Cubic critical

Percentage Accurate: 51.3% → 85.6%

Time: 17.3s

Alternatives: 16

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\frac{-1.5 \cdot a}{b}}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.4e+136)
   (/ 1.0 (/ (* -1.5 a) b))
   (if (<= b 7e-53)
     (/ (- (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 <= -7.4e+136) {
		tmp = 1.0 / ((-1.5 * a) / b);
	} else if (b <= 7e-53) {
		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 <= (-7.4d+136)) then
        tmp = 1.0d0 / (((-1.5d0) * a) / b)
    else if (b <= 7d-53) 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 <= -7.4e+136) {
		tmp = 1.0 / ((-1.5 * a) / b);
	} else if (b <= 7e-53) {
		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 <= -7.4e+136:
		tmp = 1.0 / ((-1.5 * a) / b)
	elif b <= 7e-53:
		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 <= -7.4e+136)
		tmp = Float64(1.0 / Float64(Float64(-1.5 * a) / b));
	elseif (b <= 7e-53)
		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 <= -7.4e+136)
		tmp = 1.0 / ((-1.5 * a) / b);
	elseif (b <= 7e-53)
		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, -7.4e+136], N[(1.0 / N[(N[(-1.5 * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-53], 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 < -7.4000000000000002e136

   1. Initial program 58.2%

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

      1. neg-sub0 58.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 58.2%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 58.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 58.2%

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

      5. sub-neg 58.2%

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

      6. distribute-neg-in 58.2%

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

      7. remove-double-neg 58.2%

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

      8. sqr-neg 58.2%

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

      9. associate-*l* 58.2%

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

   3. Simplified 58.2%

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

   6. Applied egg-rr 52.4%

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

      1. clear-num 52.3%

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

      2. frac-times 52.3%

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

      3. metadata-eval 52.3%

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

   8. Applied egg-rr 52.3%

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

   9. Taylor expanded in b around -inf 95.7%

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

       1. associate-*r/ 95.9%

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

   11. Simplified 95.9%

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

   if -7.4000000000000002e136 < b < 6.99999999999999987e-53

   1. Initial program 82.8%

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

   if 6.99999999999999987e-53 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 16.3%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 16.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 16.3%

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

      5. sub-neg 16.3%

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

      6. distribute-neg-in 16.3%

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

      7. remove-double-neg 16.3%

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

      8. sqr-neg 16.3%

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

      9. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around inf 92.7%

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

      1. *-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{\frac{-1.5 \cdot a}{b}}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+135}:\\ \;\;\;\;\frac{1}{\frac{-1.5 \cdot a}{b}}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e+135)
   (/ 1.0 (/ (* -1.5 a) b))
   (if (<= b 8.6e-53)
     (/ (- (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 <= -1.75e+135) {
		tmp = 1.0 / ((-1.5 * a) / b);
	} else if (b <= 8.6e-53) {
		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 <= (-1.75d+135)) then
        tmp = 1.0d0 / (((-1.5d0) * a) / b)
    else if (b <= 8.6d-53) 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 <= -1.75e+135) {
		tmp = 1.0 / ((-1.5 * a) / b);
	} else if (b <= 8.6e-53) {
		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 <= -1.75e+135:
		tmp = 1.0 / ((-1.5 * a) / b)
	elif b <= 8.6e-53:
		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 <= -1.75e+135)
		tmp = Float64(1.0 / Float64(Float64(-1.5 * a) / b));
	elseif (b <= 8.6e-53)
		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 <= -1.75e+135)
		tmp = 1.0 / ((-1.5 * a) / b);
	elseif (b <= 8.6e-53)
		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, -1.75e+135], N[(1.0 / N[(N[(-1.5 * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-53], 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 < -1.7500000000000001e135

   1. Initial program 58.2%

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

      1. neg-sub0 58.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 58.2%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 58.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 58.2%

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

      5. sub-neg 58.2%

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

      6. distribute-neg-in 58.2%

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

      7. remove-double-neg 58.2%

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

      8. sqr-neg 58.2%

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

      9. associate-*l* 58.2%

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

   3. Simplified 58.2%

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

   6. Applied egg-rr 52.4%

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

      1. clear-num 52.3%

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

      2. frac-times 52.3%

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

      3. metadata-eval 52.3%

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

   8. Applied egg-rr 52.3%

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

   9. Taylor expanded in b around -inf 95.7%

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

       1. associate-*r/ 95.9%

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

   11. Simplified 95.9%

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

   if -1.7500000000000001e135 < b < 8.5999999999999999e-53

   1. Initial program 82.8%

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

      1. neg-sub0 82.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 82.8%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 82.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 82.8%

