadata-eval 42.3%

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

      3. associate-/l* 42.3%

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

      4. associate-*r/ 42.3%

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

      5. *-commutative 42.3%

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

      6. associate-*l/ 42.3%

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

      7. associate-*r/ 42.3%

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

      8. metadata-eval 42.3%

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

      9. metadata-eval 42.3%

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

      10. times-frac 42.3%

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

      11. neg-mul-1 42.3%

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

      12. distribute-rgt-neg-in 42.3%

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

      13. times-frac 42.3%

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

      14. metadata-eval 42.3%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 42.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 42.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around inf 91.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 91.9%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      2. fma-def 92.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      3. associate-*r/ 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\frac{1.125 \cdot \left({c}^{2} \cdot a\right)}{{b}^{3}}}\right) \]

      4. *-commutative 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \color{blue}{\left(a \cdot {c}^{2}\right)}}{{b}^{3}}\right) \]

      5. unpow2 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{3}}\right) \]

   6. Simplified 92.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 92.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right)} \]

      2. expm1-udef 63.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)} - 1\right)} \]

      3. div-inv 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\left(1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \frac{1}{{b}^{3}}}\right)\right)} - 1\right) \]

      4. associate-*r* 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right)} \cdot \frac{1}{{b}^{3}}\right)\right)} - 1\right) \]

      5. pow-flip 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{{b}^{\left(-3\right)}}\right)\right)} - 1\right) \]

      6. metadata-eval 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{\color{blue}{-3}}\right)\right)} - 1\right) \]

   8. Applied egg-rr 63.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{-3}\right)\right)} - 1\right)} \]

   9. Taylor expanded in c around 0 92.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.3333333333333333 \cdot \left({c}^{2} \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. fma-def 92.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.3333333333333333 \cdot \left({c}^{2} \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)\right)\right)} \]

       2. associate-*r* 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{\left(-0.3333333333333333 \cdot {c}^{2}\right) \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)}\right) \]

       3. unpow2 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       4. fma-def 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{\mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)}\right) \]

       5. associate-*r/ 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \color{blue}{\frac{2.25 \cdot 1}{{b}^{2}}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       6. metadata-eval 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{\color{blue}{2.25}}{{b}^{2}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       7. unpow2 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{\color{blue}{b \cdot b}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       8. associate-*r/ 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \color{blue}{\frac{1.125 \cdot 1}{{b}^{2}}}\right)\right) \]

       9. metadata-eval 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{\color{blue}{1.125}}{{b}^{2}}\right)\right) \]

       10. unpow2 92.3%

           \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{\color{blue}{b \cdot b}}\right)\right) \]

   11. Simplified 92.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{b \cdot b}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\sqrt{\frac{0.1111111111111111}{a \cdot a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(\left(c \cdot c\right) \cdot -0.3333333333333333\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{b \cdot b}\right)\right)\\ \end{array} \]

Alternative 6: 85.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\frac{0.3333333333333333}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(\left(c \cdot c\right) \cdot -0.3333333333333333\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{b \cdot b}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.004)
   (*
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (exp (log (/ 0.3333333333333333 a))))
   (fma
    -0.5
    (/ c b)
    (*
     (* (* c c) -0.3333333333333333)
     (fma
      0.5
      (- (* 2.25 (/ a (pow b 3.0))) (/ 2.25 (* b b)))
      (/ 1.125 (* b b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * exp(log((0.3333333333333333 / a)));
	} else {
		tmp = fma(-0.5, (c / b), (((c * c) * -0.3333333333333333) * fma(0.5, ((2.25 * (a / pow(b, 3.0))) - (2.25 / (b * b))), (1.125 / (b * b)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * exp(log(Float64(0.3333333333333333 / a))));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(Float64(c * c) * -0.3333333333333333) * fma(0.5, Float64(Float64(2.25 * Float64(a / (b ^ 3.0))) - Float64(2.25 / Float64(b * b))), Float64(1.125 / Float64(b * b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[Exp[N[Log[N[(0.3333333333333333 / a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(0.5 * N[(N[(2.25 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.25 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.125 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0040000000000000001

