Cubic critical

Percentage Accurate: 52.5% → 85.7%

Time: 7.7s

Alternatives: 7

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: 52.5% 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: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2 \cdot \frac{b}{a}}{3}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+152)
   (/ (* -2.0 (/ b a)) 3.0)
   (if (<= b 5.5e-53)
     (/ (+ (- b) (sqrt (fma (* -3.0 a) c (* b b)))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e+152) {
		tmp = (-2.0 * (b / a)) / 3.0;
	} else if (b <= 5.5e-53) {
		tmp = (-b + sqrt(fma((-3.0 * a), c, (b * b)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e+152)
		tmp = Float64(Float64(-2.0 * Float64(b / a)) / 3.0);
	elseif (b <= 5.5e-53)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+152], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 5.5e-53], N[(N[((-b) + N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2e152

   1. Initial program 42.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 42.2

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

   4. Applied rewrites 42.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 42.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 42.2

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 42.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 42.2

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

   6. Applied rewrites 42.2%

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

   7. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot \frac{b}{a}}}{3} \]

      2. lower-/.f64 99.9

         \[\leadsto \frac{-2 \cdot \color{blue}{\frac{b}{a}}}{3} \]

   9. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{b}{a}}}{3} \]

   if -1.2e152 < b < 5.50000000000000023e-53

   1. Initial program 73.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 73.9

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

   4. Applied rewrites 73.9%

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

   if 5.50000000000000023e-53 < b

   1. Initial program 15.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 90.5

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

   5. Applied rewrites 90.5%

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

Alternative 2: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{-2 \cdot \frac{b}{a}}{3}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(c \cdot a\right) \cdot -3}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-40)
   (/ (* -2.0 (/ b a)) 3.0)
   (if (<= b 5.5e-53)
     (/ (+ (- b) (sqrt (* (* c a) -3.0))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-40) {
		tmp = (-2.0 * (b / a)) / 3.0;
	} else if (b <= 5.5e-53) {
		tmp = (-b + sqrt(((c * a) * -3.0))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.8d-40)) then
        tmp = ((-2.0d0) * (b / a)) / 3.0d0
    else if (b <= 5.5d-53) then
        tmp = (-b + sqrt(((c * a) * (-3.0d0)))) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-40) {
		tmp = (-2.0 * (b / a)) / 3.0;
	} else if (b <= 5.5e-53) {
		tmp = (-b + Math.sqrt(((c * a) * -3.0))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-40:
		tmp = (-2.0 * (b / a)) / 3.0
	elif b <= 5.5e-53:
		tmp = (-b + math.sqrt(((c * a) * -3.0))) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-40)
		tmp = Float64(Float64(-2.0 * Float64(b / a)) / 3.0);
	elseif (b <= 5.5e-53)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(c * a) * -3.0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.8e-40)
		tmp = (-2.0 * (b / a)) / 3.0;
	elseif (b <= 5.5e-53)
		tmp = (-b + sqrt(((c * a) * -3.0))) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-40], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 5.5e-53], N[(N[((-b) + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7999999999999999e-40

   1. Initial program 62.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 62.4

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

   4. Applied rewrites 62.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 62.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 62.4

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 62.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 62.4

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

   6. Applied rewrites 62.4%

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

   7. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot \frac{b}{a}}}{3} \]

      2. lower-/.f64 82.0

         \[\leadsto \frac{-2 \cdot \color{blue}{\frac{b}{a}}}{3} \]

   9. Applied rewrites 82.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{b}{a}}}{3} \]

   if -3.7999999999999999e-40 < b < 5.50000000000000023e-53

   1. Initial program 72.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if 5.50000000000000023e-53 < b

   1. Initial program 15.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 90.5

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

   5. Applied rewrites 90.5%

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

Alternative 3: 66.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-303}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1e-303) (/ (* -2.0 b) (* 3.0 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1e-303) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.1d-303) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1e-303) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.1e-303:
		tmp = (-2.0 * b) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1e-303)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.1e-303)
		tmp = (-2.0 * b) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1e-303], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1e-303

   1. Initial program 68.5%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 54.7

         \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]

   5. Applied rewrites 54.7%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]

   if 2.1e-303 < b

   1. Initial program 30.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

Alternative 4: 66.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-303}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1e-303) (/ (* -0.6666666666666666 b) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1e-303) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.1d-303) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1e-303) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.1e-303:
		tmp = (-0.6666666666666666 * b) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1e-303)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.1e-303)
		tmp = (-0.6666666666666666 * b) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1e-303], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1e-303

   1. Initial program 68.5%

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

   4. Applied rewrites 35.7%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \frac{\color{blue}{\frac{-2}{3} \cdot b}}{a} \]
   6. Step-by-step derivation

      1. lower-*.f64 54.6

         \[\leadsto \frac{\color{blue}{-0.6666666666666666 \cdot b}}{a} \]

   7. Applied rewrites 54.6%

      \[\leadsto \frac{\color{blue}{-0.6666666666666666 \cdot b}}{a} \]

   if 2.1e-303 < b

   1. Initial program 30.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

Alternative 5: 66.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-303}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1e-303) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1e-303) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.1d-303) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1e-303) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.1e-303:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1e-303)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.1e-303)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1e-303], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1e-303

   1. Initial program 68.5%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]

      2. lower-/.f64 54.6

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{b}{a}} \]

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   if 2.1e-303 < b

   1. Initial program 30.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

Alternative 6: 66.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-303}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1e-303) (* -0.6666666666666666 (/ b a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1e-303) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.1d-303) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1e-303) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.1e-303:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1e-303)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.1e-303)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1e-303], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1e-303

   1. Initial program 68.5%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]

      2. lower-/.f64 54.6

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{b}{a}} \]

   5. Applied rewrites 54.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   if 2.1e-303 < b

   1. Initial program 30.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.6%

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

Alternative 7: 34.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.6666666666666666 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.6666666666666666 (/ b a)))

C

double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.6666666666666666d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Python

def code(a, b, c):
	return -0.6666666666666666 * (b / a)

Julia

function code(a, b, c)
	return Float64(-0.6666666666666666 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.6666666666666666 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.1%

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

3. Taylor expanded in b around -inf

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

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]

   2. lower-/.f64 26.2

      \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{b}{a}} \]

5. Applied rewrites 26.2%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024342 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))