Cubic critical, narrow range

Percentage Accurate: 55.6% → 99.5%

Time: 13.3s

Alternatives: 11

Speedup: 2.9×

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: 55.6% 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: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 49.4%

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.32:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b \cdot b}, 0.5, \frac{-0.6666666666666666}{c}\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.32)
   (/ (- (sqrt (fma b b (* (* c a) -3.0))) b) (* 3.0 a))
   (/
    0.3333333333333333
    (* (fma (/ a (* b b)) 0.5 (/ -0.6666666666666666 c)) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.32) {
		tmp = (sqrt(fma(b, b, ((c * a) * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = 0.3333333333333333 / (fma((a / (b * b)), 0.5, (-0.6666666666666666 / c)) * b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.32)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(0.3333333333333333 / Float64(fma(Float64(a / Float64(b * b)), 0.5, Float64(-0.6666666666666666 / c)) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.32], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-0.6666666666666666 / c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.32000000000000006

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   if 1.32000000000000006 < b

   1. Initial program 50.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 50.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 86.0

         \[\leadsto \frac{0.3333333333333333}{b \cdot \left(0.5 \cdot \frac{a}{b \cdot b} - \color{blue}{\frac{0.6666666666666666}{c}}\right)} \]

   7. Applied rewrites 86.0%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{b \cdot \left(0.5 \cdot \frac{a}{b \cdot b} - \frac{0.6666666666666666}{c}\right)}} \]

   8. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{1}{3}}{\color{blue}{b \cdot \left(\frac{1}{2} \cdot \frac{a}{{b}^{2}} - \frac{2}{3} \cdot \frac{1}{c}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 86.0

          \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b \cdot b}, 0.5, \color{blue}{\frac{-0.6666666666666666}{c}}\right) \cdot b} \]

   10. Applied rewrites 86.0%

       \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\frac{a}{b \cdot b}, 0.5, \frac{-0.6666666666666666}{c}\right) \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.32:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b \cdot b}, 0.5, \frac{-0.6666666666666666}{c}\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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)))