Cubic critical

Percentage Accurate: 51.7% → 84.2%

Time: 14.0s

Alternatives: 12

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: 51.7% 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: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e+153)
   (/ b (* a -1.5))
   (if (<= b 2.4e-8)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e+153) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.4e-8) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7d+153)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 2.4d-8) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e+153) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.4e-8) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7e+153:
		tmp = b / (a * -1.5)
	elif b <= 2.4e-8:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e+153)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 2.4e-8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7e+153)
		tmp = b / (a * -1.5);
	elseif (b <= 2.4e-8)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e+153], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-8], 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 / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.9999999999999998e153

   1. Initial program 29.7%

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

      1. sqr-neg 29.7%

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

      2. sqr-neg 29.7%

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

      3. associate-*l* 29.7%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in b around -inf 95.0%

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

      1. *-commutative 95.0%

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

   7. Simplified 95.0%

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

      1. *-commutative 95.0%

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

      2. clear-num 95.1%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 95.2%

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

   9. Applied egg-rr 95.2%

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

       1. associate-/r/ 95.2%

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

   11. Simplified 95.2%

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

       1. *-commutative 95.2%

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

       2. clear-num 95.0%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       3. un-div-inv 95.1%

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

       4. div-inv 95.3%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       5. metadata-eval 95.3%

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

   13. Applied egg-rr 95.3%

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

   if -6.9999999999999998e153 < b < 2.39999999999999998e-8

   1. Initial program 80.7%

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

   if 2.39999999999999998e-8 < b

   1. Initial program 14.8%

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

      1. sqr-neg 14.8%

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

      2. sqr-neg 14.8%

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

      3. associate-*l* 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in b around inf 88.0%

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

      1. *-commutative 88.0%

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

   7. Simplified 88.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{b}{-a}}{1.5}\\ \mathbf{elif}\;b \leq -3.35 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))))
   (if (<= b -1.1e-28)
     (/ (/ b (- a)) 1.5)
     (if (<= b -3.35e-90)
       t_0
       (if (<= b -5.4e-130)
         (/ b (* a -1.5))
         (if (<= b 6.6e-105)
           t_0
           (if (<= b 8.8e-25)
             (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b)))
             (if (<= b 2.4e-8)
               (/ (- (sqrt (* (* a c) -3.0)) b) (* a 3.0))
               (* (/ c b) -0.5)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	double tmp;
	if (b <= -1.1e-28) {
		tmp = (b / -a) / 1.5;
	} else if (b <= -3.35e-90) {
		tmp = t_0;
	} else if (b <= -5.4e-130) {
		tmp = b / (a * -1.5);
	} else if (b <= 6.6e-105) {
		tmp = t_0;
	} else if (b <= 8.8e-25) {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	} else if (b <= 2.4e-8) {
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    if (b <= (-1.1d-28)) then
        tmp = (b / -a) / 1.5d0
    else if (b <= (-3.35d-90)) then
        tmp = t_0
    else if (b <= (-5.4d-130)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 6.6d-105) then
        tmp = t_0
    else if (b <= 8.8d-25) then
        tmp = c * (((-0.375d0) * (a * (c / (b ** 3.0d0)))) - (0.5d0 / b))
    else if (b <= 2.4d-8) then
        tmp = (sqrt(((a * c) * (-3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	double tmp;
	if (b <= -1.1e-28) {
		tmp = (b / -a) / 1.5;
	} else if (b <= -3.35e-90) {
		tmp = t_0;
	} else if (b <= -5.4e-130) {
		tmp = b / (a * -1.5);
	} else if (b <= 6.6e-105) {
		tmp = t_0;
	} else if (b <= 8.8e-25) {
		tmp = c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
	} else if (b <= 2.4e-8) {
		tmp = (Math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	tmp = 0
	if b <= -1.1e-28:
		tmp = (b / -a) / 1.5
	elif b <= -3.35e-90:
		tmp = t_0
	elif b <= -5.4e-130:
		tmp = b / (a * -1.5)
	elif b <= 6.6e-105:
		tmp = t_0
	elif b <= 8.8e-25:
		tmp = c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))
	elif b <= 2.4e-8:
		tmp = (math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (b <= -1.1e-28)
		tmp = Float64(Float64(b / Float64(-a)) / 1.5);
	elseif (b <= -3.35e-90)
		tmp = t_0;
	elseif (b <= -5.4e-130)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 6.6e-105)
		tmp = t_0;
	elseif (b <= 8.8e-25)
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)));
	elseif (b <= 2.4e-8)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -3.0)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	tmp = 0.0;
	if (b <= -1.1e-28)
		tmp = (b / -a) / 1.5;
	elseif (b <= -3.35e-90)
		tmp = t_0;
	elseif (b <= -5.4e-130)
		tmp = b / (a * -1.5);
	elseif (b <= 6.6e-105)
		tmp = t_0;
	elseif (b <= 8.8e-25)
		tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
	elseif (b <= 2.4e-8)
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.1e-28], N[(N[(b / (-a)), $MachinePrecision] / 1.5), $MachinePrecision], If[LessEqual[b, -3.35e-90], t$95$0, If[LessEqual[b, -5.4e-130], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.6e-105], t$95$0, If[LessEqual[b, 8.8e-25], N[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-8], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if b < -1.09999999999999998e-28

