Cubic critical

Percentage Accurate: 51.4% → 85.5%

Time: 8.6s

Alternatives: 12

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: 51.4% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+153)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 8.6e-48)
     (/ (- (sqrt (fma (* -3.0 a) c (* b b))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+153) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = (sqrt(fma((-3.0 * a), c, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+153)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 8.6e-48)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+153], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-48], N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000018e153

   1. Initial program 52.2%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if -5.00000000000000018e153 < b < 8.6e-48

   1. Initial program 80.0%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 80.0

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

   4. Applied rewrites 80.0%

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

   if 8.6e-48 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in c 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 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+153)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 8.6e-48)
     (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+153) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+153)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 8.6e-48)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+153], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-48], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000018e153

   1. Initial program 52.2%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if -5.00000000000000018e153 < b < 8.6e-48

   1. Initial program 80.0%

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

   4. Applied rewrites 79.9%

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

   if 8.6e-48 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in c 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 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+153)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 8.6e-48)
     (* (- (sqrt (fma (* c a) -3.0 (* b b))) b) (/ 0.3333333333333333 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+153) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = (sqrt(fma((c * a), -3.0, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+153)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 8.6e-48)
		tmp = Float64(Float64(sqrt(fma(Float64(c * a), -3.0, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+153], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-48], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5000000000000001e153

   1. Initial program 52.2%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if -4.5000000000000001e153 < b < 8.6e-48

   1. Initial program 80.0%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 80.0

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

   4. Applied rewrites 80.0%

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

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. sub-neg N/A

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

      15. lift--.f64 N/A

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 79.8%

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

   if 8.6e-48 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in c 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 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+153)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 8.6e-48)
     (* (/ 0.3333333333333333 a) (- (sqrt (fma (* -3.0 c) a (* b b))) b))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+153) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((-3.0 * c), a, (b * b))) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+153)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 8.6e-48)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+153], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-48], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5000000000000001e153

   1. Initial program 52.2%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if -4.5000000000000001e153 < b < 8.6e-48

   1. Initial program 80.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 79.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 79.8

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

   4. Applied rewrites 79.8%

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

   if 8.6e-48 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in c 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 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.4e-45)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 8.6e-48)
     (/ (- (sqrt (* (* c a) -3.0)) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-45) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = (sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / 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 <= (-8.4d-45)) then
        tmp = ((-2.0d0) * b) / (a * 3.0d0)
    else if (b <= 8.6d-48) then
        tmp = (sqrt(((c * a) * (-3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-45) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = (Math.sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.4e-45:
		tmp = (-2.0 * b) / (a * 3.0)
	elif b <= 8.6e-48:
		tmp = (math.sqrt(((c * a) * -3.0)) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.4e-45)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 8.6e-48)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -3.0)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.4e-45)
		tmp = (-2.0 * b) / (a * 3.0);
	elseif (b <= 8.6e-48)
		tmp = (sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.4e-45], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-48], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.3999999999999998e-45

   1. Initial program 74.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 88.0

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

   5. Applied rewrites 88.0%

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

   if -8.3999999999999998e-45 < b < 8.6e-48

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.8

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

   7. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. associate-/r* N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

   9. Applied rewrites 62.8%

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

   10. Taylor expanded in c around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 62.9

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

   12. Applied rewrites 62.9%

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

   if 8.6e-48 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in c 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 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(\sqrt{\left(c \cdot a\right) \cdot -3} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.4e-45)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 8.6e-48)
     (/ (* (- (sqrt (* (* c a) -3.0)) b) 0.3333333333333333) a)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-45) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = ((sqrt(((c * a) * -3.0)) - b) * 0.3333333333333333) / a;
	} else {
		tmp = -0.5 * (c / 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 <= (-8.4d-45)) then
        tmp = ((-2.0d0) * b) / (a * 3.0d0)
    else if (b <= 8.6d-48) then
        tmp = ((sqrt(((c * a) * (-3.0d0))) - b) * 0.3333333333333333d0) / a
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-45) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = ((Math.sqrt(((c * a) * -3.0)) - b) * 0.3333333333333333) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.4e-45:
		tmp = (-2.0 * b) / (a * 3.0)
	elif b <= 8.6e-48:
		tmp = ((math.sqrt(((c * a) * -3.0)) - b) * 0.3333333333333333) / a
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.4e-45)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 8.6e-48)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(c * a) * -3.0)) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.4e-45)
		tmp = (-2.0 * b) / (a * 3.0);
	elseif (b <= 8.6e-48)
		tmp = ((sqrt(((c * a) * -3.0)) - b) * 0.3333333333333333) / a;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.4e-45], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-48], N[(N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.3999999999999998e-45

