Cubic critical

Percentage Accurate: 51.3% → 84.8%

Time: 10.6s

Alternatives: 14

Speedup: 11.6×

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: 51.3% 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: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.35e+67)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 1.28e-73)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.35e+67) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.28e-73) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} 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 <= (-2.35d+67)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 1.28d-73) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    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 <= -2.35e+67) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.28e-73) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.35e+67:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 1.28e-73:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.35e+67)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 1.28e-73)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.35e+67)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 1.28e-73)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.35e+67], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 1.28e-73], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.35000000000000009e67

   1. Initial program 46.8%

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

   4. Applied egg-rr 47.0%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{b}{a}\right)}, -3\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot b}{a}\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot b\right), a\right), -3\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot 2\right), a\right), -3\right) \]

      4. *-lowering-*.f64 96.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, 2\right), a\right), -3\right) \]

   7. Simplified 96.8%

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

   if -2.35000000000000009e67 < b < 1.2799999999999999e-73

   1. Initial program 90.2%

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

   if 1.2799999999999999e-73 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 90.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{b - \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)}}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e+74)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 2.6e-73)
     (/ (/ (- b (sqrt (+ (* b b) (* -3.0 (* a c))))) -3.0) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e+74) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.6e-73) {
		tmp = ((b - sqrt(((b * b) + (-3.0 * (a * c))))) / -3.0) / 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 <= (-7.5d+74)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 2.6d-73) then
        tmp = ((b - sqrt(((b * b) + ((-3.0d0) * (a * c))))) / (-3.0d0)) / 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 <= -7.5e+74) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.6e-73) {
		tmp = ((b - Math.sqrt(((b * b) + (-3.0 * (a * c))))) / -3.0) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e+74:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 2.6e-73:
		tmp = ((b - math.sqrt(((b * b) + (-3.0 * (a * c))))) / -3.0) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e+74)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 2.6e-73)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(Float64(b * b) + Float64(-3.0 * Float64(a * c))))) / -3.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e+74)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 2.6e-73)
		tmp = ((b - sqrt(((b * b) + (-3.0 * (a * c))))) / -3.0) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e+74], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 2.6e-73], N[(N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.5e74

   1. Initial program 45.9%

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

   4. Applied egg-rr 46.0%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{b}{a}\right)}, -3\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot b}{a}\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot b\right), a\right), -3\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot 2\right), a\right), -3\right) \]

      4. *-lowering-*.f64 96.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, 2\right), a\right), -3\right) \]

   7. Simplified 96.7%

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

   if -7.5e74 < b < 2.6000000000000001e-73

   1. Initial program 90.3%

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

   4. Applied egg-rr 90.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{-3}\right), \color{blue}{a}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right), -3\right), a\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), -3\right), a\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(-3 \cdot c\right)\right)\right)\right), -3\right), a\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(-3 \cdot c\right)\right)\right)\right)\right), -3\right), a\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(-3 \cdot c\right)\right)\right)\right)\right), -3\right), a\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(-3 \cdot c\right) \cdot a\right)\right)\right)\right), -3\right), a\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-3 \cdot \left(c \cdot a\right)\right)\right)\right)\right), -3\right), a\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-3 \cdot \left(a \cdot c\right)\right)\right)\right)\right), -3\right), a\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-3, \left(a \cdot c\right)\right)\right)\right)\right), -3\right), a\right) \]

      13. *-lowering-*.f64 90.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right), -3\right), a\right) \]

   6. Applied egg-rr 90.3%

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

   if 2.6000000000000001e-73 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 90.4%

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

Alternative 3: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-76}:\\ \;\;\;\;\frac{b - \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)}}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e+89)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 4e-76)
     (* (/ (- b (sqrt (+ (* b b) (* -3.0 (* a c))))) a) -0.3333333333333333)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e+89) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 4e-76) {
		tmp = ((b - sqrt(((b * b) + (-3.0 * (a * c))))) / a) * -0.3333333333333333;
	} 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 <= (-4.2d+89)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 4d-76) then
        tmp = ((b - sqrt(((b * b) + ((-3.0d0) * (a * c))))) / a) * (-0.3333333333333333d0)
    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 <= -4.2e+89) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 4e-76) {
		tmp = ((b - Math.sqrt(((b * b) + (-3.0 * (a * c))))) / a) * -0.3333333333333333;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e+89:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 4e-76:
		tmp = ((b - math.sqrt(((b * b) + (-3.0 * (a * c))))) / a) * -0.3333333333333333
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e+89)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 4e-76)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(Float64(b * b) + Float64(-3.0 * Float64(a * c))))) / a) * -0.3333333333333333);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.2e+89)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 4e-76)
		tmp = ((b - sqrt(((b * b) + (-3.0 * (a * c))))) / a) * -0.3333333333333333;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e+89], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 4e-76], N[(N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.19999999999999972e89