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

      5. sub-neg 82.8%

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

      6. distribute-neg-in 82.8%

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

      7. remove-double-neg 82.8%

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

      8. sqr-neg 82.8%

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

      9. associate-*l* 82.7%

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

   3. Simplified 82.7%

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

   if 8.5999999999999999e-53 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 16.3%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 16.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 16.3%

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

      5. sub-neg 16.3%

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

      6. distribute-neg-in 16.3%

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

      7. remove-double-neg 16.3%

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

      8. sqr-neg 16.3%

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

      9. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around inf 92.7%

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

      1. *-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+135}:\\ \;\;\;\;\frac{1}{\frac{-1.5 \cdot a}{b}}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{b \cdot 2}{-3}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-53}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.1e-26)
   (* (/ 1.0 a) (/ (* b 2.0) -3.0))
   (if (<= b 6.4e-53)
     (* 0.3333333333333333 (/ (+ b (sqrt (* a (* c -3.0)))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.1e-26) {
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	} else if (b <= 6.4e-53) {
		tmp = 0.3333333333333333 * ((b + sqrt((a * (c * -3.0)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.1d-26)) then
        tmp = (1.0d0 / a) * ((b * 2.0d0) / (-3.0d0))
    else if (b <= 6.4d-53) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((a * (c * (-3.0d0))))) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.1e-26) {
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	} else if (b <= 6.4e-53) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((a * (c * -3.0)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.1e-26:
		tmp = (1.0 / a) * ((b * 2.0) / -3.0)
	elif b <= 6.4e-53:
		tmp = 0.3333333333333333 * ((b + math.sqrt((a * (c * -3.0)))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.1e-26)
		tmp = Float64(Float64(1.0 / a) * Float64(Float64(b * 2.0) / -3.0));
	elseif (b <= 6.4e-53)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(a * Float64(c * -3.0)))) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.1e-26)
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	elseif (b <= 6.4e-53)
		tmp = 0.3333333333333333 * ((b + sqrt((a * (c * -3.0)))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.1e-26], N[(N[(1.0 / a), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.4e-53], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(a * N[(c * -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 < -3.09999999999999983e-26

   1. Initial program 72.9%

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

      1. neg-sub0 72.9%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 72.9%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 72.9%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 72.9%

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

      5. sub-neg 72.9%

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

      6. distribute-neg-in 72.9%

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

      7. remove-double-neg 72.9%

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

      8. sqr-neg 72.9%

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

      9. associate-*l* 72.9%

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

   3. Simplified 72.9%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in b around -inf 88.2%

      \[\leadsto \frac{\color{blue}{2 \cdot b}}{-3} \cdot \frac{1}{a} \]
   8. Step-by-step derivation

      1. *-commutative 88.2%

         \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   9. Simplified 88.2%

      \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   if -3.09999999999999983e-26 < b < 6.4000000000000002e-53

   1. Initial program 79.2%

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

      1. neg-sub0 79.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 79.2%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 79.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 79.2%

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

      5. sub-neg 79.2%

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

      6. distribute-neg-in 79.2%

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

      7. remove-double-neg 79.2%

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

      8. sqr-neg 79.2%

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

      9. associate-*l* 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in b around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. *-commutative 71.4%

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

      3. associate-*r* 71.5%

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

   7. Simplified 71.5%

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

      1. +-commutative 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. fma-define 71.5%

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

      4. *-commutative 71.5%

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

      5. add-sqr-sqrt 43.0%

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

      6. sqrt-unprod 71.2%

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

      7. sqr-neg 71.2%

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

      8. sqrt-unprod 28.5%

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

      9. add-sqr-sqrt 69.9%

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

   9. Applied egg-rr 69.9%

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

       1. fma-undefine 69.9%

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

       2. *-lft-identity 69.9%

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

       3. associate-*r* 69.7%

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

       4. *-commutative 69.7%

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

       5. *-commutative 69.7%

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

       6. *-commutative 69.7%

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

   11. Simplified 69.7%

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

       1. *-un-lft-identity 69.7%

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

       2. *-un-lft-identity 69.7%

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

       3. times-frac 69.6%

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

       4. metadata-eval 69.6%

          \[\leadsto 1 \cdot \left(\color{blue}{0.3333333333333333} \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} + b}{a}\right) \]

       5. +-commutative 69.6%

          \[\leadsto 1 \cdot \left(0.3333333333333333 \cdot \frac{\color{blue}{b + \sqrt{a \cdot \left(c \cdot -3\right)}}}{a}\right) \]

       6. *-commutative 69.6%

          \[\leadsto 1 \cdot \left(0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \color{blue}{\left(-3 \cdot c\right)}}}{a}\right) \]

   13. Applied egg-rr 69.6%

       \[\leadsto \color{blue}{1 \cdot \left(0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(-3 \cdot c\right)}}{a}\right)} \]
   14. Step-by-step derivation