   1. Initial program 78.4%

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

      1. neg-sub0 78.4%

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

      2. associate-+l- 78.4%

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

      3. sub0-neg 78.4%

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

      4. neg-mul-1 78.4%

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

      5. associate-*r/ 78.4%

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

      6. *-commutative 78.4%

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

      7. metadata-eval 78.4%

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

      8. metadata-eval 78.4%

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

      9. times-frac 78.4%

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

      10. *-commutative 78.4%

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

      11. times-frac 78.4%

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

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. add-exp-log 78.5%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{e^{\log \left(\frac{0.3333333333333333}{a}\right)}} \]

   5. Applied egg-rr 78.5%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{e^{\log \left(\frac{0.3333333333333333}{a}\right)}} \]

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. metadata-eval 42.3%

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

      3. associate-/l* 42.3%

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

      4. associate-*r/ 42.3%

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

      5. *-commutative 42.3%

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

      6. associate-*l/ 42.3%

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

      7. associate-*r/ 42.3%

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

      8. metadata-eval 42.3%

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

      9. metadata-eval 42.3%

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

      10. times-frac 42.3%

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

      11. neg-mul-1 42.3%

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

      12. distribute-rgt-neg-in 42.3%

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

      13. times-frac 42.3%

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

      14. metadata-eval 42.3%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 42.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 42.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around inf 91.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 91.9%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      2. fma-def 92.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      3. associate-*r/ 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\frac{1.125 \cdot \left({c}^{2} \cdot a\right)}{{b}^{3}}}\right) \]

      4. *-commutative 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \color{blue}{\left(a \cdot {c}^{2}\right)}}{{b}^{3}}\right) \]

      5. unpow2 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{3}}\right) \]

   6. Simplified 92.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 92.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right)} \]

      2. expm1-udef 63.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)} - 1\right)} \]

      3. div-inv 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\left(1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \frac{1}{{b}^{3}}}\right)\right)} - 1\right) \]

      4. associate-*r* 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right)} \cdot \frac{1}{{b}^{3}}\right)\right)} - 1\right) \]

      5. pow-flip 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{{b}^{\left(-3\right)}}\right)\right)} - 1\right) \]

      6. metadata-eval 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{\color{blue}{-3}}\right)\right)} - 1\right) \]

   8. Applied egg-rr 63.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{-3}\right)\right)} - 1\right)} \]

   9. Taylor expanded in c around 0 92.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.3333333333333333 \cdot \left({c}^{2} \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. fma-def 92.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.3333333333333333 \cdot \left({c}^{2} \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)\right)\right)} \]

       2. associate-*r* 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{\left(-0.3333333333333333 \cdot {c}^{2}\right) \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)}\right) \]

       3. unpow2 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       4. fma-def 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{\mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)}\right) \]

       5. associate-*r/ 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \color{blue}{\frac{2.25 \cdot 1}{{b}^{2}}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       6. metadata-eval 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{\color{blue}{2.25}}{{b}^{2}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       7. unpow2 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{\color{blue}{b \cdot b}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       8. associate-*r/ 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \color{blue}{\frac{1.125 \cdot 1}{{b}^{2}}}\right)\right) \]

       9. metadata-eval 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{\color{blue}{1.125}}{{b}^{2}}\right)\right) \]

       10. unpow2 92.3%

           \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{\color{blue}{b \cdot b}}\right)\right) \]

   11. Simplified 92.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{b \cdot b}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot e^{\log \left(\frac{0.3333333333333333}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(\left(c \cdot c\right) \cdot -0.3333333333333333\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{b \cdot b}\right)\right)\\ \end{array} \]

Alternative 7: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(\left(c \cdot c\right) \cdot -0.3333333333333333\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{b \cdot b}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.004)
   (/ (* -0.3333333333333333 (- b (sqrt (fma b b (* (* c a) -3.0))))) a)
   (fma
    -0.5
    (/ c b)
    (*
     (* (* c c) -0.3333333333333333)
     (fma
      0.5
      (- (* 2.25 (/ a (pow b 3.0))) (/ 2.25 (* b b)))
      (/ 1.125 (* b b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (-0.3333333333333333 * (b - sqrt(fma(b, b, ((c * a) * -3.0))))) / a;
	} else {
		tmp = fma(-0.5, (c / b), (((c * c) * -0.3333333333333333) * fma(0.5, ((2.25 * (a / pow(b, 3.0))) - (2.25 / (b * b))), (1.125 / (b * b)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(fma(b, b, Float64(Float64(c * a) * -3.0))))) / a);
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(Float64(c * c) * -0.3333333333333333) * fma(0.5, Float64(Float64(2.25 * Float64(a / (b ^ 3.0))) - Float64(2.25 / Float64(b * b))), Float64(1.125 / Float64(b * b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(0.5 * N[(N[(2.25 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.25 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.125 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0040000000000000001

