Cubic critical, medium range

Percentage Accurate: 31.5% → 99.2%

Time: 16.3s

Alternatives: 8

Speedup: 2.9×

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: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b + \sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)}\\ \frac{\frac{\frac{b \cdot b - b \cdot b}{t\_0} - \frac{c \cdot \left(a \cdot -3\right)}{t\_0}}{a}}{-3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ b (sqrt (fma (* a c) -3.0 (* b b))))))
   (/ (/ (- (/ (- (* b b) (* b b)) t_0) (/ (* c (* a -3.0)) t_0)) a) -3.0)))

C

double code(double a, double b, double c) {
	double t_0 = b + sqrt(fma((a * c), -3.0, (b * b)));
	return (((((b * b) - (b * b)) / t_0) - ((c * (a * -3.0)) / t_0)) / a) / -3.0;
}

Julia

function code(a, b, c)
	t_0 = Float64(b + sqrt(fma(Float64(a * c), -3.0, Float64(b * b))))
	return Float64(Float64(Float64(Float64(Float64(Float64(b * b) - Float64(b * b)) / t_0) - Float64(Float64(c * Float64(a * -3.0)) / t_0)) / a) / -3.0)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision]]

Derivation

1. Initial program 29.0%

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

4. Applied egg-rr 28.9%

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-lowering-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. *-lowering-*.f64 29.0

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

6. Applied egg-rr 29.0%

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

   1. flip-- N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. rem-square-sqrt N/A

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

   9. +-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

8. Applied egg-rr 99.2%

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

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(b + \sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)}\right)\\ \frac{\frac{b \cdot b - b \cdot b}{t\_0} - \frac{c \cdot \left(a \cdot -3\right)}{t\_0}}{-3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (+ b (sqrt (fma (* a c) -3.0 (* b b)))))))
   (/ (- (/ (- (* b b) (* b b)) t_0) (/ (* c (* a -3.0)) t_0)) -3.0)))

C

double code(double a, double b, double c) {
	double t_0 = a * (b + sqrt(fma((a * c), -3.0, (b * b))));
	return ((((b * b) - (b * b)) / t_0) - ((c * (a * -3.0)) / t_0)) / -3.0;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(b + sqrt(fma(Float64(a * c), -3.0, Float64(b * b)))))
	return Float64(Float64(Float64(Float64(Float64(b * b) - Float64(b * b)) / t_0) - Float64(Float64(c * Float64(a * -3.0)) / t_0)) / -3.0)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(b * b), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]]

Derivation

1. Initial program 29.0%

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

4. Applied egg-rr 28.9%

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-lowering-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. *-lowering-*.f64 29.0

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

6. Applied egg-rr 29.0%

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. flip-- N/A

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

   5. associate-/l/ N/A

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

   6. rem-square-sqrt N/A

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

   7. +-commutative N/A

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

   8. associate--r+ N/A

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

   9. div-sub N/A

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

8. Applied egg-rr 99.2%

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

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)}\right)\right)\\ \frac{b \cdot b - b \cdot b}{t\_0} - \frac{c \cdot \left(a \cdot -3\right)}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* -3.0 (+ b (sqrt (fma (* a c) -3.0 (* b b))))))))
   (- (/ (- (* b b) (* b b)) t_0) (/ (* c (* a -3.0)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = a * (-3.0 * (b + sqrt(fma((a * c), -3.0, (b * b)))));
	return (((b * b) - (b * b)) / t_0) - ((c * (a * -3.0)) / t_0);
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(-3.0 * Float64(b + sqrt(fma(Float64(a * c), -3.0, Float64(b * b))))))
	return Float64(Float64(Float64(Float64(b * b) - Float64(b * b)) / t_0) - Float64(Float64(c * Float64(a * -3.0)) / t_0))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(-3.0 * N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(b * b), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.0%

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

4. Applied egg-rr 28.9%

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-lowering-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. *-lowering-*.f64 29.0

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

6. Applied egg-rr 29.0%

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

   1. associate-/r* N/A

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

8. Applied egg-rr 99.1%

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

Alternative 4: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(c, c \cdot -0.375, \frac{a \cdot \left(-0.5625 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  a
  (/
   (fma c (* c -0.375) (/ (* a (* -0.5625 (* c (* c c)))) (* b b)))
   (* b (* b b)))
  (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	return fma(a, (fma(c, (c * -0.375), ((a * (-0.5625 * (c * (c * c)))) / (b * b))) / (b * (b * b))), (-0.5 * (c / b)));
}

Julia

function code(a, b, c)
	return fma(a, Float64(fma(c, Float64(c * -0.375), Float64(Float64(a * Float64(-0.5625 * Float64(c * Float64(c * c)))) / Float64(b * b))) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)))
end