       1. *-lft-identity 69.6%

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

       2. associate-*r* 69.7%

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

       3. *-commutative 69.7%

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

       4. *-commutative 69.7%

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

       5. associate-*r* 69.6%

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

       6. *-commutative 69.6%

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

   15. Simplified 69.6%

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

   if 6.4000000000000002e-53 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 16.3%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 16.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 16.3%

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

      5. sub-neg 16.3%

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

      6. distribute-neg-in 16.3%

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

      7. remove-double-neg 16.3%

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

      8. sqr-neg 16.3%

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

      9. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around inf 92.7%

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

      1. *-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{b \cdot 2}{-3}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-53}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{b \cdot 2}{-3}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e-24)
   (* (/ 1.0 a) (/ (* b 2.0) -3.0))
   (if (<= b 5.2e-53)
     (/ (+ b (sqrt (* a (* c -3.0)))) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-24) {
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	} else if (b <= 5.2e-53) {
		tmp = (b + sqrt((a * (c * -3.0)))) / (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 <= (-1.6d-24)) then
        tmp = (1.0d0 / a) * ((b * 2.0d0) / (-3.0d0))
    else if (b <= 5.2d-53) then
        tmp = (b + sqrt((a * (c * (-3.0d0))))) / (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 <= -1.6e-24) {
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	} else if (b <= 5.2e-53) {
		tmp = (b + Math.sqrt((a * (c * -3.0)))) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e-24:
		tmp = (1.0 / a) * ((b * 2.0) / -3.0)
	elif b <= 5.2e-53:
		tmp = (b + math.sqrt((a * (c * -3.0)))) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e-24)
		tmp = Float64(Float64(1.0 / a) * Float64(Float64(b * 2.0) / -3.0));
	elseif (b <= 5.2e-53)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(c * -3.0)))) / 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 <= -1.6e-24)
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	elseif (b <= 5.2e-53)
		tmp = (b + sqrt((a * (c * -3.0)))) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e-24], N[(N[(1.0 / a), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-53], N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 < -1.60000000000000006e-24

   1. Initial program 72.9%

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

      1. neg-sub0 72.9%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 72.9%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 72.9%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 72.9%

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

      5. sub-neg 72.9%

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

      6. distribute-neg-in 72.9%

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

      7. remove-double-neg 72.9%

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

      8. sqr-neg 72.9%

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

      9. associate-*l* 72.9%

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

   3. Simplified 72.9%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in b around -inf 88.2%

      \[\leadsto \frac{\color{blue}{2 \cdot b}}{-3} \cdot \frac{1}{a} \]
   8. Step-by-step derivation

      1. *-commutative 88.2%

         \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   9. Simplified 88.2%

      \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   if -1.60000000000000006e-24 < b < 5.19999999999999993e-53

   1. Initial program 79.2%

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

      1. neg-sub0 79.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 79.2%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 79.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 79.2%

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

      5. sub-neg 79.2%

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

      6. distribute-neg-in 79.2%

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

      7. remove-double-neg 79.2%

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

      8. sqr-neg 79.2%

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

      9. associate-*l* 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in b around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. *-commutative 71.4%

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

      3. associate-*r* 71.5%

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

   7. Simplified 71.5%

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

      1. +-commutative 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. fma-define 71.5%

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

      4. *-commutative 71.5%

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

      5. add-sqr-sqrt 43.0%

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

      6. sqrt-unprod 71.2%

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

      7. sqr-neg 71.2%

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

      8. sqrt-unprod 28.5%

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

      9. add-sqr-sqrt 69.9%

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

   9. Applied egg-rr 69.9%

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

       1. fma-undefine 69.9%

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

       2. *-lft-identity 69.9%

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

       3. associate-*r* 69.7%

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

       4. *-commutative 69.7%

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

       5. *-commutative 69.7%

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

       6. *-commutative 69.7%

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

   11. Simplified 69.7%

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

   if 5.19999999999999993e-53 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 16.3%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 16.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 16.3%

