Cubic critical

Percentage Accurate: 51.6% → 85.7%

Time: 8.0s

Alternatives: 11

Speedup: 2.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+117)
   (/ (/ (- (- b) b) a) 3.0)
   (if (<= b 6.2e-57)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+117) {
		tmp = ((-b - b) / a) / 3.0;
	} else if (b <= 6.2e-57) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d+117)) then
        tmp = ((-b - b) / a) / 3.0d0
    else if (b <= 6.2d-57) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+117) {
		tmp = ((-b - b) / a) / 3.0;
	} else if (b <= 6.2e-57) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+117:
		tmp = ((-b - b) / a) / 3.0
	elif b <= 6.2e-57:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+117)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / a) / 3.0);
	elseif (b <= 6.2e-57)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+117)
		tmp = ((-b - b) / a) / 3.0;
	elseif (b <= 6.2e-57)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+117], N[(N[(N[((-b) - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 6.2e-57], 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[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e117

   1. Initial program 43.3%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 43.5

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

   5. Applied rewrites 43.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 43.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot b} - b}{a}}{3} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.4

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

   10. Applied rewrites 96.4%

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

   if -2.0000000000000001e117 < b < 6.19999999999999952e-57

   1. Initial program 81.5%

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

   if 6.19999999999999952e-57 < b

   1. Initial program 10.7%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+117)
   (/ (/ (- (- b) b) a) 3.0)
   (if (<= b 6.2e-57)
     (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+117) {
		tmp = ((-b - b) / a) / 3.0;
	} else if (b <= 6.2e-57) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+117)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / a) / 3.0);
	elseif (b <= 6.2e-57)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+117], N[(N[(N[((-b) - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 6.2e-57], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e117

   1. Initial program 43.3%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 43.5

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

   5. Applied rewrites 43.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 43.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot b} - b}{a}}{3} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.4

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

   10. Applied rewrites 96.4%

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

   if -2.0000000000000001e117 < b < 6.19999999999999952e-57

   1. Initial program 81.5%

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

   4. Applied rewrites 81.5%

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

   if 6.19999999999999952e-57 < b

   1. Initial program 10.7%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e+153)
   (/ (/ (- (- b) b) a) 3.0)
   (if (<= b 6.2e-57)
     (/ (* 0.3333333333333333 (- (sqrt (fma (* -3.0 c) a (* b b))) b)) a)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e+153) {
		tmp = ((-b - b) / a) / 3.0;
	} else if (b <= 6.2e-57) {
		tmp = (0.3333333333333333 * (sqrt(fma((-3.0 * c), a, (b * b))) - b)) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e+153)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / a) / 3.0);
	elseif (b <= 6.2e-57)
		tmp = Float64(Float64(0.3333333333333333 * Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b)) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e+153], N[(N[(N[((-b) - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 6.2e-57], N[(N[(0.3333333333333333 * N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.1999999999999998e153

   1. Initial program 34.3%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 34.4

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

   5. Applied rewrites 34.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 34.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 34.4

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

   7. Applied rewrites 34.4%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot b} - b}{a}}{3} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.9

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

   10. Applied rewrites 97.9%

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

   if -5.1999999999999998e153 < b < 6.19999999999999952e-57

   1. Initial program 82.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 82.0%

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

   if 6.19999999999999952e-57 < b

   1. Initial program 10.7%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e+117)
   (/ (/ (- (- b) b) a) 3.0)
   (if (<= b 6.2e-57)
     (* (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a) 0.3333333333333333)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e+117) {
		tmp = ((-b - b) / a) / 3.0;
	} else if (b <= 6.2e-57) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e+117)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / a) / 3.0);
	elseif (b <= 6.2e-57)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e+117], N[(N[(N[((-b) - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 6.2e-57], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.05e117

   1. Initial program 43.3%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 43.5

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

   5. Applied rewrites 43.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 43.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot b} - b}{a}}{3} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.4

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

   10. Applied rewrites 96.4%

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

   if -2.05e117 < b < 6.19999999999999952e-57

   1. Initial program 81.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 81.4%

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

   if 6.19999999999999952e-57 < b

   1. Initial program 10.7%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-122)
   (/ (/ (- (- b) b) a) 3.0)
   (if (<= b 6.2e-57)
     (/ (- (sqrt (* (* c a) -3.0)) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-122) {
		tmp = ((-b - b) / a) / 3.0;
	} else if (b <= 6.2e-57) {
		tmp = (sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.3d-122)) then
        tmp = ((-b - b) / a) / 3.0d0
    else if (b <= 6.2d-57) then
        tmp = (sqrt(((c * a) * (-3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-122) {
		tmp = ((-b - b) / a) / 3.0;
	} else if (b <= 6.2e-57) {
		tmp = (Math.sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e-122:
		tmp = ((-b - b) / a) / 3.0
	elif b <= 6.2e-57:
		tmp = (math.sqrt(((c * a) * -3.0)) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-122)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / a) / 3.0);
	elseif (b <= 6.2e-57)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -3.0)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e-122)
		tmp = ((-b - b) / a) / 3.0;
	elseif (b <= 6.2e-57)
		tmp = (sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-122], N[(N[(N[((-b) - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[b, 6.2e-57], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.30000000000000007e-122