   1. Initial program 78.4%

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

      1. /-rgt-identity 78.4%

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

      2. metadata-eval 78.4%

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

      3. associate-/r/ 78.4%

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

      4. metadata-eval 78.4%

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

      5. metadata-eval 78.4%

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

      6. times-frac 78.4%

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

      7. *-commutative 78.4%

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

      8. times-frac 78.4%

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

      9. *-commutative 78.4%

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

      10. associate-/r* 78.4%

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

      11. associate-*l/ 78.4%

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

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. metadata-eval 42.3%

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

      3. associate-/l* 42.3%

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

      4. associate-*r/ 42.3%

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

      5. *-commutative 42.3%

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

      6. associate-*l/ 42.3%

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

      7. associate-*r/ 42.3%

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

      8. metadata-eval 42.3%

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

      9. metadata-eval 42.3%

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

      10. times-frac 42.3%

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

      11. neg-mul-1 42.3%

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

      12. distribute-rgt-neg-in 42.3%

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

      13. times-frac 42.3%

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

      14. metadata-eval 42.3%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 42.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 42.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around inf 91.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 91.9%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      2. fma-def 92.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      3. associate-*r/ 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\frac{1.125 \cdot \left({c}^{2} \cdot a\right)}{{b}^{3}}}\right) \]

      4. *-commutative 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \color{blue}{\left(a \cdot {c}^{2}\right)}}{{b}^{3}}\right) \]

      5. unpow2 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{3}}\right) \]

   6. Simplified 92.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 92.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)\right)} \]

      2. expm1-udef 63.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\right)} - 1\right)} \]

      3. div-inv 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\left(1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \frac{1}{{b}^{3}}}\right)\right)} - 1\right) \]

      4. associate-*r* 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right)} \cdot \frac{1}{{b}^{3}}\right)\right)} - 1\right) \]

      5. pow-flip 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{{b}^{\left(-3\right)}}\right)\right)} - 1\right) \]

      6. metadata-eval 63.0%

         \[\leadsto -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{\color{blue}{-3}}\right)\right)} - 1\right) \]

   8. Applied egg-rr 63.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(1.5, \frac{c}{b}, \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{-3}\right)\right)} - 1\right)} \]

   9. Taylor expanded in c around 0 92.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.3333333333333333 \cdot \left({c}^{2} \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. fma-def 92.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.3333333333333333 \cdot \left({c}^{2} \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)\right)\right)} \]

       2. associate-*r* 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{\left(-0.3333333333333333 \cdot {c}^{2}\right) \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)}\right) \]

       3. unpow2 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \left(0.5 \cdot \left(2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}\right) + 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       4. fma-def 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{\mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - 2.25 \cdot \frac{1}{{b}^{2}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)}\right) \]

       5. associate-*r/ 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \color{blue}{\frac{2.25 \cdot 1}{{b}^{2}}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       6. metadata-eval 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{\color{blue}{2.25}}{{b}^{2}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       7. unpow2 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{\color{blue}{b \cdot b}}, 1.125 \cdot \frac{1}{{b}^{2}}\right)\right) \]

       8. associate-*r/ 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \color{blue}{\frac{1.125 \cdot 1}{{b}^{2}}}\right)\right) \]

       9. metadata-eval 92.3%

          \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{\color{blue}{1.125}}{{b}^{2}}\right)\right) \]

       10. unpow2 92.3%

           \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{\color{blue}{b \cdot b}}\right)\right) \]

   11. Simplified 92.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(-0.3333333333333333 \cdot \left(c \cdot c\right)\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{b \cdot b}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \left(\left(c \cdot c\right) \cdot -0.3333333333333333\right) \cdot \mathsf{fma}\left(0.5, 2.25 \cdot \frac{a}{{b}^{3}} - \frac{2.25}{b \cdot b}, \frac{1.125}{b \cdot b}\right)\right)\\ \end{array} \]