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

      5. sub-neg 16.3%

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

      6. distribute-neg-in 16.3%

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

      7. remove-double-neg 16.3%

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

      8. sqr-neg 16.3%

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

      9. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around inf 92.7%

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

      1. *-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{b \cdot 2}{-3}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{b \cdot 2}{-3}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-25)
   (* (/ 1.0 a) (/ (* b 2.0) -3.0))
   (if (<= b 5.6e-53)
     (/ (+ b (sqrt (* c (* a -3.0)))) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-25) {
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	} else if (b <= 5.6e-53) {
		tmp = (b + sqrt((c * (a * -3.0)))) / (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 <= (-6d-25)) then
        tmp = (1.0d0 / a) * ((b * 2.0d0) / (-3.0d0))
    else if (b <= 5.6d-53) then
        tmp = (b + sqrt((c * (a * (-3.0d0))))) / (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 <= -6e-25) {
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	} else if (b <= 5.6e-53) {
		tmp = (b + Math.sqrt((c * (a * -3.0)))) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-25:
		tmp = (1.0 / a) * ((b * 2.0) / -3.0)
	elif b <= 5.6e-53:
		tmp = (b + math.sqrt((c * (a * -3.0)))) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-25)
		tmp = Float64(Float64(1.0 / a) * Float64(Float64(b * 2.0) / -3.0));
	elseif (b <= 5.6e-53)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / 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 <= -6e-25)
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	elseif (b <= 5.6e-53)
		tmp = (b + sqrt((c * (a * -3.0)))) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-25], N[(N[(1.0 / a), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.6e-53], N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 < -5.9999999999999995e-25

   1. Initial program 72.9%

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

      1. neg-sub0 72.9%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 72.9%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 72.9%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 72.9%

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

      5. sub-neg 72.9%

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

      6. distribute-neg-in 72.9%

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

      7. remove-double-neg 72.9%

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

      8. sqr-neg 72.9%

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

      9. associate-*l* 72.9%

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

   3. Simplified 72.9%

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in b around -inf 88.2%

      \[\leadsto \frac{\color{blue}{2 \cdot b}}{-3} \cdot \frac{1}{a} \]
   8. Step-by-step derivation

      1. *-commutative 88.2%

         \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   9. Simplified 88.2%

      \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   if -5.9999999999999995e-25 < b < 5.59999999999999971e-53

   1. Initial program 79.2%

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

      1. neg-sub0 79.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 79.2%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 79.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 79.2%

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

      5. sub-neg 79.2%

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

      6. distribute-neg-in 79.2%

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

      7. remove-double-neg 79.2%

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

      8. sqr-neg 79.2%

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

      9. associate-*l* 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in b around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. *-commutative 71.4%

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

      3. associate-*r* 71.5%

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

   7. Simplified 71.5%

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

      1. +-commutative 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. fma-define 71.5%

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

      4. *-commutative 71.5%

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

      5. add-sqr-sqrt 43.0%

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

      6. sqrt-unprod 71.2%

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

      7. sqr-neg 71.2%

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

      8. sqrt-unprod 28.5%

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

      9. add-sqr-sqrt 69.9%

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

   9. Applied egg-rr 69.9%

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

       1. fma-undefine 69.9%

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

       2. *-lft-identity 69.9%

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

       3. *-commutative 69.9%

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

   11. Simplified 69.9%

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

   if 5.59999999999999971e-53 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 16.3%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 16.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 16.3%

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

      5. sub-neg 16.3%

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

      6. distribute-neg-in 16.3%

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

      7. remove-double-neg 16.3%

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

      8. sqr-neg 16.3%

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

      9. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around inf 92.7%

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

      1. *-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{b \cdot 2}{-3}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{b \cdot 2}{-3}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-12)
   (* (/ 1.0 a) (/ (* b 2.0) -3.0))
   (if (<= b 5.2e-53)
     (/ (- (sqrt (* (* a c) -3.0)) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-12) {
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	} else if (b <= 5.2e-53) {
		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) :: tmp
    if (b <= (-1d-12)) then
        tmp = (1.0d0 / a) * ((b * 2.0d0) / (-3.0d0))
    else if (b <= 5.2d-53) 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 tmp;
	if (b <= -1e-12) {
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	} else if (b <= 5.2e-53) {
		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):
	tmp = 0
	if b <= -1e-12:
		tmp = (1.0 / a) * ((b * 2.0) / -3.0)
	elif b <= 5.2e-53:
		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)
	tmp = 0.0
	if (b <= -1e-12)
		tmp = Float64(Float64(1.0 / a) * Float64(Float64(b * 2.0) / -3.0));
	elseif (b <= 5.2e-53)
		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)
	tmp = 0.0;
	if (b <= -1e-12)
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	elseif (b <= 5.2e-53)
		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_] := If[LessEqual[b, -1e-12], N[(N[(1.0 / a), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-53], 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 3 regimes

2. if b < -9.9999999999999998e-13

   1. Initial program 72.2%

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

      1. neg-sub0 72.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 72.2%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 72.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 72.2%

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

      5. sub-neg 72.2%

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

      6. distribute-neg-in 72.2%

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

      7. remove-double-neg 72.2%

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

      8. sqr-neg 72.2%

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

      9. associate-*l* 72.2%

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

   3. Simplified 72.2%

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

   6. Applied egg-rr 56.4%

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

   7. Taylor expanded in b around -inf 89.1%

      \[\leadsto \frac{\color{blue}{2 \cdot b}}{-3} \cdot \frac{1}{a} \]
   8. Step-by-step derivation