   1. Initial program 60.8%

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

      1. sqr-neg 60.8%

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

      2. sqr-neg 60.8%

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

      3. associate-*l* 60.7%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in b around -inf 85.6%

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

      1. *-commutative 85.6%

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

   7. Simplified 85.6%

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

      1. *-commutative 85.6%

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

      2. clear-num 85.5%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 85.6%

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

   9. Applied egg-rr 85.6%

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

       1. associate-/r/ 85.6%

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

   11. Simplified 85.6%

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

       1. associate-/r/ 85.6%

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

       2. metadata-eval 85.6%

          \[\leadsto \frac{\color{blue}{\frac{1}{-1.5}}}{\frac{a}{b}} \]

       3. associate-/r* 85.7%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

       4. *-commutative 85.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} \cdot -1.5}} \]

       5. associate-*l/ 85.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot -1.5}{b}}} \]

   13. Applied egg-rr 85.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -1.5}{b}}} \]
   14. Step-by-step derivation

       1. *-commutative 85.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{-1.5 \cdot a}}{b}} \]

       2. associate-/l* 85.7%

          \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

       3. metadata-eval 85.7%

          \[\leadsto \frac{1}{\color{blue}{\left(-1.5\right)} \cdot \frac{a}{b}} \]

       4. distribute-lft-neg-in 85.7%

          \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

       5. *-commutative 85.7%

          \[\leadsto \frac{1}{-\color{blue}{\frac{a}{b} \cdot 1.5}} \]

       6. distribute-frac-neg2 85.7%

          \[\leadsto \color{blue}{-\frac{1}{\frac{a}{b} \cdot 1.5}} \]

       7. associate-/r* 85.6%

          \[\leadsto -\color{blue}{\frac{\frac{1}{\frac{a}{b}}}{1.5}} \]

       8. distribute-neg-frac 85.6%

          \[\leadsto \color{blue}{\frac{-\frac{1}{\frac{a}{b}}}{1.5}} \]

       9. distribute-frac-neg2 85.6%

          \[\leadsto \frac{\color{blue}{\frac{1}{-\frac{a}{b}}}}{1.5} \]

       10. distribute-frac-neg2 85.6%

           \[\leadsto \frac{\frac{1}{\color{blue}{\frac{a}{-b}}}}{1.5} \]

       11. associate-/r/ 85.7%

           \[\leadsto \frac{\color{blue}{\frac{1}{a} \cdot \left(-b\right)}}{1.5} \]

       12. distribute-rgt-neg-in 85.7%

           \[\leadsto \frac{\color{blue}{-\frac{1}{a} \cdot b}}{1.5} \]

       13. associate-*l/ 85.7%

           \[\leadsto \frac{-\color{blue}{\frac{1 \cdot b}{a}}}{1.5} \]

       14. *-lft-identity 85.7%

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

       15. distribute-frac-neg2 85.7%

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

   15. Simplified 85.7%

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

   if -1.09999999999999998e-28 < b < -3.3500000000000002e-90 or -5.39999999999999982e-130 < b < 6.5999999999999997e-105

   1. Initial program 83.0%

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

      1. sqr-neg 83.0%

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

      2. sqr-neg 83.0%

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

      3. associate-*l* 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in b around 0 73.7%

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

      1. *-commutative 73.7%

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

      2. *-commutative 73.7%

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

      3. associate-*r* 73.9%

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

   7. Simplified 73.9%

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

      1. +-commutative 73.9%

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

      2. unsub-neg 73.9%

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

   9. Applied egg-rr 73.9%

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

   if -3.3500000000000002e-90 < b < -5.39999999999999982e-130

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

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

      2. sqr-neg 99.7%

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

      3. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in b around -inf 99.5%

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

      1. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-/r/ 98.7%

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

   11. Simplified 98.7%

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

       1. *-commutative 98.7%

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

       2. clear-num 99.0%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       3. un-div-inv 99.5%

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

       4. div-inv 99.7%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       5. metadata-eval 99.7%

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

   13. Applied egg-rr 99.7%

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

   if 6.5999999999999997e-105 < b < 8.8000000000000008e-25

   1. Initial program 36.9%

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

      1. sqr-neg 36.9%

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

      2. sqr-neg 36.9%

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

      3. associate-*l* 36.9%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in c around 0 57.0%

      \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 57.0%

         \[\leadsto c \cdot \left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{3}}\right)} - 0.5 \cdot \frac{1}{b}\right) \]

      2. associate-*r/ 57.0%

         \[\leadsto c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

      3. metadata-eval 57.0%

         \[\leadsto c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{\color{blue}{0.5}}{b}\right) \]

   7. Simplified 57.0%

      \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)} \]

   if 8.8000000000000008e-25 < b < 2.39999999999999998e-8

   1. Initial program 83.7%

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

      1. sqr-neg 83.7%

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

      2. sqr-neg 83.7%

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

      3. associate-*l* 84.0%

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

   3. Simplified 84.0%

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

   5. Taylor expanded in b around 0 83.3%

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

   if 2.39999999999999998e-8 < b

   1. Initial program 14.8%

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

      1. sqr-neg 14.8%

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

      2. sqr-neg 14.8%

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

      3. associate-*l* 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in b around inf 88.0%