   1. Initial program 43.0%

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

   4. Applied egg-rr 43.1%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{b}{a}\right)}, -3\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot b}{a}\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot b\right), a\right), -3\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot 2\right), a\right), -3\right) \]

      4. *-lowering-*.f64 96.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, 2\right), a\right), -3\right) \]

   7. Simplified 96.5%

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

   if -4.19999999999999972e89 < b < 3.99999999999999971e-76

   1. Initial program 90.5%

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

   4. Applied egg-rr 90.4%

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

      1. div-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{a}\right), \color{blue}{\left(\frac{1}{-3}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right), a\right), \left(\frac{\color{blue}{1}}{-3}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(-3 \cdot c\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(-3 \cdot c\right)\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(-3 \cdot c\right)\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(-3 \cdot c\right) \cdot a\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-3 \cdot \left(c \cdot a\right)\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-3 \cdot \left(a \cdot c\right)\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-3, \left(a \cdot c\right)\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      13. metadata-eval 90.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right), a\right), \frac{-1}{3}\right) \]

   6. Applied egg-rr 90.2%

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

   if 3.99999999999999971e-76 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 90.4%

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

Alternative 4: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.8e-44)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 1.4e-74)
     (/ (- (sqrt (* a (* -3.0 c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.8e-44) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.4e-74) {
		tmp = (sqrt((a * (-3.0 * c))) - b) / (a * 3.0);
	} 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 <= (-6.8d-44)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 1.4d-74) then
        tmp = (sqrt((a * ((-3.0d0) * c))) - b) / (a * 3.0d0)
    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 <= -6.8e-44) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.4e-74) {
		tmp = (Math.sqrt((a * (-3.0 * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.8e-44:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 1.4e-74:
		tmp = (math.sqrt((a * (-3.0 * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.8e-44)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 1.4e-74)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-3.0 * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.8e-44)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 1.4e-74)
		tmp = (sqrt((a * (-3.0 * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.8e-44], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 1.4e-74], N[(N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.80000000000000033e-44

   1. Initial program 60.2%

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

   4. Applied egg-rr 60.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{b}{a}\right)}, -3\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot b}{a}\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot b\right), a\right), -3\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot 2\right), a\right), -3\right) \]

      4. *-lowering-*.f64 90.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, 2\right), a\right), -3\right) \]

   7. Simplified 90.5%

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

   if -6.80000000000000033e-44 < b < 1.39999999999999994e-74

   1. Initial program 87.7%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -3\right)\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(a \cdot \left(c \cdot -3\right)\right)\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(a \cdot \left(-3 \cdot c\right)\right)\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(-3 \cdot c\right)\right)\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot -3\right)\right)\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

      6. *-lowering-*.f64 84.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right)\right), \mathsf{*.f64}\left(3, a\right)\right) \]

   5. Simplified 84.5%

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

   if 1.39999999999999994e-74 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 90.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-44)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 5.8e-79)
     (/ (/ (- b (sqrt (* c (* a -3.0)))) a) -3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-44) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 5.8e-79) {
		tmp = ((b - sqrt((c * (a * -3.0)))) / a) / -3.0;
	} 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 <= (-5d-44)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 5.8d-79) then
        tmp = ((b - sqrt((c * (a * (-3.0d0))))) / a) / (-3.0d0)
    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 <= -5e-44) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 5.8e-79) {
		tmp = ((b - Math.sqrt((c * (a * -3.0)))) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-44:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 5.8e-79:
		tmp = ((b - math.sqrt((c * (a * -3.0)))) / a) / -3.0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-44)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 5.8e-79)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) / a) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-44)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 5.8e-79)
		tmp = ((b - sqrt((c * (a * -3.0)))) / a) / -3.0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-44], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 5.8e-79], N[(N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000039e-44

   1. Initial program 60.2%

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

   4. Applied egg-rr 60.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{b}{a}\right)}, -3\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot b}{a}\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot b\right), a\right), -3\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot 2\right), a\right), -3\right) \]

      4. *-lowering-*.f64 90.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, 2\right), a\right), -3\right) \]