   1. Initial program 74.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 88.0

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

   5. Applied rewrites 88.0%

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

   if -8.3999999999999998e-45 < b < 8.6e-48

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.8

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

   7. Applied rewrites 62.8%

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

   if 8.6e-48 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in c 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 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(\sqrt{\left(c \cdot a\right) \cdot -3} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot a\right) \cdot -3} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.4e-45)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 8.6e-48)
     (* (- (sqrt (* (* c a) -3.0)) b) (/ 0.3333333333333333 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-45) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = (sqrt(((c * a) * -3.0)) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / 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 <= (-8.4d-45)) then
        tmp = ((-2.0d0) * b) / (a * 3.0d0)
    else if (b <= 8.6d-48) then
        tmp = (sqrt(((c * a) * (-3.0d0))) - b) * (0.3333333333333333d0 / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-45) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = (Math.sqrt(((c * a) * -3.0)) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.4e-45:
		tmp = (-2.0 * b) / (a * 3.0)
	elif b <= 8.6e-48:
		tmp = (math.sqrt(((c * a) * -3.0)) - b) * (0.3333333333333333 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.4e-45)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 8.6e-48)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -3.0)) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.4e-45)
		tmp = (-2.0 * b) / (a * 3.0);
	elseif (b <= 8.6e-48)
		tmp = (sqrt(((c * a) * -3.0)) - b) * (0.3333333333333333 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.4e-45], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-48], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.3999999999999998e-45

   1. Initial program 74.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 88.0

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

   5. Applied rewrites 88.0%

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

   if -8.3999999999999998e-45 < b < 8.6e-48

   1. Initial program 70.7%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 70.7

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

   4. Applied rewrites 70.7%

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

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. sub-neg N/A

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

      15. lift--.f64 N/A

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 70.5%

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

   7. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 62.8

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

   9. Applied rewrites 62.8%

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

   if 8.6e-48 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in c 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 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot a\right) \cdot -3} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\left(\sqrt{\left(-3 \cdot c\right) \cdot a} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.4e-45)
   (/ (* -2.0 b) (* a 3.0))
   (if (<= b 8.6e-48)
     (* (- (sqrt (* (* -3.0 c) a)) b) (/ 0.3333333333333333 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-45) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = (sqrt(((-3.0 * c) * a)) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / 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 <= (-8.4d-45)) then
        tmp = ((-2.0d0) * b) / (a * 3.0d0)
    else if (b <= 8.6d-48) then
        tmp = (sqrt((((-3.0d0) * c) * a)) - b) * (0.3333333333333333d0 / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-45) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else if (b <= 8.6e-48) {
		tmp = (Math.sqrt(((-3.0 * c) * a)) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.4e-45:
		tmp = (-2.0 * b) / (a * 3.0)
	elif b <= 8.6e-48:
		tmp = (math.sqrt(((-3.0 * c) * a)) - b) * (0.3333333333333333 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.4e-45)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	elseif (b <= 8.6e-48)
		tmp = Float64(Float64(sqrt(Float64(Float64(-3.0 * c) * a)) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.4e-45)
		tmp = (-2.0 * b) / (a * 3.0);
	elseif (b <= 8.6e-48)
		tmp = (sqrt(((-3.0 * c) * a)) - b) * (0.3333333333333333 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.4e-45], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-48], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.3999999999999998e-45

   1. Initial program 74.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 88.0

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

   5. Applied rewrites 88.0%

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

   if -8.3999999999999998e-45 < b < 8.6e-48

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.8

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

   7. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. associate-/r* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 62.7%

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

   if 8.6e-48 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in c 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 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\left(\sqrt{\left(-3 \cdot c\right) \cdot a} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.4e-303) (/ (* -2.0 b) (* a 3.0)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.4e-303) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / 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 <= 9.4d-303) then
        tmp = ((-2.0d0) * b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.4e-303) {
		tmp = (-2.0 * b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9.4e-303:
		tmp = (-2.0 * b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.4e-303)
		tmp = Float64(Float64(-2.0 * b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9.4e-303)
		tmp = (-2.0 * b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.4e-303], N[(N[(-2.0 * b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.3999999999999994e-303

   1. Initial program 75.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 69.3

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

   5. Applied rewrites 69.3%

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

   if 9.3999999999999994e-303 < b

   1. Initial program 31.6%

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

   3. Taylor expanded in c 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 65.9

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

   5. Applied rewrites 65.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{-2 \cdot b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.4e-303) (* (/ b a) -0.6666666666666666) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.4e-303) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = -0.5 * (c / 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 <= 9.4d-303) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.4e-303) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9.4e-303:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.4e-303)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9.4e-303)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.4e-303], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.3999999999999994e-303

   1. Initial program 75.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 69.3

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

   5. Applied rewrites 69.3%

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

   if 9.3999999999999994e-303 < b

   1. Initial program 31.6%

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

   3. Taylor expanded in c 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 65.9

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

   5. Applied rewrites 65.9%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.4e-303) (* (/ b a) -0.6666666666666666) (* (/ -0.5 b) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.4e-303) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (-0.5 / b) * c;
	}
	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 <= 9.4d-303) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = ((-0.5d0) / b) * c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.4e-303) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (-0.5 / b) * c;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9.4e-303:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (-0.5 / b) * c
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.4e-303)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(-0.5 / b) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9.4e-303)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (-0.5 / b) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.4e-303], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(-0.5 / b), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.3999999999999994e-303

   1. Initial program 75.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 69.3

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

   5. Applied rewrites 69.3%

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

   if 9.3999999999999994e-303 < b

   1. Initial program 31.6%

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

   3. Taylor expanded in c 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 65.9

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

   5. Applied rewrites 65.9%

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

   7. Applied rewrites 65.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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 / a) * (-0.6666666666666666d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in 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 34.9

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

5. Applied rewrites 34.9%

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

6. Final simplification 34.9%

   \[\leadsto \frac{b}{a} \cdot -0.6666666666666666 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024256 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))