Cubic critical, medium range

Percentage Accurate: 31.5% → 99.6%

Time: 11.5s

Alternatives: 7

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.5% 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: 99.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (* c (* a 3.0)) (* a 3.0))
  (- (- b) (sqrt (fma a (* c -3.0) (pow b 2.0))))))

C

double code(double a, double b, double c) {
	return ((c * (a * 3.0)) / (a * 3.0)) / (-b - sqrt(fma(a, (c * -3.0), pow(b, 2.0))));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(a * 3.0)) / Float64(a * 3.0)) / Float64(Float64(-b) - sqrt(fma(a, Float64(c * -3.0), (b ^ 2.0)))))
end

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. add-cube-cbrt 31.1%

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

   2. pow3 31.1%

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

   3. associate-*l* 31.1%

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

3. Applied egg-rr 31.1%

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

   1. flip-+ 31.1%

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

5. Applied egg-rr 32.1%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(a \cdot 3\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}}}}{3 \cdot a} \]
6. Step-by-step derivation

   1. associate--r- 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in b around 0 99.3%

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

   1. fma-def 99.3%

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

   2. *-commutative 99.3%

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

10. Simplified 99.3%

    \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}}}}{3 \cdot a} \]
11. Step-by-step derivation

    1. expm1-log1p-u 86.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}}}{3 \cdot a}\right)\right)} \]

    2. expm1-udef 39.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}}}{3 \cdot a}\right)} - 1} \]

12. Applied egg-rr 39.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(c, a \cdot 3, {b}^{2} - {b}^{2}\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}\right)}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 86.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(c, a \cdot 3, {b}^{2} - {b}^{2}\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}\right)}\right)\right)} \]

    2. expm1-log1p 99.3%

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

    3. associate-/r* 99.6%

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

    4. fma-udef 99.6%

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

    5. +-inverses 99.6%

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

    6. +-rgt-identity 99.6%

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

    7. fma-def 99.6%

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

    8. associate-*r* 99.6%

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

    9. *-commutative 99.6%

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

    10. *-commutative 99.6%

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

    11. fma-def 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 2: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -6e-7)
   (* 0.3333333333333333 (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6e-7) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (c * (a * -3.0)))) - b) / 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(a * 3.0)))) - b) / Float64(a * 3.0)) <= -6e-7)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -6e-7], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / 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)) < -5.9999999999999997e-7

   1. Initial program 68.5%

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

      1. add-cube-cbrt 68.5%

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

      2. pow3 68.5%

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

      3. associate-*l* 68.5%

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

   3. Applied egg-rr 68.5%

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

      1. expm1-log1p-u 38.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{3 \cdot \left(a \cdot c\right)}\right)}^{3}}}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 37.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{3 \cdot \left(a \cdot c\right)}\right)}^{3}}}{3 \cdot a}\right)} - 1} \]

   5. Applied egg-rr 37.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}{a \cdot 3}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 38.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}{a \cdot 3}\right)\right)} \]

      2. expm1-log1p 68.5%

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

      3. *-lft-identity 68.5%

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

      4. *-commutative 68.5%

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

      5. times-frac 68.6%

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

      6. metadata-eval 68.6%

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

   7. Simplified 68.6%

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

   if -5.9999999999999997e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 15.3%

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

   2. Taylor expanded in b around inf 92.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -6e-7)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6e-7) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -6e-7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -6e-7], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-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)) < -5.9999999999999997e-7

   1. Initial program 68.5%

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

      1. add-cube-cbrt 68.5%

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

      2. pow3 68.5%

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

      3. associate-*l* 68.5%

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

   3. Applied egg-rr 68.5%

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

      1. expm1-log1p-u 42.2%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{3 \cdot \left(a \cdot c\right)}\right)}^{3}}\right)\right)}}{3 \cdot a} \]

      2. expm1-udef 30.3%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{3 \cdot \left(a \cdot c\right)}\right)}^{3}}\right)} - 1}}{3 \cdot a} \]

      3. neg-mul-1 30.3%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - {\left(\sqrt[3]{3 \cdot \left(a \cdot c\right)}\right)}^{3}}\right)} - 1}{3 \cdot a} \]

      4. fma-def 30.3%

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

      5. pow2 30.3%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2}} - {\left(\sqrt[3]{3 \cdot \left(a \cdot c\right)}\right)}^{3}}\right)\right)} - 1}{3 \cdot a} \]

      6. rem-cube-cbrt 30.3%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{3 \cdot \left(a \cdot c\right)}}\right)\right)} - 1}{3 \cdot a} \]

      7. associate-*r* 30.3%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(3 \cdot a\right) \cdot c}}\right)\right)} - 1}{3 \cdot a} \]

      8. *-commutative 30.3%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot 3\right)} \cdot c}\right)\right)} - 1}{3 \cdot a} \]

   5. Applied egg-rr 30.3%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)\right)} - 1}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. expm1-def 42.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)\right)\right)}}{3 \cdot a} \]

      2. expm1-log1p 68.5%

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

      3. fma-udef 68.5%

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

      4. neg-mul-1 68.5%

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

      5. +-commutative 68.5%

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

      6. unsub-neg 68.5%

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

      7. unpow2 68.5%

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

      8. fma-neg 68.6%

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

      9. *-commutative 68.6%

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

      10. distribute-rgt-neg-in 68.6%

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

      11. distribute-rgt-neg-in 68.6%

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

      12. metadata-eval 68.6%

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

   7. Simplified 68.6%

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

   if -5.9999999999999997e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 15.3%

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

   2. Taylor expanded in b around inf 92.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -6e-7)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6e-7) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-6d-7)) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a * 3.0d0)
    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 * (a * 3.0)))) - b) / (a * 3.0)) <= -6e-7) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6e-7:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -6e-7)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0));
	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 * (a * 3.0)))) - b) / (a * 3.0)) <= -6e-7)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	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[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -6e-7], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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)) < -5.9999999999999997e-7

   1. Initial program 68.5%

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

   2. Taylor expanded in a around 0 68.5%

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

      1. associate-*r* 68.5%

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

      2. *-commutative 68.5%

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

      3. associate-*l* 68.6%

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

   4. Simplified 68.6%

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

   if -5.9999999999999997e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 15.3%

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

   2. Taylor expanded in b around inf 92.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5: 90.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

2. Taylor expanded in b around inf 89.9%

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

3. Final simplification 89.9%

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

Alternative 6: 81.2% 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 31.1%

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

2. Taylor expanded in b around inf 81.4%

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

3. Final simplification 81.4%

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

Alternative 7: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. add-cube-cbrt 31.1%

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

   2. pow3 31.1%

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

   3. associate-*l* 31.1%

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

3. Applied egg-rr 31.1%

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

   1. div-inv 31.1%

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

   2. neg-mul-1 31.1%

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

   3. fma-def 31.1%

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

   4. pow2 31.1%

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

   5. rem-cube-cbrt 31.1%

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

   6. associate-*r* 31.1%

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

   7. *-commutative 31.1%

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

   8. *-commutative 31.1%

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

5. Applied egg-rr 31.1%

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

6. Taylor expanded in a around 0 3.2%

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

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

8. Simplified 3.2%

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

9. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2023333 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))