   7. Simplified 90.5%

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

   if -5.00000000000000039e-44 < b < 5.8000000000000001e-79

   1. Initial program 87.7%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right)\right), a\right), -3\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -3\right)\right)\right), a\right), -3\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(c \cdot a\right) \cdot -3\right)\right)\right), a\right), -3\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -3\right)\right)\right)\right), a\right), -3\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-3 \cdot a\right)\right)\right)\right), a\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-3 \cdot a\right)\right)\right)\right), a\right), -3\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -3\right)\right)\right)\right), a\right), -3\right) \]

      7. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), a\right), -3\right) \]

   7. Simplified 84.4%

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

   if 5.8000000000000001e-79 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 90.4%

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

Alternative 6: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-73}:\\ \;\;\;\;\frac{b - \sqrt{-3 \cdot \left(a \cdot c\right)}}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-44)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 1.26e-73)
     (/ (- b (sqrt (* -3.0 (* a c)))) (* a -3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-44) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.26e-73) {
		tmp = (b - sqrt((-3.0 * (a * c)))) / (a * -3.0);
	} 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 <= (-3.7d-44)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 1.26d-73) then
        tmp = (b - sqrt(((-3.0d0) * (a * c)))) / (a * (-3.0d0))
    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 <= -3.7e-44) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.26e-73) {
		tmp = (b - Math.sqrt((-3.0 * (a * c)))) / (a * -3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-44:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 1.26e-73:
		tmp = (b - math.sqrt((-3.0 * (a * c)))) / (a * -3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-44)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 1.26e-73)
		tmp = Float64(Float64(b - sqrt(Float64(-3.0 * Float64(a * c)))) / Float64(a * -3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.7e-44)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 1.26e-73)
		tmp = (b - sqrt((-3.0 * (a * c)))) / (a * -3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-44], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 1.26e-73], N[(N[(b - N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7e-44

   1. Initial program 60.2%

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

   4. Applied egg-rr 60.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{b}{a}\right)}, -3\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot b}{a}\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot b\right), a\right), -3\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot 2\right), a\right), -3\right) \]

      4. *-lowering-*.f64 90.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, 2\right), a\right), -3\right) \]

   7. Simplified 90.5%

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

   if -3.7e-44 < b < 1.26000000000000001e-73

   1. Initial program 87.7%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right)\right), a\right), -3\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -3\right)\right)\right), a\right), -3\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(c \cdot a\right) \cdot -3\right)\right)\right), a\right), -3\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -3\right)\right)\right)\right), a\right), -3\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-3 \cdot a\right)\right)\right)\right), a\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-3 \cdot a\right)\right)\right)\right), a\right), -3\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -3\right)\right)\right)\right), a\right), -3\right) \]

      7. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), a\right), -3\right) \]

   7. Simplified 84.4%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right), \color{blue}{\left(a \cdot -3\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{c \cdot \left(a \cdot -3\right)}\right)\right), \left(\color{blue}{a} \cdot -3\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{\left(c \cdot a\right) \cdot -3}\right)\right), \left(a \cdot -3\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{\left(a \cdot c\right) \cdot -3}\right)\right), \left(a \cdot -3\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{-3 \cdot \left(a \cdot c\right)}\right)\right), \left(a \cdot -3\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-3 \cdot \left(a \cdot c\right)\right)\right)\right), \left(a \cdot -3\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \left(a \cdot c\right)\right)\right)\right), \left(a \cdot -3\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(a, c\right)\right)\right)\right), \left(a \cdot -3\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(a, c\right)\right)\right)\right), \left(-3 \cdot \color{blue}{a}\right)\right) \]

      11. *-lowering-*.f64 84.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(a, c\right)\right)\right)\right), \mathsf{*.f64}\left(-3, \color{blue}{a}\right)\right) \]

   9. Applied egg-rr 84.4%

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

   if 1.26000000000000001e-73 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 90.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-73}:\\ \;\;\;\;\frac{b - \sqrt{-3 \cdot \left(a \cdot c\right)}}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-73}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-43)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 1.28e-73)
     (/ (* -0.3333333333333333 (- b (sqrt (* c (* a -3.0))))) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-43) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.28e-73) {
		tmp = (-0.3333333333333333 * (b - sqrt((c * (a * -3.0))))) / 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 <= (-1.8d-43)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 1.28d-73) then
        tmp = ((-0.3333333333333333d0) * (b - sqrt((c * (a * (-3.0d0)))))) / 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 <= -1.8e-43) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.28e-73) {
		tmp = (-0.3333333333333333 * (b - Math.sqrt((c * (a * -3.0))))) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.8e-43:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 1.28e-73:
		tmp = (-0.3333333333333333 * (b - math.sqrt((c * (a * -3.0))))) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e-43)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 1.28e-73)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(Float64(c * Float64(a * -3.0))))) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.8e-43)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 1.28e-73)
		tmp = (-0.3333333333333333 * (b - sqrt((c * (a * -3.0))))) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e-43], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 1.28e-73], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7999999999999999e-43

