Cubic critical

Percentage Accurate: 52.9% → 85.5%

Time: 12.2s

Alternatives: 15

Speedup: 2.2×

Specification

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: 52.9% 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: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-3}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e+67)
   (/ (/ (- b (- b)) -3.0) a)
   (if (<= b 1.06e-54)
     (/ 1.0 (* a (/ -3.0 (- b (sqrt (fma b b (* c (* a -3.0))))))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e+67) {
		tmp = ((b - -b) / -3.0) / a;
	} else if (b <= 1.06e-54) {
		tmp = 1.0 / (a * (-3.0 / (b - sqrt(fma(b, b, (c * (a * -3.0)))))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e+67)
		tmp = Float64(Float64(Float64(b - Float64(-b)) / -3.0) / a);
	elseif (b <= 1.06e-54)
		tmp = Float64(1.0 / Float64(a * Float64(-3.0 / Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))))));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e+67], N[(N[(N[(b - (-b)), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.06e-54], N[(1.0 / N[(a * N[(-3.0 / N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.2000000000000001e67

   1. Initial program 64.7%

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

   4. Applied rewrites 64.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower-/.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 64.8

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

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 64.6

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

   6. Applied rewrites 64.6%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.9

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

   9. Applied rewrites 97.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. un-div-inv N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-/r* N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 98.1

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

   11. Applied rewrites 98.1%

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

   if -5.2000000000000001e67 < b < 1.0600000000000001e-54

   1. Initial program 76.3%

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

   4. Applied rewrites 76.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower-/.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 76.2

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

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 76.2

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

   6. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 76.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 76.4%

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

   if 1.0600000000000001e-54 < b

   1. Initial program 23.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{b - \left(-b\right)}{-3}}{a}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-3}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+101}:\\ \;\;\;\;\left(b - \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e+101)
   (* (- b (- b)) (/ (/ 1.0 a) -3.0))
   (if (<= b 1.06e-54)
     (/ (/ (- b (sqrt (fma a (* -3.0 c) (* b b)))) a) -3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e+101) {
		tmp = (b - -b) * ((1.0 / a) / -3.0);
	} else if (b <= 1.06e-54) {
		tmp = ((b - sqrt(fma(a, (-3.0 * c), (b * b)))) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e+101)
		tmp = Float64(Float64(b - Float64(-b)) * Float64(Float64(1.0 / a) / -3.0));
	elseif (b <= 1.06e-54)
		tmp = Float64(Float64(Float64(b - sqrt(fma(a, Float64(-3.0 * c), Float64(b * b)))) / a) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e+101], N[(N[(b - (-b)), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.06e-54], N[(N[(N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.6000000000000003e101

   1. Initial program 53.5%

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

   4. Applied rewrites 53.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower-/.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 53.6

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

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 53.4

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.2

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

   9. Applied rewrites 95.2%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 95.3

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

   11. Applied rewrites 95.3%

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

   if -4.6000000000000003e101 < b < 1.0600000000000001e-54

   1. Initial program 80.1%

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

   4. Applied rewrites 80.0%

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

   if 1.0600000000000001e-54 < b

   1. Initial program 16.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b} \]

      4. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+101}:\\ \;\;\;\;\left(b - \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))