      1. *-commutative 89.1%

         \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   9. Simplified 89.1%

      \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   if -9.9999999999999998e-13 < b < 5.19999999999999993e-53

   1. Initial program 79.6%

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

      1. /-rgt-identity 79.6%

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

      2. metadata-eval 79.6%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in a around inf 79.5%

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

   6. Taylor expanded in a around inf 71.1%

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

   if 5.19999999999999993e-53 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 16.3%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 16.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 16.3%

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

      5. sub-neg 16.3%

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

      6. distribute-neg-in 16.3%

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

      7. remove-double-neg 16.3%

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

      8. sqr-neg 16.3%

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

      9. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around inf 92.7%

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

      1. *-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{b \cdot 2}{-3}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{b \cdot 2}{-3}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e-13)
   (* (/ 1.0 a) (/ (* b 2.0) -3.0))
   (if (<= b 7.5e-53)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e-13) {
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	} else if (b <= 7.5e-53) {
		tmp = (sqrt((c * (a * -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) :: tmp
    if (b <= (-7.2d-13)) then
        tmp = (1.0d0 / a) * ((b * 2.0d0) / (-3.0d0))
    else if (b <= 7.5d-53) then
        tmp = (sqrt((c * (a * (-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 tmp;
	if (b <= -7.2e-13) {
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	} else if (b <= 7.5e-53) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.2e-13:
		tmp = (1.0 / a) * ((b * 2.0) / -3.0)
	elif b <= 7.5e-53:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e-13)
		tmp = Float64(Float64(1.0 / a) * Float64(Float64(b * 2.0) / -3.0));
	elseif (b <= 7.5e-53)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -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)
	tmp = 0.0;
	if (b <= -7.2e-13)
		tmp = (1.0 / a) * ((b * 2.0) / -3.0);
	elseif (b <= 7.5e-53)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e-13], N[(N[(1.0 / a), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e-53], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $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 < -7.1999999999999996e-13

   1. Initial program 72.2%

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

      1. neg-sub0 72.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 72.2%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 72.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 72.2%

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

      5. sub-neg 72.2%

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

      6. distribute-neg-in 72.2%

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

      7. remove-double-neg 72.2%

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

      8. sqr-neg 72.2%

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

      9. associate-*l* 72.2%

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

   3. Simplified 72.2%

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

   6. Applied egg-rr 56.4%

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

   7. Taylor expanded in b around -inf 89.1%

      \[\leadsto \frac{\color{blue}{2 \cdot b}}{-3} \cdot \frac{1}{a} \]
   8. Step-by-step derivation

      1. *-commutative 89.1%

         \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   9. Simplified 89.1%

      \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   if -7.1999999999999996e-13 < b < 7.5000000000000001e-53

   1. Initial program 79.6%

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

      1. neg-sub0 79.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 79.6%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 79.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 79.6%

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

      5. sub-neg 79.6%

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

      6. distribute-neg-in 79.6%

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

      7. remove-double-neg 79.6%

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

      8. sqr-neg 79.6%

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

      9. associate-*l* 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in b around 0 71.1%

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

      1. *-commutative 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-*r* 71.2%

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

   7. Simplified 71.2%

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

      1. *-commutative 71.2%

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

      2. sqrt-prod 39.1%

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

      3. *-commutative 39.1%

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

   9. Applied egg-rr 39.1%

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

       1. +-commutative 39.1%

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

       2. unsub-neg 39.1%

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

       3. *-commutative 39.1%

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

       4. sqrt-unprod 71.2%

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

       5. *-commutative 71.2%

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

   11. Applied egg-rr 71.2%

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

   if 7.5000000000000001e-53 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 16.3%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 16.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 16.3%

