Cubic critical, narrow range

Percentage Accurate: 55.1% → 99.3%

Time: 14.4s

Alternatives: 15

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: 55.1% 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: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \frac{-0.3333333333333333}{a}}{\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* c (/ -0.3333333333333333 a))
  (* (/ 0.3333333333333333 a) (+ b (sqrt (fma c (* a -3.0) (* b b)))))))

C

double code(double a, double b, double c) {
	return (c * (-0.3333333333333333 / a)) / ((0.3333333333333333 / a) * (b + sqrt(fma(c, (a * -3.0), (b * b)))));
}

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-0.3333333333333333 / a)) / Float64(Float64(0.3333333333333333 / a) * Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b))))))
end

Wolfram

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

Derivation

1. Initial program 56.4%

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

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 55.9%

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

6. Applied rewrites 57.0%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{\left(a \cdot a\right) \cdot 9} - \frac{b \cdot b}{\left(a \cdot a\right) \cdot 9}}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} + b\right)}} \]

7. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.1

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

9. Applied rewrites 99.1%

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

11. Applied rewrites 99.3%

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

12. Final simplification 99.3%

    \[\leadsto \frac{c \cdot \frac{-0.3333333333333333}{a}}{\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} \]
13. Add Preprocessing

Alternative 2: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.048:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot -0.3333333333333333}{a}}{b \cdot \mathsf{fma}\left(-0.5, \frac{c}{b \cdot b}, \frac{0.6666666666666666}{a}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.048)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/
    (/ (* c -0.3333333333333333) a)
    (* b (fma -0.5 (/ c (* b b)) (/ 0.6666666666666666 a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.048) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = ((c * -0.3333333333333333) / a) / (b * fma(-0.5, (c / (b * b)), (0.6666666666666666 / a)));
	}
	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)) <= -0.048)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(c * -0.3333333333333333) / a) / Float64(b * fma(-0.5, Float64(c / Float64(b * b)), Float64(0.6666666666666666 / a))));
	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], -0.048], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.3333333333333333), $MachinePrecision] / a), $MachinePrecision] / N[(b * N[(-0.5 * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / a), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.048000000000000001

   1. Initial program 79.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 79.6

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

   4. Applied rewrites 79.6%

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

   if -0.048000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 48.2%

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

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 47.5%

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

   6. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{\left(a \cdot a\right) \cdot 9} - \frac{b \cdot b}{\left(a \cdot a\right) \cdot 9}}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} + b\right)}} \]

   7. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.1

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f64 87.3

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

   12. Applied rewrites 87.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.048:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot -0.3333333333333333}{a}}{b \cdot \mathsf{fma}\left(-0.5, \frac{c}{b \cdot b}, \frac{0.6666666666666666}{a}\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(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)))