   1. Initial program 68.0%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 56.9

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

   5. Applied rewrites 56.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 56.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 56.9

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

   7. Applied rewrites 56.9%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot b} - b}{a}}{3} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.8

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

   10. Applied rewrites 85.8%

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

   if -2.30000000000000007e-122 < b < 6.19999999999999952e-57

   1. Initial program 70.5%

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 70.3

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

   7. Applied rewrites 70.3%

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

   if 6.19999999999999952e-57 < b

   1. Initial program 10.7%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 93.3

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

   5. Applied rewrites 93.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (/ (- (- b) b) a) 3.0) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = ((-b - b) / a) / 3.0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = ((-b - b) / a) / 3.0d0
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = ((-b - b) / a) / 3.0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = ((-b - b) / a) / 3.0
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(Float64(Float64(-b) - b) / a) / 3.0);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = ((-b - b) / a) / 3.0;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(N[((-b) - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 69.1%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 45.2

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

   5. Applied rewrites 45.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 45.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 45.2

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

   7. Applied rewrites 45.2%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot b} - b}{a}}{3} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.3

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

   10. Applied rewrites 69.3%

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

   if -1.999999999999994e-310 < b

   1. Initial program 26.6%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (- b) (* 1.5 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = -b / (1.5 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = -b / (1.5d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = -b / (1.5 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -b / (1.5 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(-b) / Float64(1.5 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = -b / (1.5 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[((-b) / N[(1.5 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 69.1%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 69.0

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 69.2%

      \[\leadsto \frac{-b}{\color{blue}{a \cdot 1.5}} \]

   if -1.999999999999994e-310 < b

   1. Initial program 26.6%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (* -0.6666666666666666 b) a) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = ((-0.6666666666666666d0) * b) / a
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (-0.6666666666666666 * b) / a
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(-0.6666666666666666 * b) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (-0.6666666666666666 * b) / a;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 69.1%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 69.0

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 69.1%

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

   if -1.999999999999994e-310 < b

   1. Initial program 26.6%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (* (/ b a) -0.6666666666666666) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 69.1%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 69.0

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

   5. Applied rewrites 69.0%

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

   if -1.999999999999994e-310 < b

   1. Initial program 26.6%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.6e+74) (* (/ b a) -0.6666666666666666) (* (/ 0.5 b) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.6e+74) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (0.5 / b) * c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.6d+74) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = (0.5d0 / b) * c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.6e+74) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (0.5 / b) * c;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.6e+74:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (0.5 / b) * c
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.6e+74)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(0.5 / b) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.6e+74)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (0.5 / b) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.6e+74], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5999999999999997e74

   1. Initial program 63.5%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 49.8

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

   5. Applied rewrites 49.8%

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

   if 4.5999999999999997e74 < b

   1. Initial program 10.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 2.5

          \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, -0.5, \color{blue}{\frac{0.6666666666666666}{a}}\right) \]

   5. Applied rewrites 2.5%

      \[\leadsto \color{blue}{\left(-b\right) \cdot \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, -0.5, \frac{0.6666666666666666}{a}\right)} \]

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 41.9%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 10.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{b} \cdot c \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (0.5d0 / b) * c
end function

Java

public static double code(double a, double b, double c) {
	return (0.5 / b) * c;
}

Python

def code(a, b, c):
	return (0.5 / b) * c

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (0.5 / b) * c;
end

Wolfram

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

Derivation

1. Initial program 48.0%

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

3. Taylor expanded in b around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 35.8

       \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, -0.5, \color{blue}{\frac{0.6666666666666666}{a}}\right) \]

5. Applied rewrites 35.8%

   \[\leadsto \color{blue}{\left(-b\right) \cdot \mathsf{fma}\left(\frac{\frac{c}{b}}{b}, -0.5, \frac{0.6666666666666666}{a}\right)} \]

6. Taylor expanded in c around inf

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

8. Applied rewrites 14.5%

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

Reproduce

herbie shell --seed 2024254 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))