   1. Initial program 60.2%

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

   4. Applied egg-rr 60.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{b}{a}\right)}, -3\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot b}{a}\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot b\right), a\right), -3\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot 2\right), a\right), -3\right) \]

      4. *-lowering-*.f64 90.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, 2\right), a\right), -3\right) \]

   7. Simplified 90.5%

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

   if -1.7999999999999999e-43 < b < 1.2799999999999999e-73

   1. Initial program 87.7%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right)\right), a\right), -3\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -3\right)\right)\right), a\right), -3\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(c \cdot a\right) \cdot -3\right)\right)\right), a\right), -3\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -3\right)\right)\right)\right), a\right), -3\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-3 \cdot a\right)\right)\right)\right), a\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-3 \cdot a\right)\right)\right)\right), a\right), -3\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -3\right)\right)\right)\right), a\right), -3\right) \]

      7. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), a\right), -3\right) \]

   7. Simplified 84.4%

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

      1. div-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\right), \color{blue}{\left(\frac{1}{-3}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right), a\right), \left(\frac{\color{blue}{1}}{-3}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{c \cdot \left(a \cdot -3\right)}\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{\left(c \cdot a\right) \cdot -3}\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{\left(a \cdot c\right) \cdot -3}\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{-3 \cdot \left(a \cdot c\right)}\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-3 \cdot \left(a \cdot c\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \left(a \cdot c\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(a, c\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      11. metadata-eval 84.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(a, c\right)\right)\right)\right), a\right), \frac{-1}{3}\right) \]

   9. Applied egg-rr 84.2%

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

       1. associate-*l/ N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(\left(b - \sqrt{-3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-1}{3}\right), \color{blue}{a}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(b - \sqrt{-3 \cdot \left(a \cdot c\right)}\right), \frac{-1}{3}\right), a\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{-3 \cdot \left(a \cdot c\right)}\right)\right), \frac{-1}{3}\right), a\right) \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-3 \cdot \left(a \cdot c\right)\right)\right)\right), \frac{-1}{3}\right), a\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(-3 \cdot a\right) \cdot c\right)\right)\right), \frac{-1}{3}\right), a\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-3 \cdot a\right)\right)\right)\right), \frac{-1}{3}\right), a\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-3 \cdot a\right)\right)\right)\right), \frac{-1}{3}\right), a\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -3\right)\right)\right)\right), \frac{-1}{3}\right), a\right) \]

       10. *-lowering-*.f64 84.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), \frac{-1}{3}\right), a\right) \]

   11. Applied egg-rr 84.4%

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

   if 1.2799999999999999e-73 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 90.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-73}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 3.05 \cdot 10^{-75}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{a}{b - \sqrt{-3 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.6e-44)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 3.05e-75)
     (/ -0.3333333333333333 (/ a (- b (sqrt (* -3.0 (* a c))))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e-44) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 3.05e-75) {
		tmp = -0.3333333333333333 / (a / (b - sqrt((-3.0 * (a * c)))));
	} 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 <= (-6.6d-44)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 3.05d-75) then
        tmp = (-0.3333333333333333d0) / (a / (b - sqrt(((-3.0d0) * (a * c)))))
    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 <= -6.6e-44) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 3.05e-75) {
		tmp = -0.3333333333333333 / (a / (b - Math.sqrt((-3.0 * (a * c)))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.6e-44:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 3.05e-75:
		tmp = -0.3333333333333333 / (a / (b - math.sqrt((-3.0 * (a * c)))))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.6e-44)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 3.05e-75)
		tmp = Float64(-0.3333333333333333 / Float64(a / Float64(b - sqrt(Float64(-3.0 * Float64(a * c))))));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.6e-44)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 3.05e-75)
		tmp = -0.3333333333333333 / (a / (b - sqrt((-3.0 * (a * c)))));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.6e-44], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 3.05e-75], N[(-0.3333333333333333 / N[(a / N[(b - N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.60000000000000011e-44