Alternative 8: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.004)
   (/ (* -0.3333333333333333 (- b (sqrt (fma b b (* (* c a) -3.0))))) a)
   (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (-0.3333333333333333 * (b - sqrt(fma(b, b, ((c * a) * -3.0))))) / a;
	} else {
		tmp = fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(fma(b, b, Float64(Float64(c * a) * -3.0))))) / a);
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $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)) < -0.0040000000000000001

   1. Initial program 78.4%

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

      1. /-rgt-identity 78.4%

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

      2. metadata-eval 78.4%

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

      3. associate-/r/ 78.4%

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

      4. metadata-eval 78.4%

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

      5. metadata-eval 78.4%

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

      6. times-frac 78.4%

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

      7. *-commutative 78.4%

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

      8. times-frac 78.4%

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

      9. *-commutative 78.4%

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

      10. associate-/r* 78.4%

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

      11. associate-*l/ 78.4%

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

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. metadata-eval 42.3%

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

      3. associate-/r/ 42.3%

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

      4. metadata-eval 42.3%

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

      5. metadata-eval 42.3%

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

      6. times-frac 42.3%

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

      7. *-commutative 42.3%

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

      8. times-frac 42.3%

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

      9. *-commutative 42.3%

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

      10. associate-/r* 42.3%

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

      11. associate-*l/ 42.3%

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

   3. Simplified 42.5%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

   4. Taylor expanded in b around inf 92.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   5. Step-by-step derivation

      1. fma-def 92.3%

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

      2. associate-*r/ 92.3%

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

      3. *-commutative 92.3%

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

      4. unpow2 92.3%

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

   6. Simplified 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 9: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot 1.5 + \left(\left(c \cdot c\right) \cdot \left(a \cdot 1.125\right)\right) \cdot {b}^{-3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.004)
   (* -0.3333333333333333 (/ (- b (sqrt (fma b b (* a (* c -3.0))))) a))
   (*
    -0.3333333333333333
    (+ (* (/ c b) 1.5) (* (* (* c c) (* a 1.125)) (pow b -3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.004) {
		tmp = -0.3333333333333333 * ((b - sqrt(fma(b, b, (a * (c * -3.0))))) / a);
	} else {
		tmp = -0.3333333333333333 * (((c / b) * 1.5) + (((c * c) * (a * 1.125)) * pow(b, -3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))) / a));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(c / b) * 1.5) + Float64(Float64(Float64(c * c) * Float64(a * 1.125)) * (b ^ -3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(c / b), $MachinePrecision] * 1.5), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a * 1.125), $MachinePrecision]), $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0040000000000000001

   1. Initial program 78.4%

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

      1. /-rgt-identity 78.4%

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

      2. metadata-eval 78.4%

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

      3. associate-/l* 78.4%

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

      4. associate-*r/ 78.4%

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

      5. *-commutative 78.4%

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

      6. associate-*l/ 78.4%

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

      7. associate-*r/ 78.4%

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

      8. metadata-eval 78.4%

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

      9. metadata-eval 78.4%

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

      10. times-frac 78.4%

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

      11. neg-mul-1 78.4%

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

      12. distribute-rgt-neg-in 78.4%

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

      13. times-frac 78.4%

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

      14. metadata-eval 78.4%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 78.4%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. metadata-eval 42.3%

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

      3. associate-/l* 42.3%

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

      4. associate-*r/ 42.3%

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

      5. *-commutative 42.3%

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

      6. associate-*l/ 42.3%

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

      7. associate-*r/ 42.3%

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

      8. metadata-eval 42.3%

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

      9. metadata-eval 42.3%

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

      10. times-frac 42.3%

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

      11. neg-mul-1 42.3%

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

      12. distribute-rgt-neg-in 42.3%

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

      13. times-frac 42.3%

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

      14. metadata-eval 42.3%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 42.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 42.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around inf 91.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 91.9%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      2. fma-def 92.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      3. associate-*r/ 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\frac{1.125 \cdot \left({c}^{2} \cdot a\right)}{{b}^{3}}}\right) \]

      4. *-commutative 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \color{blue}{\left(a \cdot {c}^{2}\right)}}{{b}^{3}}\right) \]

      5. unpow2 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{3}}\right) \]

   6. Simplified 92.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]
   7. Step-by-step derivation

      1. fma-udef 91.9%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]

      2. div-inv 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \color{blue}{\left(1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \frac{1}{{b}^{3}}}\right) \]