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

      1. *-commutative 88.0%

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

   7. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 6 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{b}{-a}}{1.5}\\ \mathbf{elif}\;b \leq -3.35 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b - \sqrt{\left(a \cdot c\right) \cdot -3}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{if}\;b \leq -9.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{b}{-a}}{1.5}\\ \mathbf{elif}\;b \leq -4.8 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (- b (sqrt (* (* a c) -3.0))) (/ -0.3333333333333333 a))))
   (if (<= b -9.2e-31)
     (/ (/ b (- a)) 1.5)
     (if (<= b -4.8e-89)
       t_0
       (if (<= b -2e-129)
         (/ b (* a -1.5))
         (if (<= b 3.7e-113) t_0 (* (/ c b) -0.5)))))))

C

double code(double a, double b, double c) {
	double t_0 = (b - sqrt(((a * c) * -3.0))) * (-0.3333333333333333 / a);
	double tmp;
	if (b <= -9.2e-31) {
		tmp = (b / -a) / 1.5;
	} else if (b <= -4.8e-89) {
		tmp = t_0;
	} else if (b <= -2e-129) {
		tmp = b / (a * -1.5);
	} else if (b <= 3.7e-113) {
		tmp = t_0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b - sqrt(((a * c) * (-3.0d0)))) * ((-0.3333333333333333d0) / a)
    if (b <= (-9.2d-31)) then
        tmp = (b / -a) / 1.5d0
    else if (b <= (-4.8d-89)) then
        tmp = t_0
    else if (b <= (-2d-129)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 3.7d-113) then
        tmp = t_0
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b - Math.sqrt(((a * c) * -3.0))) * (-0.3333333333333333 / a);
	double tmp;
	if (b <= -9.2e-31) {
		tmp = (b / -a) / 1.5;
	} else if (b <= -4.8e-89) {
		tmp = t_0;
	} else if (b <= -2e-129) {
		tmp = b / (a * -1.5);
	} else if (b <= 3.7e-113) {
		tmp = t_0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (b - math.sqrt(((a * c) * -3.0))) * (-0.3333333333333333 / a)
	tmp = 0
	if b <= -9.2e-31:
		tmp = (b / -a) / 1.5
	elif b <= -4.8e-89:
		tmp = t_0
	elif b <= -2e-129:
		tmp = b / (a * -1.5)
	elif b <= 3.7e-113:
		tmp = t_0
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b - sqrt(Float64(Float64(a * c) * -3.0))) * Float64(-0.3333333333333333 / a))
	tmp = 0.0
	if (b <= -9.2e-31)
		tmp = Float64(Float64(b / Float64(-a)) / 1.5);
	elseif (b <= -4.8e-89)
		tmp = t_0;
	elseif (b <= -2e-129)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 3.7e-113)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b - sqrt(((a * c) * -3.0))) * (-0.3333333333333333 / a);
	tmp = 0.0;
	if (b <= -9.2e-31)
		tmp = (b / -a) / 1.5;
	elseif (b <= -4.8e-89)
		tmp = t_0;
	elseif (b <= -2e-129)
		tmp = b / (a * -1.5);
	elseif (b <= 3.7e-113)
		tmp = t_0;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -9.2e-31], N[(N[(b / (-a)), $MachinePrecision] / 1.5), $MachinePrecision], If[LessEqual[b, -4.8e-89], t$95$0, If[LessEqual[b, -2e-129], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.7e-113], t$95$0, N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -9.1999999999999994e-31

   1. Initial program 60.8%

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

      1. sqr-neg 60.8%

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

      2. sqr-neg 60.8%

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

      3. associate-*l* 60.7%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in b around -inf 85.6%

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

      1. *-commutative 85.6%

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

   7. Simplified 85.6%

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

      1. *-commutative 85.6%

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

      2. clear-num 85.5%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 85.6%

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

   9. Applied egg-rr 85.6%

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

       1. associate-/r/ 85.6%

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

   11. Simplified 85.6%

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

       1. associate-/r/ 85.6%

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

       2. metadata-eval 85.6%

          \[\leadsto \frac{\color{blue}{\frac{1}{-1.5}}}{\frac{a}{b}} \]

       3. associate-/r* 85.7%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

       4. *-commutative 85.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} \cdot -1.5}} \]

       5. associate-*l/ 85.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot -1.5}{b}}} \]

   13. Applied egg-rr 85.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -1.5}{b}}} \]
   14. Step-by-step derivation

       1. *-commutative 85.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{-1.5 \cdot a}}{b}} \]

       2. associate-/l* 85.7%

          \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

       3. metadata-eval 85.7%

          \[\leadsto \frac{1}{\color{blue}{\left(-1.5\right)} \cdot \frac{a}{b}} \]

       4. distribute-lft-neg-in 85.7%

          \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

       5. *-commutative 85.7%

          \[\leadsto \frac{1}{-\color{blue}{\frac{a}{b} \cdot 1.5}} \]

       6. distribute-frac-neg2 85.7%

          \[\leadsto \color{blue}{-\frac{1}{\frac{a}{b} \cdot 1.5}} \]

       7. associate-/r* 85.6%

          \[\leadsto -\color{blue}{\frac{\frac{1}{\frac{a}{b}}}{1.5}} \]

       8. distribute-neg-frac 85.6%

          \[\leadsto \color{blue}{\frac{-\frac{1}{\frac{a}{b}}}{1.5}} \]