Wolfram

code[a_, b_, c_] := N[(a * N[(N[(c * N[(c * -0.375), $MachinePrecision] + N[(N[(a * N[(-0.5625 * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.0%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Simplified 96.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

8. Simplified 94.8%

   \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\mathsf{fma}\left(c, c \cdot -0.375, \frac{a \cdot \left(-0.5625 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]
9. Add Preprocessing

Alternative 5: 93.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, \frac{-0.5625 \cdot \left(a \cdot \left(a \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (fma
    c
    (fma
     -0.375
     (/ a (* b b))
     (/ (* -0.5625 (* a (* a c))) (* (* b b) (* b b))))
    -0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * fma(c, fma(-0.375, (a / (b * b)), ((-0.5625 * (a * (a * c))) / ((b * b) * (b * b)))), -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(c, fma(-0.375, Float64(a / Float64(b * b)), Float64(Float64(-0.5625 * Float64(a * Float64(a * c))) / Float64(Float64(b * b) * Float64(b * b)))), -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(c * N[(-0.375 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5625 * N[(a * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 29.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

5. Simplified 94.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)\right)}{b}} \]

6. Taylor expanded in c around 0

   \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) - \frac{1}{2}\right)}}{b} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) - \frac{1}{2}\right)}}{b} \]

   2. sub-neg N/A

      \[\leadsto \frac{c \cdot \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}{b} \]

   3. metadata-eval N/A

      \[\leadsto \frac{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) + \color{blue}{\frac{-1}{2}}\right)}{b} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{c \cdot \color{blue}{\mathsf{fma}\left(c, \frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}, \frac{-1}{2}\right)}}{b} \]

8. Simplified 94.7%

   \[\leadsto \frac{\color{blue}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, \frac{-0.5625 \cdot \left(a \cdot \left(a \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), -0.5\right)}}{b} \]
9. Add Preprocessing

Alternative 6: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma a (/ (* -0.375 (* c c)) (* b b)) (* c -0.5)) b))

C

double code(double a, double b, double c) {
	return fma(a, ((-0.375 * (c * c)) / (b * b)), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * b)), Float64(c * -0.5)) / b)
end

Wolfram

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

Derivation

1. Initial program 29.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

5. Simplified 91.9%

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

6. Final simplification 91.9%

   \[\leadsto \frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b} \]
7. Add Preprocessing

Alternative 7: 90.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(c, -0.375 \cdot \frac{a}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (fma c (* -0.375 (/ a (* b b))) -0.5)) b))

C

double code(double a, double b, double c) {
	return (c * fma(c, (-0.375 * (a / (b * b))), -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(c, Float64(-0.375 * Float64(a / Float64(b * b))), -0.5)) / b)
end

Wolfram

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

Derivation

1. Initial program 29.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

5. Simplified 91.9%

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

6. Taylor expanded in c around 0

   \[\leadsto \frac{\color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}}{b} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}}} - \frac{1}{2}\right)}{b} \]

   2. associate-*r* N/A

      \[\leadsto \frac{c \cdot \left(\frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{2}} - \frac{1}{2}\right)}{b} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{2}} \cdot c} - \frac{1}{2}\right)}{b} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{c \cdot \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right)} \cdot c - \frac{1}{2}\right)}{b} \]

   5. sub-neg N/A

      \[\leadsto \frac{c \cdot \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}{b} \]

   6. metadata-eval N/A

      \[\leadsto \frac{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) \cdot c + \color{blue}{\frac{-1}{2}}\right)}{b} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) \cdot c + \frac{-1}{2}\right)}}{b} \]

   8. *-commutative N/A

      \[\leadsto \frac{c \cdot \left(\color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right)} + \frac{-1}{2}\right)}{b} \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{c \cdot \color{blue}{\mathsf{fma}\left(c, \frac{-3}{8} \cdot \frac{a}{{b}^{2}}, \frac{-1}{2}\right)}}{b} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \color{blue}{\frac{-3}{8} \cdot \frac{a}{{b}^{2}}}, \frac{-1}{2}\right)}{b} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{-3}{8} \cdot \color{blue}{\frac{a}{{b}^{2}}}, \frac{-1}{2}\right)}{b} \]

   12. unpow2 N/A

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \frac{-3}{8} \cdot \frac{a}{\color{blue}{b \cdot b}}, \frac{-1}{2}\right)}{b} \]

   13. *-lowering-*.f64 91.8

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

8. Simplified 91.8%

   \[\leadsto \frac{\color{blue}{c \cdot \mathsf{fma}\left(c, -0.375 \cdot \frac{a}{b \cdot b}, -0.5\right)}}{b} \]
9. Add Preprocessing

Alternative 8: 81.1% accurate, 2.9× 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 29.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. *-lowering-*.f64 N/A

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

   2. /-lowering-/.f64 83.2

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

5. Simplified 83.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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)))