   1. Initial program 60.2%

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

   4. Applied egg-rr 60.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{b}{a}\right)}, -3\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot b}{a}\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot b\right), a\right), -3\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot 2\right), a\right), -3\right) \]

      4. *-lowering-*.f64 90.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, 2\right), a\right), -3\right) \]

   7. Simplified 90.5%

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

   if -6.60000000000000011e-44 < b < 3.05000000000000021e-75

   1. Initial program 87.7%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right)\right), a\right), -3\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -3\right)\right)\right), a\right), -3\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(c \cdot a\right) \cdot -3\right)\right)\right), a\right), -3\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -3\right)\right)\right)\right), a\right), -3\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-3 \cdot a\right)\right)\right)\right), a\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-3 \cdot a\right)\right)\right)\right), a\right), -3\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -3\right)\right)\right)\right), a\right), -3\right) \]

      7. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), a\right), -3\right) \]

   7. Simplified 84.4%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{-3}\right), \color{blue}{\left(\frac{a}{b - \sqrt{c \cdot \left(a \cdot -3\right)}}\right)}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \left(\frac{\color{blue}{a}}{b - \sqrt{c \cdot \left(a \cdot -3\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(a, \color{blue}{\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b, \color{blue}{\left(\sqrt{c \cdot \left(a \cdot -3\right)}\right)}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b, \left(\sqrt{\left(c \cdot a\right) \cdot -3}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b, \left(\sqrt{\left(a \cdot c\right) \cdot -3}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b, \left(\sqrt{-3 \cdot \left(a \cdot c\right)}\right)\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-3 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 84.3%

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 84.3%

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

   if 3.05000000000000021e-75 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 90.4%

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

Alternative 9: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-43)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 2.6e-73)
     (* -0.3333333333333333 (/ (- b (sqrt (* -3.0 (* a c)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-43) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.6e-73) {
		tmp = -0.3333333333333333 * ((b - sqrt((-3.0 * (a * c)))) / 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 <= (-1.15d-43)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 2.6d-73) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt(((-3.0d0) * (a * c)))) / 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 <= -1.15e-43) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.6e-73) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt((-3.0 * (a * c)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-43:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 2.6e-73:
		tmp = -0.3333333333333333 * ((b - math.sqrt((-3.0 * (a * c)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-43)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 2.6e-73)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(-3.0 * Float64(a * c)))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-43)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 2.6e-73)
		tmp = -0.3333333333333333 * ((b - sqrt((-3.0 * (a * c)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-43], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 2.6e-73], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1499999999999999e-43

   1. Initial program 60.2%

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

   4. Applied egg-rr 60.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{b}{a}\right)}, -3\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot b}{a}\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot b\right), a\right), -3\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot 2\right), a\right), -3\right) \]

      4. *-lowering-*.f64 90.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, 2\right), a\right), -3\right) \]

   7. Simplified 90.5%

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

   if -1.1499999999999999e-43 < b < 2.6000000000000001e-73

   1. Initial program 87.7%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-3 \cdot \left(a \cdot c\right)\right)}\right)\right), a\right), -3\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -3\right)\right)\right), a\right), -3\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(c \cdot a\right) \cdot -3\right)\right)\right), a\right), -3\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -3\right)\right)\right)\right), a\right), -3\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-3 \cdot a\right)\right)\right)\right), a\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-3 \cdot a\right)\right)\right)\right), a\right), -3\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -3\right)\right)\right)\right), a\right), -3\right) \]

      7. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), a\right), -3\right) \]

   7. Simplified 84.4%

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

      1. div-inv N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\right), \color{blue}{\left(\frac{1}{-3}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right), a\right), \left(\frac{\color{blue}{1}}{-3}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{c \cdot \left(a \cdot -3\right)}\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{\left(c \cdot a\right) \cdot -3}\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{\left(a \cdot c\right) \cdot -3}\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\sqrt{-3 \cdot \left(a \cdot c\right)}\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-3 \cdot \left(a \cdot c\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \left(a \cdot c\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(a, c\right)\right)\right)\right), a\right), \left(\frac{1}{-3}\right)\right) \]

      11. metadata-eval 84.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-3, \mathsf{*.f64}\left(a, c\right)\right)\right)\right), a\right), \frac{-1}{3}\right) \]