       9. distribute-frac-neg2 85.6%

          \[\leadsto \frac{\color{blue}{\frac{1}{-\frac{a}{b}}}}{1.5} \]

       10. distribute-frac-neg2 85.6%

           \[\leadsto \frac{\frac{1}{\color{blue}{\frac{a}{-b}}}}{1.5} \]

       11. associate-/r/ 85.7%

           \[\leadsto \frac{\color{blue}{\frac{1}{a} \cdot \left(-b\right)}}{1.5} \]

       12. distribute-rgt-neg-in 85.7%

           \[\leadsto \frac{\color{blue}{-\frac{1}{a} \cdot b}}{1.5} \]

       13. associate-*l/ 85.7%

           \[\leadsto \frac{-\color{blue}{\frac{1 \cdot b}{a}}}{1.5} \]

       14. *-lft-identity 85.7%

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

       15. distribute-frac-neg2 85.7%

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

   15. Simplified 85.7%

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

   if -9.1999999999999994e-31 < b < -4.80000000000000032e-89 or -1.9999999999999999e-129 < b < 3.6999999999999998e-113

   1. Initial program 83.0%

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

      1. sqr-neg 83.0%

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

      2. sqr-neg 83.0%

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

      3. associate-*l* 82.8%

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

   3. Simplified 82.8%

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

      1. frac-2neg 82.8%

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

      2. div-inv 82.8%

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

   6. Applied egg-rr 82.9%

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

   7. Taylor expanded in a around 0 82.7%

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

   8. Taylor expanded in b around 0 73.6%

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

   if -4.80000000000000032e-89 < b < -1.9999999999999999e-129

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

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

      2. sqr-neg 99.7%

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

      3. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in b around -inf 99.5%

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

      1. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-/r/ 98.7%

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

   11. Simplified 98.7%

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

       1. *-commutative 98.7%

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

       2. clear-num 99.0%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       3. un-div-inv 99.5%

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

       4. div-inv 99.7%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       5. metadata-eval 99.7%

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

   13. Applied egg-rr 99.7%

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

   if 3.6999999999999998e-113 < b

   1. Initial program 22.2%

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

      1. sqr-neg 22.2%

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

      2. sqr-neg 22.2%

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

      3. associate-*l* 22.1%

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

   3. Simplified 22.1%

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

   5. Taylor expanded in b around inf 78.7%

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

      1. *-commutative 78.7%

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

   7. Simplified 78.7%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{b}{-a}}{1.5}\\ \mathbf{elif}\;b \leq -4.8 \cdot 10^{-89}:\\ \;\;\;\;\left(b - \sqrt{\left(a \cdot c\right) \cdot -3}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-113}:\\ \;\;\;\;\left(b - \sqrt{\left(a \cdot c\right) \cdot -3}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{b}{-a}}{1.5}\\ \mathbf{elif}\;b \leq -2.9 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-106}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))))
   (if (<= b -2.5e-30)
     (/ (/ b (- a)) 1.5)
     (if (<= b -2.9e-90)
       t_0
       (if (<= b -2e-129)
         (/ b (* a -1.5))
         (if (<= b 1.3e-106) t_0 (* (/ c b) -0.5)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	double tmp;
	if (b <= -2.5e-30) {
		tmp = (b / -a) / 1.5;
	} else if (b <= -2.9e-90) {
		tmp = t_0;
	} else if (b <= -2e-129) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.3e-106) {
		tmp = t_0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    if (b <= (-2.5d-30)) then
        tmp = (b / -a) / 1.5d0
    else if (b <= (-2.9d-90)) then
        tmp = t_0
    else if (b <= (-2d-129)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.3d-106) then
        tmp = t_0
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	double tmp;
	if (b <= -2.5e-30) {
		tmp = (b / -a) / 1.5;
	} else if (b <= -2.9e-90) {
		tmp = t_0;
	} else if (b <= -2e-129) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.3e-106) {
		tmp = t_0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	tmp = 0
	if b <= -2.5e-30:
		tmp = (b / -a) / 1.5
	elif b <= -2.9e-90:
		tmp = t_0
	elif b <= -2e-129:
		tmp = b / (a * -1.5)
	elif b <= 1.3e-106:
		tmp = t_0
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (b <= -2.5e-30)
		tmp = Float64(Float64(b / Float64(-a)) / 1.5);
	elseif (b <= -2.9e-90)
		tmp = t_0;
	elseif (b <= -2e-129)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.3e-106)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	tmp = 0.0;
	if (b <= -2.5e-30)
		tmp = (b / -a) / 1.5;
	elseif (b <= -2.9e-90)
		tmp = t_0;
	elseif (b <= -2e-129)
		tmp = b / (a * -1.5);
	elseif (b <= 1.3e-106)
		tmp = t_0;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.5e-30], N[(N[(b / (-a)), $MachinePrecision] / 1.5), $MachinePrecision], If[LessEqual[b, -2.9e-90], t$95$0, If[LessEqual[b, -2e-129], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e-106], t$95$0, N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.49999999999999986e-30

   1. Initial program 60.8%

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

      1. sqr-neg 60.8%

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

      2. sqr-neg 60.8%

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

      3. associate-*l* 60.7%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in b around -inf 85.6%