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

      5. sub-neg 16.3%

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

      6. distribute-neg-in 16.3%

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

      7. remove-double-neg 16.3%

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

      8. sqr-neg 16.3%

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

      9. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around inf 92.7%

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

      1. *-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{b \cdot 2}{-3}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-13}:\\ \;\;\;\;b \cdot \left(0.6666666666666666 \cdot \frac{-1}{a} - -0.5 \cdot \frac{c}{{b}^{2}}\right)\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e-13)
   (* b (- (* 0.6666666666666666 (/ -1.0 a)) (* -0.5 (/ c (pow b 2.0)))))
   (if (<= b 6.5e-53)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-13) {
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / pow(b, 2.0))));
	} else if (b <= 6.5e-53) {
		tmp = (sqrt((c * (a * -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) :: tmp
    if (b <= (-6.5d-13)) then
        tmp = b * ((0.6666666666666666d0 * ((-1.0d0) / a)) - ((-0.5d0) * (c / (b ** 2.0d0))))
    else if (b <= 6.5d-53) then
        tmp = (sqrt((c * (a * (-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 tmp;
	if (b <= -6.5e-13) {
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / Math.pow(b, 2.0))));
	} else if (b <= 6.5e-53) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e-13:
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / math.pow(b, 2.0))))
	elif b <= 6.5e-53:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-13)
		tmp = Float64(b * Float64(Float64(0.6666666666666666 * Float64(-1.0 / a)) - Float64(-0.5 * Float64(c / (b ^ 2.0)))));
	elseif (b <= 6.5e-53)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -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)
	tmp = 0.0;
	if (b <= -6.5e-13)
		tmp = b * ((0.6666666666666666 * (-1.0 / a)) - (-0.5 * (c / (b ^ 2.0))));
	elseif (b <= 6.5e-53)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-13], N[(b * N[(N[(0.6666666666666666 * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e-53], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $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 < -6.49999999999999957e-13

   1. Initial program 72.2%

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

      1. neg-sub0 72.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 72.2%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 72.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 72.2%

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

      5. sub-neg 72.2%

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

      6. distribute-neg-in 72.2%

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

      7. remove-double-neg 72.2%

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

      8. sqr-neg 72.2%

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

      9. associate-*l* 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in b around -inf 89.5%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   if -6.49999999999999957e-13 < b < 6.4999999999999997e-53

   1. Initial program 79.6%

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

      1. neg-sub0 79.6%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 79.6%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 79.6%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 79.6%

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

      5. sub-neg 79.6%

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

      6. distribute-neg-in 79.6%

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

      7. remove-double-neg 79.6%

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

      8. sqr-neg 79.6%

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

      9. associate-*l* 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in b around 0 71.1%

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

      1. *-commutative 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-*r* 71.2%

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

   7. Simplified 71.2%

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

      1. *-commutative 71.2%

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

      2. sqrt-prod 39.1%

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

      3. *-commutative 39.1%

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

   9. Applied egg-rr 39.1%

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

       1. +-commutative 39.1%

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

       2. unsub-neg 39.1%

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

       3. *-commutative 39.1%

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

       4. sqrt-unprod 71.2%

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

       5. *-commutative 71.2%

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

   11. Applied egg-rr 71.2%

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

   if 6.4999999999999997e-53 < b

   1. Initial program 16.3%

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

      1. neg-sub0 16.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 16.3%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 16.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 16.3%

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

      5. sub-neg 16.3%

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

      6. distribute-neg-in 16.3%

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

      7. remove-double-neg 16.3%

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

      8. sqr-neg 16.3%

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

      9. associate-*l* 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b around inf 92.7%

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

      1. *-commutative 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-13}:\\ \;\;\;\;b \cdot \left(0.6666666666666666 \cdot \frac{-1}{a} - -0.5 \cdot \frac{c}{{b}^{2}}\right)\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;\frac{1}{-1.5 \cdot \frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.16e-262) (/ 1.0 (* -1.5 (/ a b))) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.16e-262) {
		tmp = 1.0 / (-1.5 * (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 <= 1.16d-262) then
        tmp = 1.0d0 / ((-1.5d0) * (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 <= 1.16e-262) {
		tmp = 1.0 / (-1.5 * (a / b));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.16e-262:
		tmp = 1.0 / (-1.5 * (a / b))
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.16e-262)
		tmp = Float64(1.0 / Float64(-1.5 * 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 <= 1.16e-262)
		tmp = 1.0 / (-1.5 * (a / b));
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.16e-262], N[(1.0 / N[(-1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.16000000000000001e-262

   1. Initial program 77.5%

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

      1. neg-sub0 77.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 77.5%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 77.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 77.5%

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

      5. sub-neg 77.5%

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

      6. distribute-neg-in 77.5%

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

      7. remove-double-neg 77.5%

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

      8. sqr-neg 77.5%

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

      9. associate-*l* 77.5%

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

   3. Simplified 77.5%

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

   6. Applied egg-rr 67.4%

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

      1. clear-num 67.3%

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

      2. frac-times 67.4%

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

      3. metadata-eval 67.4%

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

   8. Applied egg-rr 67.4%

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

   9. Taylor expanded in b around -inf 60.5%

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

   if 1.16000000000000001e-262 < b

   1. Initial program 31.5%

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

      1. neg-sub0 31.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 31.5%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 31.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 31.5%

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

      5. sub-neg 31.5%

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

      6. distribute-neg-in 31.5%

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

      7. remove-double-neg 31.5%

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

      8. sqr-neg 31.5%

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

      9. associate-*l* 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in b around inf 73.1%