      3. associate-*r* 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \color{blue}{\left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right)} \cdot \frac{1}{{b}^{3}}\right) \]

      4. pow-flip 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{{b}^{\left(-3\right)}}\right) \]

      5. metadata-eval 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{\color{blue}{-3}}\right) \]

   8. Applied egg-rr 91.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{-3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot 1.5 + \left(\left(c \cdot c\right) \cdot \left(a \cdot 1.125\right)\right) \cdot {b}^{-3}\right)\\ \end{array} \]

Alternative 10: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot 1.5 + \left(\left(c \cdot c\right) \cdot \left(a \cdot 1.125\right)\right) \cdot {b}^{-3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.004)
   (* (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ 0.3333333333333333 a))
   (*
    -0.3333333333333333
    (+ (* (/ c b) 1.5) (* (* (* c c) (* a 1.125)) (pow b -3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.3333333333333333 * (((c / b) * 1.5) + (((c * c) * (a * 1.125)) * pow(b, -3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(c / b) * 1.5) + Float64(Float64(Float64(c * c) * Float64(a * 1.125)) * (b ^ -3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(c / b), $MachinePrecision] * 1.5), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a * 1.125), $MachinePrecision]), $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0040000000000000001

   1. Initial program 78.4%

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

      1. neg-sub0 78.4%

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

      2. associate-+l- 78.4%

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

      3. sub0-neg 78.4%

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

      4. neg-mul-1 78.4%

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

      5. associate-*r/ 78.4%

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

      6. *-commutative 78.4%

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

      7. metadata-eval 78.4%

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

      8. metadata-eval 78.4%

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

      9. times-frac 78.4%

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

      10. *-commutative 78.4%

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

      11. times-frac 78.4%

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

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. metadata-eval 42.3%

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

      3. associate-/l* 42.3%

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

      4. associate-*r/ 42.3%

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

      5. *-commutative 42.3%

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

      6. associate-*l/ 42.3%

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

      7. associate-*r/ 42.3%

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

      8. metadata-eval 42.3%

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

      9. metadata-eval 42.3%

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

      10. times-frac 42.3%

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

      11. neg-mul-1 42.3%

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

      12. distribute-rgt-neg-in 42.3%

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

      13. times-frac 42.3%

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

      14. metadata-eval 42.3%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 42.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 42.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around inf 91.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 91.9%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      2. fma-def 92.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      3. associate-*r/ 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\frac{1.125 \cdot \left({c}^{2} \cdot a\right)}{{b}^{3}}}\right) \]

      4. *-commutative 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \color{blue}{\left(a \cdot {c}^{2}\right)}}{{b}^{3}}\right) \]

      5. unpow2 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{3}}\right) \]

   6. Simplified 92.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]
   7. Step-by-step derivation

      1. fma-udef 91.9%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]

      2. div-inv 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \color{blue}{\left(1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \frac{1}{{b}^{3}}}\right) \]

      3. associate-*r* 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \color{blue}{\left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right)} \cdot \frac{1}{{b}^{3}}\right) \]

      4. pow-flip 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{{b}^{\left(-3\right)}}\right) \]

      5. metadata-eval 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{\color{blue}{-3}}\right) \]

   8. Applied egg-rr 91.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{-3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot 1.5 + \left(\left(c \cdot c\right) \cdot \left(a \cdot 1.125\right)\right) \cdot {b}^{-3}\right)\\ \end{array} \]

Alternative 11: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot 1.5 + \left(\left(c \cdot c\right) \cdot \left(a \cdot 1.125\right)\right) \cdot {b}^{-3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.004)
   (/ (* -0.3333333333333333 (- b (sqrt (fma b b (* (* c a) -3.0))))) a)
   (*
    -0.3333333333333333
    (+ (* (/ c b) 1.5) (* (* (* c c) (* a 1.125)) (pow b -3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (-0.3333333333333333 * (b - sqrt(fma(b, b, ((c * a) * -3.0))))) / a;
	} else {
		tmp = -0.3333333333333333 * (((c / b) * 1.5) + (((c * c) * (a * 1.125)) * pow(b, -3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(fma(b, b, Float64(Float64(c * a) * -3.0))))) / a);
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(c / b) * 1.5) + Float64(Float64(Float64(c * c) * Float64(a * 1.125)) * (b ^ -3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(c / b), $MachinePrecision] * 1.5), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a * 1.125), $MachinePrecision]), $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0040000000000000001