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

      1. *-commutative 85.6%

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

   7. Simplified 85.6%

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

      1. *-commutative 85.6%

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

      2. clear-num 85.5%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 85.6%

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

   9. Applied egg-rr 85.6%

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

       1. associate-/r/ 85.6%

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

   11. Simplified 85.6%

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

       1. associate-/r/ 85.6%

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

       2. metadata-eval 85.6%

          \[\leadsto \frac{\color{blue}{\frac{1}{-1.5}}}{\frac{a}{b}} \]

       3. associate-/r* 85.7%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

       4. *-commutative 85.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} \cdot -1.5}} \]

       5. associate-*l/ 85.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot -1.5}{b}}} \]

   13. Applied egg-rr 85.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -1.5}{b}}} \]
   14. Step-by-step derivation

       1. *-commutative 85.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{-1.5 \cdot a}}{b}} \]

       2. associate-/l* 85.7%

          \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

       3. metadata-eval 85.7%

          \[\leadsto \frac{1}{\color{blue}{\left(-1.5\right)} \cdot \frac{a}{b}} \]

       4. distribute-lft-neg-in 85.7%

          \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

       5. *-commutative 85.7%

          \[\leadsto \frac{1}{-\color{blue}{\frac{a}{b} \cdot 1.5}} \]

       6. distribute-frac-neg2 85.7%

          \[\leadsto \color{blue}{-\frac{1}{\frac{a}{b} \cdot 1.5}} \]

       7. associate-/r* 85.6%

          \[\leadsto -\color{blue}{\frac{\frac{1}{\frac{a}{b}}}{1.5}} \]

       8. distribute-neg-frac 85.6%

          \[\leadsto \color{blue}{\frac{-\frac{1}{\frac{a}{b}}}{1.5}} \]

       9. distribute-frac-neg2 85.6%

          \[\leadsto \frac{\color{blue}{\frac{1}{-\frac{a}{b}}}}{1.5} \]

       10. distribute-frac-neg2 85.6%

           \[\leadsto \frac{\frac{1}{\color{blue}{\frac{a}{-b}}}}{1.5} \]

       11. associate-/r/ 85.7%

           \[\leadsto \frac{\color{blue}{\frac{1}{a} \cdot \left(-b\right)}}{1.5} \]

       12. distribute-rgt-neg-in 85.7%

           \[\leadsto \frac{\color{blue}{-\frac{1}{a} \cdot b}}{1.5} \]

       13. associate-*l/ 85.7%

           \[\leadsto \frac{-\color{blue}{\frac{1 \cdot b}{a}}}{1.5} \]

       14. *-lft-identity 85.7%

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

       15. distribute-frac-neg2 85.7%

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

   15. Simplified 85.7%

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

   if -2.49999999999999986e-30 < b < -2.89999999999999983e-90 or -1.9999999999999999e-129 < b < 1.3e-106

   1. Initial program 83.0%

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

      1. sqr-neg 83.0%

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

      2. sqr-neg 83.0%

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

      3. associate-*l* 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in b around 0 73.7%

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

      1. *-commutative 73.7%

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

      2. *-commutative 73.7%

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

      3. associate-*r* 73.9%

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

   7. Simplified 73.9%

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

      1. +-commutative 73.9%

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

      2. unsub-neg 73.9%

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

   9. Applied egg-rr 73.9%

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

   if -2.89999999999999983e-90 < b < -1.9999999999999999e-129

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

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

      2. sqr-neg 99.7%

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

      3. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in b around -inf 99.5%

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

      1. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. clear-num 99.5%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-/r/ 98.7%

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

   11. Simplified 98.7%

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

       1. *-commutative 98.7%

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

       2. clear-num 99.0%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       3. un-div-inv 99.5%

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

       4. div-inv 99.7%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       5. metadata-eval 99.7%

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

   13. Applied egg-rr 99.7%

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

   if 1.3e-106 < b

   1. Initial program 22.2%

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

      1. sqr-neg 22.2%

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

      2. sqr-neg 22.2%

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

      3. associate-*l* 22.1%

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

   3. Simplified 22.1%

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

   5. Taylor expanded in b around inf 78.7%

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

      1. *-commutative 78.7%

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

   7. Simplified 78.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{b}{-a}}{1.5}\\ \mathbf{elif}\;b \leq -2.9 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+154)
   (/ b (* a -1.5))
   (if (<= b 2.4e-8)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+154) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.4e-8) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.5d+154)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 2.4d-8) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+154) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.4e-8) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e+154:
		tmp = b / (a * -1.5)
	elif b <= 2.4e-8:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+154)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 2.4e-8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.5e+154)
		tmp = b / (a * -1.5);
	elseif (b <= 2.4e-8)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+154], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-8], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.50000000000000013e154

   1. Initial program 29.7%

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

      1. sqr-neg 29.7%

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

      2. sqr-neg 29.7%

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

      3. associate-*l* 29.7%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in b around -inf 95.0%

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

      1. *-commutative 95.0%

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

   7. Simplified 95.0%

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

      1. *-commutative 95.0%

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

      2. clear-num 95.1%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 95.2%

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

   9. Applied egg-rr 95.2%

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

       1. associate-/r/ 95.2%

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

   11. Simplified 95.2%

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

       1. *-commutative 95.2%

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

       2. clear-num 95.0%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       3. un-div-inv 95.1%

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

       4. div-inv 95.3%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       5. metadata-eval 95.3%