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

      1. *-commutative 73.1%

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

   7. Simplified 73.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;\frac{1}{-1.5 \cdot \frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;\frac{1}{\frac{-1.5 \cdot a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.16e-262) (/ 1.0 (/ (* -1.5 a) b)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.16e-262) {
		tmp = 1.0 / ((-1.5 * 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 <= 1.16d-262) then
        tmp = 1.0d0 / (((-1.5d0) * 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 <= 1.16e-262) {
		tmp = 1.0 / ((-1.5 * a) / b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.16e-262:
		tmp = 1.0 / ((-1.5 * a) / b)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.16e-262)
		tmp = Float64(1.0 / Float64(Float64(-1.5 * 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 <= 1.16e-262)
		tmp = 1.0 / ((-1.5 * a) / b);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.16e-262], N[(1.0 / N[(N[(-1.5 * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.16000000000000001e-262

   1. Initial program 77.5%

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

      1. neg-sub0 77.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 77.5%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 77.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 77.5%

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

      5. sub-neg 77.5%

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

      6. distribute-neg-in 77.5%

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

      7. remove-double-neg 77.5%

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

      8. sqr-neg 77.5%

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

      9. associate-*l* 77.5%

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

   3. Simplified 77.5%

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

   6. Applied egg-rr 67.4%

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

      1. clear-num 67.3%

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

      2. frac-times 67.4%

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

      3. metadata-eval 67.4%

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

   8. Applied egg-rr 67.4%

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

   9. Taylor expanded in b around -inf 60.5%

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

       1. associate-*r/ 60.5%

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

   11. Simplified 60.5%

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

   if 1.16000000000000001e-262 < b

   1. Initial program 31.5%

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

      1. neg-sub0 31.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 31.5%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 31.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 31.5%

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

      5. sub-neg 31.5%

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

      6. distribute-neg-in 31.5%

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

      7. remove-double-neg 31.5%

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

      8. sqr-neg 31.5%

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

      9. associate-*l* 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in b around inf 73.1%

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

      1. *-commutative 73.1%

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

   7. Simplified 73.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;\frac{1}{\frac{-1.5 \cdot a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.16e-262) (/ (* b -2.0) (* a 3.0)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.16e-262) {
		tmp = (b * -2.0) / (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 <= 1.16d-262) then
        tmp = (b * (-2.0d0)) / (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 <= 1.16e-262) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.16e-262:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.16e-262)
		tmp = Float64(Float64(b * -2.0) / 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 <= 1.16e-262)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.16e-262], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.16000000000000001e-262

   1. Initial program 77.5%

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

      1. neg-sub0 77.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 77.5%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 77.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 77.5%

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

      5. sub-neg 77.5%

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

      6. distribute-neg-in 77.5%

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

      7. remove-double-neg 77.5%

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

      8. sqr-neg 77.5%

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

      9. associate-*l* 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in b around -inf 60.6%

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

      1. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   if 1.16000000000000001e-262 < b

   1. Initial program 31.5%

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

      1. neg-sub0 31.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 31.5%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 31.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 31.5%

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

      5. sub-neg 31.5%

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

      6. distribute-neg-in 31.5%

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

      7. remove-double-neg 31.5%

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

      8. sqr-neg 31.5%

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

      9. associate-*l* 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in b around inf 73.1%

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

      1. *-commutative 73.1%

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

   7. Simplified 73.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.16e-262) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.16e-262) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.16d-262) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.16e-262) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.16e-262:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.16e-262)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.16e-262)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.16e-262], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.16000000000000001e-262

   1. Initial program 77.5%

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

      1. neg-sub0 77.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 77.5%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 77.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 77.5%

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

      5. sub-neg 77.5%

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

      6. distribute-neg-in 77.5%

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

      7. remove-double-neg 77.5%

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

      8. sqr-neg 77.5%

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

      9. associate-*l* 77.5%

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

   3. Simplified 77.5%

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in b around -inf 60.4%

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

      1. *-commutative 60.4%

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

      2. associate-*l/ 60.4%

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

      3. associate-/l* 60.4%

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

   9. Simplified 60.4%

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

   if 1.16000000000000001e-262 < b

   1. Initial program 31.5%

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

      1. neg-sub0 31.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 31.5%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 31.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 31.5%

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

      5. sub-neg 31.5%

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

      6. distribute-neg-in 31.5%

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

      7. remove-double-neg 31.5%

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

      8. sqr-neg 31.5%

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

      9. associate-*l* 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in c around 0 65.7%

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

      1. associate-*r/ 65.7%

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

      2. metadata-eval 65.7%

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

   7. Simplified 65.7%

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

   8. Taylor expanded in a around 0 72.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.16e-262) (* b (/ -0.6666666666666666 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.16e-262) {
		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 <= 1.16d-262) 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 <= 1.16e-262) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.16e-262:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.16e-262)
		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 <= 1.16e-262)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.16e-262], 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 < 1.16000000000000001e-262