   1. Initial program 78.4%

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

      1. /-rgt-identity 78.4%

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

      2. metadata-eval 78.4%

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

      3. associate-/r/ 78.4%

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

      4. metadata-eval 78.4%

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

      5. metadata-eval 78.4%

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

      6. times-frac 78.4%

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

      7. *-commutative 78.4%

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

      8. times-frac 78.4%

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

      9. *-commutative 78.4%

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

      10. associate-/r* 78.4%

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

      11. associate-*l/ 78.4%

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

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. metadata-eval 42.3%

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

      3. associate-/l* 42.3%

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

      4. associate-*r/ 42.3%

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

      5. *-commutative 42.3%

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

      6. associate-*l/ 42.3%

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

      7. associate-*r/ 42.3%

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

      8. metadata-eval 42.3%

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

      9. metadata-eval 42.3%

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

      10. times-frac 42.3%

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

      11. neg-mul-1 42.3%

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

      12. distribute-rgt-neg-in 42.3%

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

      13. times-frac 42.3%

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

      14. metadata-eval 42.3%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 42.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 42.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around inf 91.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 91.9%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      2. fma-def 92.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      3. associate-*r/ 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\frac{1.125 \cdot \left({c}^{2} \cdot a\right)}{{b}^{3}}}\right) \]

      4. *-commutative 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \color{blue}{\left(a \cdot {c}^{2}\right)}}{{b}^{3}}\right) \]

      5. unpow2 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{3}}\right) \]

   6. Simplified 92.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]
   7. Step-by-step derivation

      1. fma-udef 91.9%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]

      2. div-inv 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \color{blue}{\left(1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \frac{1}{{b}^{3}}}\right) \]

      3. associate-*r* 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \color{blue}{\left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right)} \cdot \frac{1}{{b}^{3}}\right) \]

      4. pow-flip 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{{b}^{\left(-3\right)}}\right) \]

      5. metadata-eval 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{\color{blue}{-3}}\right) \]

   8. Applied egg-rr 91.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{-3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot 1.5 + \left(\left(c \cdot c\right) \cdot \left(a \cdot 1.125\right)\right) \cdot {b}^{-3}\right)\\ \end{array} \]

Alternative 12: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot 1.5 + \left(\left(c \cdot c\right) \cdot \left(a \cdot 1.125\right)\right) \cdot {b}^{-3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.004)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* 3.0 a))
   (*
    -0.3333333333333333
    (+ (* (/ c b) 1.5) (* (* (* c c) (* a 1.125)) (pow b -3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} else {
		tmp = -0.3333333333333333 * (((c / b) * 1.5) + (((c * c) * (a * 1.125)) * pow(b, -3.0)));
	}
	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) :: tmp
    if (((sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)) <= (-0.004d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.3333333333333333d0) * (((c / b) * 1.5d0) + (((c * c) * (a * 1.125d0)) * (b ** (-3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.004) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} else {
		tmp = -0.3333333333333333 * (((c / b) * 1.5) + (((c * c) * (a * 1.125)) * Math.pow(b, -3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.004:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a)
	else:
		tmp = -0.3333333333333333 * (((c / b) * 1.5) + (((c * c) * (a * 1.125)) * math.pow(b, -3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.004)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(Float64(c / b) * 1.5) + Float64(Float64(Float64(c * c) * Float64(a * 1.125)) * (b ^ -3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.004)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	else
		tmp = -0.3333333333333333 * (((c / b) * 1.5) + (((c * c) * (a * 1.125)) * (b ^ -3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[(c / b), $MachinePrecision] * 1.5), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a * 1.125), $MachinePrecision]), $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0040000000000000001