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

   13. Applied egg-rr 95.3%

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

   if -1.50000000000000013e154 < b < 2.39999999999999998e-8

   1. Initial program 80.7%

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

      1. sqr-neg 80.7%

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

      2. sqr-neg 80.7%

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

      3. associate-*l* 80.6%

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

   3. Simplified 80.6%

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

   if 2.39999999999999998e-8 < b

   1. Initial program 14.8%

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

      1. sqr-neg 14.8%

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

      2. sqr-neg 14.8%

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

      3. associate-*l* 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in b around inf 88.0%

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

      1. *-commutative 88.0%

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

   7. Simplified 88.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{b}{-a}}{1.5}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.2e-134)
   (/ (/ b (- a)) 1.5)
   (if (<= b 1.55e-108)
     (* 0.3333333333333333 (/ (+ b (sqrt (* c (* a -3.0)))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.2e-134) {
		tmp = (b / -a) / 1.5;
	} else if (b <= 1.55e-108) {
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9.2d-134)) then
        tmp = (b / -a) / 1.5d0
    else if (b <= 1.55d-108) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((c * (a * (-3.0d0))))) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.2e-134) {
		tmp = (b / -a) / 1.5;
	} else if (b <= 1.55e-108) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.2e-134:
		tmp = (b / -a) / 1.5
	elif b <= 1.55e-108:
		tmp = 0.3333333333333333 * ((b + math.sqrt((c * (a * -3.0)))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.2e-134)
		tmp = Float64(Float64(b / Float64(-a)) / 1.5);
	elseif (b <= 1.55e-108)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.2e-134)
		tmp = (b / -a) / 1.5;
	elseif (b <= 1.55e-108)
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.2e-134], N[(N[(b / (-a)), $MachinePrecision] / 1.5), $MachinePrecision], If[LessEqual[b, 1.55e-108], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.2000000000000001e-134

   1. Initial program 67.7%

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

      1. sqr-neg 67.7%

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

      2. sqr-neg 67.7%

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

      3. associate-*l* 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in b around -inf 79.5%

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

      1. *-commutative 79.5%

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

   7. Simplified 79.5%

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

      1. *-commutative 79.5%

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

      2. clear-num 79.4%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 79.5%

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

   9. Applied egg-rr 79.5%

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

       1. associate-/r/ 79.5%

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

   11. Simplified 79.5%

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

       1. associate-/r/ 79.5%

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

       2. metadata-eval 79.5%

          \[\leadsto \frac{\color{blue}{\frac{1}{-1.5}}}{\frac{a}{b}} \]

       3. associate-/r* 79.6%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

       4. *-commutative 79.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} \cdot -1.5}} \]

       5. associate-*l/ 79.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot -1.5}{b}}} \]

   13. Applied egg-rr 79.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -1.5}{b}}} \]
   14. Step-by-step derivation

       1. *-commutative 79.5%

          \[\leadsto \frac{1}{\frac{\color{blue}{-1.5 \cdot a}}{b}} \]

       2. associate-/l* 79.6%

          \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

       3. metadata-eval 79.6%

          \[\leadsto \frac{1}{\color{blue}{\left(-1.5\right)} \cdot \frac{a}{b}} \]

       4. distribute-lft-neg-in 79.6%

          \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

       5. *-commutative 79.6%

          \[\leadsto \frac{1}{-\color{blue}{\frac{a}{b} \cdot 1.5}} \]

       6. distribute-frac-neg2 79.6%

          \[\leadsto \color{blue}{-\frac{1}{\frac{a}{b} \cdot 1.5}} \]

       7. associate-/r* 79.5%

          \[\leadsto -\color{blue}{\frac{\frac{1}{\frac{a}{b}}}{1.5}} \]

       8. distribute-neg-frac 79.5%

          \[\leadsto \color{blue}{\frac{-\frac{1}{\frac{a}{b}}}{1.5}} \]

       9. distribute-frac-neg2 79.5%

          \[\leadsto \frac{\color{blue}{\frac{1}{-\frac{a}{b}}}}{1.5} \]

       10. distribute-frac-neg2 79.5%

           \[\leadsto \frac{\frac{1}{\color{blue}{\frac{a}{-b}}}}{1.5} \]

       11. associate-/r/ 79.6%

           \[\leadsto \frac{\color{blue}{\frac{1}{a} \cdot \left(-b\right)}}{1.5} \]

       12. distribute-rgt-neg-in 79.6%

           \[\leadsto \frac{\color{blue}{-\frac{1}{a} \cdot b}}{1.5} \]

       13. associate-*l/ 79.7%

           \[\leadsto \frac{-\color{blue}{\frac{1 \cdot b}{a}}}{1.5} \]

       14. *-lft-identity 79.7%

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

       15. distribute-frac-neg2 79.7%

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

   15. Simplified 79.7%

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

   if -9.2000000000000001e-134 < b < 1.55000000000000007e-108

   1. Initial program 78.8%

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

      1. sqr-neg 78.8%

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

      2. sqr-neg 78.8%

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

      3. associate-*l* 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in b around 0 76.8%