   1. Initial program 77.5%

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

      1. neg-sub0 77.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 77.5%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 77.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 77.5%

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

      5. sub-neg 77.5%

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

      6. distribute-neg-in 77.5%

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

      7. remove-double-neg 77.5%

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

      8. sqr-neg 77.5%

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

      9. associate-*l* 77.5%

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

   3. Simplified 77.5%

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in b around -inf 60.4%

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

      1. *-commutative 60.4%

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

      2. associate-*l/ 60.4%

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

      3. associate-/l* 60.4%

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

   9. Simplified 60.4%

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

   if 1.16000000000000001e-262 < b

   1. Initial program 31.5%

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

      1. neg-sub0 31.5%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. sqr-neg 31.5%

         \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. associate-+l- 31.5%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      4. sub0-neg 31.5%

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

      5. sub-neg 31.5%

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

      6. distribute-neg-in 31.5%

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

      7. remove-double-neg 31.5%

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

      8. sqr-neg 31.5%

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

      9. associate-*l* 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in b around inf 73.1%

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

      1. *-commutative 73.1%

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

   7. Simplified 73.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 2.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.3%

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

   1. /-rgt-identity 56.3%

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

   2. metadata-eval 56.3%

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

3. Simplified 55.8%

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

   1. sub-neg 55.8%

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

   2. fma-undefine 55.8%

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

   3. add-sqr-sqrt 46.8%

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

   4. hypot-define 51.8%

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

   5. add-sqr-sqrt 35.5%

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

   6. sqrt-unprod 43.9%

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

   7. sqr-neg 43.9%

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

   8. sqrt-prod 11.3%

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

   9. add-sqr-sqrt 27.8%

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

6. Applied egg-rr 27.8%

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

7. Taylor expanded in b around inf 2.4%

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

8. Final simplification 2.4%

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

Alternative 15: 2.5% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* 1.3333333333333333 (/ b a)))

C

double code(double a, double b, double c) {
	return 1.3333333333333333 * (b / a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.3333333333333333d0 * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return 1.3333333333333333 * (b / a);
}

Python

def code(a, b, c):
	return 1.3333333333333333 * (b / a)

Julia

function code(a, b, c)
	return Float64(1.3333333333333333 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.3333333333333333 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(1.3333333333333333 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.3%

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

   1. /-rgt-identity 56.3%

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

   2. metadata-eval 56.3%

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

3. Simplified 55.8%

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

   1. *-un-lft-identity 55.8%

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

   2. *-un-lft-identity 55.8%

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

   3. prod-diff 55.8%

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

   4. *-commutative 55.8%

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

   5. *-un-lft-identity 55.8%

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

   6. fma-define 55.8%

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

   7. *-un-lft-identity 55.8%

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

   8. +-commutative 55.8%

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

   9. add-sqr-sqrt 40.5%

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

   10. sqrt-unprod 52.0%

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

   11. sqr-neg 52.0%

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

   12. sqrt-prod 11.5%

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

   13. add-sqr-sqrt 36.4%

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

   14. fma-undefine 36.4%

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

   15. add-sqr-sqrt 33.2%

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

   16. hypot-define 27.8%

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

6. Applied egg-rr 27.4%

   \[\leadsto \frac{\color{blue}{\left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) + \mathsf{fma}\left(b, 1, b\right)}}{3 \cdot a} \]
7. Step-by-step derivation

   1. +-commutative 27.4%

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

   2. associate-+l+ 27.4%

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

   3. associate-*r* 27.4%

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

   4. *-commutative 27.4%

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

   5. associate-*r* 27.5%

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

   6. fma-undefine 27.5%

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

   7. *-rgt-identity 27.5%

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

8. Simplified 27.5%

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

9. Taylor expanded in b around inf 2.4%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{b}{a}} \]

10. Final simplification 2.4%

    \[\leadsto 1.3333333333333333 \cdot \frac{b}{a} \]
11. Add Preprocessing

Alternative 16: 35.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.3%

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

   1. neg-sub0 56.3%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. sqr-neg 56.3%

      \[\leadsto \frac{\left(0 - b\right) + \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   3. associate-+l- 56.3%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

   4. sub0-neg 56.3%

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

   5. sub-neg 56.3%

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

   6. distribute-neg-in 56.3%

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

   7. remove-double-neg 56.3%

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

   8. sqr-neg 56.3%

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

   9. associate-*l* 56.2%

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

3. Simplified 56.2%

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

6. Applied egg-rr 51.8%

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

7. Taylor expanded in b around -inf 33.8%

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

   1. *-commutative 33.8%

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

   2. associate-*l/ 33.8%

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

   3. associate-/l* 33.8%

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

9. Simplified 33.8%

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

10. Final simplification 33.8%

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

Reproduce

herbie shell --seed 2024085 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))