   1. Initial program 78.4%

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

      1. neg-sub0 78.4%

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

      2. associate-+l- 78.4%

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

      3. sub0-neg 78.4%

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

      4. neg-mul-1 78.4%

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

      5. associate-*r/ 78.4%

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

      6. metadata-eval 78.4%

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

      7. metadata-eval 78.4%

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

      8. times-frac 78.4%

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

      9. *-commutative 78.4%

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

      10. times-frac 78.4%

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

      11. associate-*l/ 78.4%

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

   3. Simplified 78.4%

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

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. metadata-eval 42.3%

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

      3. associate-/l* 42.3%

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

      4. associate-*r/ 42.3%

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

      5. *-commutative 42.3%

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

      6. associate-*l/ 42.3%

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

      7. associate-*r/ 42.3%

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

      8. metadata-eval 42.3%

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

      9. metadata-eval 42.3%

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

      10. times-frac 42.3%

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

      11. neg-mul-1 42.3%

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

      12. distribute-rgt-neg-in 42.3%

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

      13. times-frac 42.3%

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

      14. metadata-eval 42.3%

          \[\leadsto \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]

      15. neg-mul-1 42.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Simplified 42.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]

   4. Taylor expanded in b around inf 91.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 1.5 \cdot \frac{c}{b}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 91.9%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      2. fma-def 92.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, 1.125 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]

      3. associate-*r/ 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \color{blue}{\frac{1.125 \cdot \left({c}^{2} \cdot a\right)}{{b}^{3}}}\right) \]

      4. *-commutative 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \color{blue}{\left(a \cdot {c}^{2}\right)}}{{b}^{3}}\right) \]

      5. unpow2 92.0%

         \[\leadsto -0.3333333333333333 \cdot \mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{3}}\right) \]

   6. Simplified 92.0%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b}, \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]
   7. Step-by-step derivation

      1. fma-udef 91.9%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + \frac{1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)} \]

      2. div-inv 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \color{blue}{\left(1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \frac{1}{{b}^{3}}}\right) \]

      3. associate-*r* 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \color{blue}{\left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right)} \cdot \frac{1}{{b}^{3}}\right) \]

      4. pow-flip 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{{b}^{\left(-3\right)}}\right) \]

      5. metadata-eval 91.9%

         \[\leadsto -0.3333333333333333 \cdot \left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{\color{blue}{-3}}\right) \]

   8. Applied egg-rr 91.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(1.5 \cdot \frac{c}{b} + \left(\left(1.125 \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot {b}^{-3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.004:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot 1.5 + \left(\left(c \cdot c\right) \cdot \left(a \cdot 1.125\right)\right) \cdot {b}^{-3}\right)\\ \end{array} \]

Alternative 13: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.000172:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.000172)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.000172) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} 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) :: tmp
    if (((sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)) <= (-0.000172d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.000172) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.000172:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a)
	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(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.000172)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.000172)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.000172], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $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)) < -1.7200000000000001e-4

   1. Initial program 76.1%

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

      1. neg-sub0 76.1%

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

      2. associate-+l- 76.1%

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

      3. sub0-neg 76.1%

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

      4. neg-mul-1 76.1%

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

      5. associate-*r/ 76.1%

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

      6. metadata-eval 76.1%

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

      7. metadata-eval 76.1%

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

      8. times-frac 76.1%

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

      9. *-commutative 76.1%

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

      10. times-frac 76.2%

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

      11. associate-*l/ 76.1%

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

   3. Simplified 76.2%

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

   if -1.7200000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 38.9%

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

      1. /-rgt-identity 38.9%

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

      2. metadata-eval 38.9%

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

      3. associate-/r/ 38.9%

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

      4. metadata-eval 38.9%

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

      5. metadata-eval 38.9%

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

      6. times-frac 38.9%

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

      7. *-commutative 38.9%

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

      8. times-frac 38.9%

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

      9. *-commutative 38.9%

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

      10. associate-/r* 38.9%

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

      11. associate-*l/ 38.9%

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

   3. Simplified 39.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

   4. Taylor expanded in b around inf 79.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.000172:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 14: 64.7% accurate, 23.2× speedup

Math

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

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(-0.5 * Float64(c / b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. /-rgt-identity 54.3%

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

   2. metadata-eval 54.3%

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

   3. associate-/r/ 54.3%

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

   4. metadata-eval 54.3%

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

   5. metadata-eval 54.3%

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

   6. times-frac 54.3%

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

   7. *-commutative 54.3%

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

   8. times-frac 54.3%

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

   9. *-commutative 54.3%

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

   10. associate-/r* 54.3%

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

   11. associate-*l/ 54.3%

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

3. Simplified 54.4%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]

4. Taylor expanded in b around inf 65.8%

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

5. Final simplification 65.8%

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

Reproduce

herbie shell --seed 2023214 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))