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

      1. *-commutative 76.8%

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

      2. *-commutative 76.8%

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

      3. associate-*r* 77.0%

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

   7. Simplified 77.0%

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

      1. *-un-lft-identity 77.0%

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

      2. times-frac 76.9%

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

      3. metadata-eval 76.9%

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

      4. add-sqr-sqrt 36.7%

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

      5. sqrt-unprod 76.4%

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

      6. sqr-neg 76.4%

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

      7. sqrt-unprod 40.3%

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

      8. add-sqr-sqrt 75.9%

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

   9. Applied egg-rr 75.9%

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

   if 1.55000000000000007e-108 < b

   1. Initial program 22.2%

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

      1. sqr-neg 22.2%

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

      2. sqr-neg 22.2%

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

      3. associate-*l* 22.1%

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

   3. Simplified 22.1%

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

   5. Taylor expanded in b around inf 78.7%

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

      1. *-commutative 78.7%

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

   7. Simplified 78.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{b}{-a}}{1.5}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-254}:\\ \;\;\;\;\frac{\frac{b}{-a}}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.4e-254) (/ (/ b (- a)) 1.5) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e-254) {
		tmp = (b / -a) / 1.5;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.4d-254) then
        tmp = (b / -a) / 1.5d0
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e-254) {
		tmp = (b / -a) / 1.5;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.4e-254:
		tmp = (b / -a) / 1.5
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.4e-254)
		tmp = Float64(Float64(b / Float64(-a)) / 1.5);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.4e-254)
		tmp = (b / -a) / 1.5;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.4e-254], N[(N[(b / (-a)), $MachinePrecision] / 1.5), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.4000000000000002e-254

   1. Initial program 70.5%

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

      1. sqr-neg 70.5%

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

      2. sqr-neg 70.5%

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

      3. associate-*l* 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in b around -inf 66.7%

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

      1. *-commutative 66.7%

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

   7. Simplified 66.7%

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

      1. *-commutative 66.7%

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

      2. clear-num 66.6%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 66.7%

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

   9. Applied egg-rr 66.7%

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

       1. associate-/r/ 66.7%

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

   11. Simplified 66.7%

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

       1. associate-/r/ 66.7%

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

       2. metadata-eval 66.7%

          \[\leadsto \frac{\color{blue}{\frac{1}{-1.5}}}{\frac{a}{b}} \]

       3. associate-/r* 66.7%

          \[\leadsto \color{blue}{\frac{1}{-1.5 \cdot \frac{a}{b}}} \]

       4. *-commutative 66.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} \cdot -1.5}} \]

       5. associate-*l/ 66.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot -1.5}{b}}} \]

   13. Applied egg-rr 66.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -1.5}{b}}} \]
   14. Step-by-step derivation

       1. *-commutative 66.7%

          \[\leadsto \frac{1}{\frac{\color{blue}{-1.5 \cdot a}}{b}} \]

       2. associate-/l* 66.7%

          \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

       3. metadata-eval 66.7%

          \[\leadsto \frac{1}{\color{blue}{\left(-1.5\right)} \cdot \frac{a}{b}} \]

       4. distribute-lft-neg-in 66.7%

          \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

       5. *-commutative 66.7%

          \[\leadsto \frac{1}{-\color{blue}{\frac{a}{b} \cdot 1.5}} \]

       6. distribute-frac-neg2 66.7%

          \[\leadsto \color{blue}{-\frac{1}{\frac{a}{b} \cdot 1.5}} \]

       7. associate-/r* 66.7%

          \[\leadsto -\color{blue}{\frac{\frac{1}{\frac{a}{b}}}{1.5}} \]

       8. distribute-neg-frac 66.7%

          \[\leadsto \color{blue}{\frac{-\frac{1}{\frac{a}{b}}}{1.5}} \]

       9. distribute-frac-neg2 66.7%

          \[\leadsto \frac{\color{blue}{\frac{1}{-\frac{a}{b}}}}{1.5} \]

       10. distribute-frac-neg2 66.7%

           \[\leadsto \frac{\frac{1}{\color{blue}{\frac{a}{-b}}}}{1.5} \]

       11. associate-/r/ 66.8%

           \[\leadsto \frac{\color{blue}{\frac{1}{a} \cdot \left(-b\right)}}{1.5} \]

       12. distribute-rgt-neg-in 66.8%

           \[\leadsto \frac{\color{blue}{-\frac{1}{a} \cdot b}}{1.5} \]

       13. associate-*l/ 66.8%

           \[\leadsto \frac{-\color{blue}{\frac{1 \cdot b}{a}}}{1.5} \]

       14. *-lft-identity 66.8%

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

       15. distribute-frac-neg2 66.8%

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

   15. Simplified 66.8%

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

   if 4.4000000000000002e-254 < b

   1. Initial program 30.3%

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

      1. sqr-neg 30.3%

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

      2. sqr-neg 30.3%

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

      3. associate-*l* 30.2%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in b around inf 69.4%

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

      1. *-commutative 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-254}:\\ \;\;\;\;\frac{\frac{b}{-a}}{1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7200000:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7200000.0) (* b (/ -0.6666666666666666 a)) (* (/ c b) 0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7200000.0) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 7200000.0d0) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c / b) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7200000.0) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7200000.0:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * 0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7200000.0)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c / b) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7200000.0)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7200000.0], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.2e6

   1. Initial program 68.0%

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

      1. sqr-neg 68.0%

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

      2. sqr-neg 68.0%

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

      3. associate-*l* 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in b around -inf 53.2%