   9. Applied egg-rr 84.2%

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

   if 2.6000000000000001e-73 < b

   1. Initial program 16.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 90.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 90.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-299}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.7e-299) (/ (/ (* b 2.0) a) -3.0) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.7e-299) {
		tmp = ((b * 2.0) / a) / -3.0;
	} 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 <= 1.7d-299) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    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 <= 1.7e-299) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.7e-299:
		tmp = ((b * 2.0) / a) / -3.0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.7e-299)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.7e-299)
		tmp = ((b * 2.0) / a) / -3.0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.7e-299], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6999999999999999e-299

   1. Initial program 71.6%

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

   4. Applied egg-rr 71.5%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{b}{a}\right)}, -3\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot b}{a}\right), -3\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot b\right), a\right), -3\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot 2\right), a\right), -3\right) \]

      4. *-lowering-*.f64 63.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, 2\right), a\right), -3\right) \]

   7. Simplified 63.1%

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

   if 1.6999999999999999e-299 < b

   1. Initial program 33.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 70.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 70.5%

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

Alternative 11: 67.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-299}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.55e-299) (* (/ b a) -0.6666666666666666) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.55e-299) {
		tmp = (b / a) * -0.6666666666666666;
	} 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 <= 1.55d-299) then
        tmp = (b / a) * (-0.6666666666666666d0)
    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 <= 1.55e-299) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.55e-299:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.55e-299)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.55e-299)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.55e-299], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.55e-299

   1. Initial program 71.6%

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{b}{a}\right), \color{blue}{\frac{-2}{3}}\right) \]

      3. /-lowering-/.f64 63.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, a\right), \frac{-2}{3}\right) \]

   5. Simplified 63.0%

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

   if 1.55e-299 < b

   1. Initial program 33.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 70.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 70.5%

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

Alternative 12: 67.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-299}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.4e-299) (* (/ b a) -0.6666666666666666) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.4e-299) {
		tmp = (b / a) * -0.6666666666666666;
	} 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 <= 1.4d-299) then
        tmp = (b / a) * (-0.6666666666666666d0)
    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 <= 1.4e-299) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.4e-299:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.4e-299)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	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 <= 1.4e-299)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.4e-299], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4000000000000001e-299

   1. Initial program 71.6%

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{b}{a}\right), \color{blue}{\frac{-2}{3}}\right) \]

      3. /-lowering-/.f64 63.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, a\right), \frac{-2}{3}\right) \]

   5. Simplified 63.0%

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

   if 1.4000000000000001e-299 < b

   1. Initial program 33.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 70.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 70.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \color{blue}{c}\right) \]

      4. /-lowering-/.f64 70.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), c\right) \]

   7. Applied egg-rr 70.3%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-299}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-299}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.55e-299) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.55e-299) {
		tmp = b * (-0.6666666666666666 / 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 <= 1.55d-299) then
        tmp = b * ((-0.6666666666666666d0) / 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 <= 1.55e-299) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.55e-299:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.55e-299)
		tmp = Float64(b * Float64(-0.6666666666666666 / 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 <= 1.55e-299)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.55e-299], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.55e-299

   1. Initial program 71.6%

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{b}{a}\right), \color{blue}{\frac{-2}{3}}\right) \]

      3. /-lowering-/.f64 63.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, a\right), \frac{-2}{3}\right) \]

   5. Simplified 63.0%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{\frac{-2}{3}}{a}\right)}\right) \]

      4. /-lowering-/.f64 62.9%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{/.f64}\left(\frac{-2}{3}, \color{blue}{a}\right)\right) \]

   7. Applied egg-rr 62.9%

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

   if 1.55e-299 < b

   1. Initial program 33.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

      4. *-lowering-*.f64 70.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

   5. Simplified 70.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \color{blue}{c}\right) \]

      4. /-lowering-/.f64 70.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), c\right) \]

   7. Applied egg-rr 70.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-299}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / 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 * ((-0.6666666666666666d0) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.3%

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

3. Taylor expanded in b around -inf

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

   1. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{b}{a}\right), \color{blue}{\frac{-2}{3}}\right) \]

   3. /-lowering-/.f64 31.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, a\right), \frac{-2}{3}\right) \]

5. Simplified 31.2%

   \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
6. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{\frac{-2}{3}}{a}\right)}\right) \]

   4. /-lowering-/.f64 31.2%

      \[\leadsto \mathsf{*.f64}\left(b, \mathsf{/.f64}\left(\frac{-2}{3}, \color{blue}{a}\right)\right) \]

7. Applied egg-rr 31.2%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))