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

      1. *-commutative 53.2%

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

   7. Simplified 53.2%

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

      1. *-commutative 53.2%

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

      2. clear-num 53.2%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 53.2%

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

   9. Applied egg-rr 53.2%

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

       1. associate-/r/ 53.2%

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

   11. Simplified 53.2%

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

   if 7.2e6 < b

   1. Initial program 15.1%

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

      1. sqr-neg 15.1%

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

      2. sqr-neg 15.1%

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

      3. associate-*l* 15.1%

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

   3. Simplified 15.1%

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

      1. *-un-lft-identity 15.1%

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

      2. times-frac 15.1%

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

      3. metadata-eval 15.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 8.8%

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

      6. sqr-neg 8.8%

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

      7. sqrt-prod 8.8%

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

      8. add-sqr-sqrt 8.8%

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

      9. fma-neg 8.8%

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

      10. associate-*r* 8.8%

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

      11. *-commutative 8.8%

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

      12. distribute-rgt-neg-in 8.8%

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

      13. *-commutative 8.8%

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

      14. distribute-rgt-neg-in 8.8%

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

      15. metadata-eval 8.8%

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

   6. Applied egg-rr 8.8%

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

   7. Taylor expanded in b around -inf 23.0%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7200000:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-254}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.4e-254) (* b (/ -0.6666666666666666 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e-254) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.4d-254) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e-254) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.4e-254:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.4e-254)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.4e-254)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.4e-254], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.4000000000000002e-254

   1. Initial program 70.5%

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

      1. sqr-neg 70.5%

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

      2. sqr-neg 70.5%

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

      3. associate-*l* 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in b around -inf 66.7%

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

      1. *-commutative 66.7%

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

   7. Simplified 66.7%

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

      1. *-commutative 66.7%

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

      2. clear-num 66.6%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 66.7%

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

   9. Applied egg-rr 66.7%

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

       1. associate-/r/ 66.7%

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

   11. Simplified 66.7%

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

   if 4.4000000000000002e-254 < b

   1. Initial program 30.3%

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

      1. sqr-neg 30.3%

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

      2. sqr-neg 30.3%

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

      3. associate-*l* 30.2%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in b around inf 69.4%

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

      1. *-commutative 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-254}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-254}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.4e-254) (/ -0.6666666666666666 (/ a b)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e-254) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.4d-254) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e-254) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.4e-254:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.4e-254)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.4e-254)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.4e-254], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.4000000000000002e-254

   1. Initial program 70.5%

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

      1. sqr-neg 70.5%

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

      2. sqr-neg 70.5%

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

      3. associate-*l* 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in b around -inf 66.7%

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

      1. *-commutative 66.7%

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

   7. Simplified 66.7%

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

      1. *-commutative 66.7%

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

      2. clear-num 66.6%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 66.7%

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

   9. Applied egg-rr 66.7%

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

   if 4.4000000000000002e-254 < b

   1. Initial program 30.3%

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

      1. sqr-neg 30.3%

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

      2. sqr-neg 30.3%

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

      3. associate-*l* 30.2%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in b around inf 69.4%

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

      1. *-commutative 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-254}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-254}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.4e-254) (/ b (* a -1.5)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e-254) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.4d-254) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e-254) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.4e-254:
		tmp = b / (a * -1.5)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.4e-254)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.4e-254)
		tmp = b / (a * -1.5);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.4e-254], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.4000000000000002e-254

   1. Initial program 70.5%

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

      1. sqr-neg 70.5%

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

      2. sqr-neg 70.5%

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

      3. associate-*l* 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in b around -inf 66.7%

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

      1. *-commutative 66.7%

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

   7. Simplified 66.7%

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

      1. *-commutative 66.7%

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

      2. clear-num 66.6%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 66.7%

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

   9. Applied egg-rr 66.7%

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

       1. associate-/r/ 66.7%

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

   11. Simplified 66.7%

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

       1. *-commutative 66.7%

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

       2. clear-num 66.6%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       3. un-div-inv 66.7%

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

       4. div-inv 66.8%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       5. metadata-eval 66.8%

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

   13. Applied egg-rr 66.8%

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

   if 4.4000000000000002e-254 < b

   1. Initial program 30.3%

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

      1. sqr-neg 30.3%

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

      2. sqr-neg 30.3%

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

      3. associate-*l* 30.2%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in b around inf 69.4%

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

      1. *-commutative 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-254}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 10.9% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

   1. sqr-neg 52.7%

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

   2. sqr-neg 52.7%

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

   3. associate-*l* 52.7%

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

3. Simplified 52.7%

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

   1. *-un-lft-identity 52.7%

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

   2. times-frac 52.6%

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

   3. metadata-eval 52.6%

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

   4. add-sqr-sqrt 37.3%

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

   5. sqrt-unprod 49.7%

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

   6. sqr-neg 49.7%

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

   7. sqrt-prod 12.5%

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

   8. add-sqr-sqrt 29.3%

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

   9. fma-neg 29.3%

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

   10. associate-*r* 29.4%

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

   11. *-commutative 29.4%

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

   12. distribute-rgt-neg-in 29.4%

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

   13. *-commutative 29.4%

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

   14. distribute-rgt-neg-in 29.4%

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

   15. metadata-eval 29.4%

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

6. Applied egg-rr 29.4%

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

7. Taylor expanded in b around -inf 9.0%

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

8. Final simplification 9.0%

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

Reproduce

herbie shell --seed 2024072 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))