Cubic critical

Percentage Accurate: 52.1% → 86.0%

Time: 13.2s

Alternatives: 17

Speedup: 16.4×

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: 52.1% 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: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+151)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 2.05e-47)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+151) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.05e-47) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d+151)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 2.05d-47) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+151) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.05e-47) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+151:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 2.05e-47:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+151)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2.05e-47)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+151)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 2.05e-47)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+151], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.05e-47], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000002e151

   1. Initial program 39.7%

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

      1. sqr-neg 39.7%

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

      2. sqr-neg 39.7%

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

      3. associate-*l* 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in b around -inf 99.8%

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

      1. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if -1.00000000000000002e151 < b < 2.05000000000000001e-47

   1. Initial program 88.7%

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

   if 2.05000000000000001e-47 < b

   1. Initial program 18.6%

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

      1. sqr-neg 18.6%

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

      2. sqr-neg 18.6%

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

      3. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in b around inf 85.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-47}:\\ \;\;\;\;\left(\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e+140)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 8.5e-47)
     (* (- (sqrt (- (* b b) (* a (* 3.0 c)))) b) (/ 0.3333333333333333 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+140) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 8.5e-47) {
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.6d+140)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 8.5d-47) then
        tmp = (sqrt(((b * b) - (a * (3.0d0 * c)))) - b) * (0.3333333333333333d0 / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+140) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 8.5e-47) {
		tmp = (Math.sqrt(((b * b) - (a * (3.0 * c)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e+140:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 8.5e-47:
		tmp = (math.sqrt(((b * b) - (a * (3.0 * c)))) - b) * (0.3333333333333333 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e+140)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 8.5e-47)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(3.0 * c)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.6e+140)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 8.5e-47)
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) * (0.3333333333333333 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e+140], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e-47], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.6e140

   1. Initial program 42.7%

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

      1. sqr-neg 42.7%

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

      2. sqr-neg 42.7%

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

      3. associate-*l* 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in b around -inf 99.8%

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

      1. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if -3.6e140 < b < 8.4999999999999999e-47

   1. Initial program 88.5%

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

      1. neg-sub0 88.5%

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

      2. sqr-neg 88.5%

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

      3. associate-+l- 88.5%

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

      4. sub0-neg 88.5%

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

      5. neg-mul-1 88.5%

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

   3. Simplified 88.4%

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

      1. associate-*r* 88.4%

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

      2. metadata-eval 88.4%

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

      3. distribute-rgt-neg-in 88.4%

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

      4. *-commutative 88.4%

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

      5. fma-neg 88.4%

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

      6. associate-*r* 88.4%

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

      7. *-commutative 88.4%

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

      8. associate-*l* 88.4%

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

   5. Applied egg-rr 88.4%

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

      1. *-commutative 88.4%

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

   7. Simplified 88.4%

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

      1. div-inv 88.2%

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

      2. clear-num 88.3%

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

   9. Applied egg-rr 88.3%

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

   if 8.4999999999999999e-47 < b

   1. Initial program 18.6%

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

      1. sqr-neg 18.6%

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

      2. sqr-neg 18.6%

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

      3. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in b around inf 85.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-47}:\\ \;\;\;\;\left(\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+108}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e+108)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (if (<= b 4.3e-47)
     (* (/ (- (sqrt (- (* b b) (* a (* 3.0 c)))) b) a) 0.3333333333333333)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e+108) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 4.3e-47) {
		tmp = ((sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7d+108)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + ((c / b) * 0.5d0)
    else if (b <= 4.3d-47) then
        tmp = ((sqrt(((b * b) - (a * (3.0d0 * c)))) - b) / a) * 0.3333333333333333d0
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e+108) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else if (b <= 4.3e-47) {
		tmp = ((Math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7e+108:
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5)
	elif b <= 4.3e-47:
		tmp = ((math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e+108)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 4.3e-47)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(3.0 * c)))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7e+108)
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	elseif (b <= 4.3e-47)
		tmp = ((sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e+108], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.3e-47], N[(N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.0000000000000005e108

   1. Initial program 51.2%

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

      1. sqr-neg 51.2%

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

      2. sqr-neg 51.2%

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

      3. associate-*l* 51.2%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in b around -inf 99.5%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -7.0000000000000005e108 < b < 4.2999999999999998e-47

   1. Initial program 87.9%

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

      1. neg-sub0 87.9%

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

      2. sqr-neg 87.9%

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

      3. associate-+l- 87.9%

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

      4. sub0-neg 87.9%

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

      5. neg-mul-1 87.9%

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

   3. Simplified 87.7%

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

      1. associate-*r* 87.7%

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

      2. metadata-eval 87.7%

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

      3. distribute-rgt-neg-in 87.7%

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

      4. *-commutative 87.7%

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

      5. fma-neg 87.7%

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

      6. associate-*r* 87.7%

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

      7. *-commutative 87.7%

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

      8. associate-*l* 87.7%

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

   5. Applied egg-rr 87.7%

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

      1. *-commutative 87.7%

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

   7. Simplified 87.7%

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

      1. associate-/r/ 87.8%

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

   9. Applied egg-rr 87.8%

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

   if 4.2999999999999998e-47 < b

   1. Initial program 18.6%

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

      1. sqr-neg 18.6%

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

      2. sqr-neg 18.6%

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

      3. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in b around inf 85.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+108}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{+105}:\\ \;\;\;\;\frac{b \cdot -2 + \left|\frac{a}{b} \cdot \left(c \cdot 1.5\right)\right|}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e+105)
   (/ (+ (* b -2.0) (fabs (* (/ a b) (* c 1.5)))) (* 3.0 a))
   (if (<= b 1.75e-47)
     (* (/ (- (sqrt (- (* b b) (* a (* 3.0 c)))) b) a) 0.3333333333333333)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e+105) {
		tmp = ((b * -2.0) + fabs(((a / b) * (c * 1.5)))) / (3.0 * a);
	} else if (b <= 1.75e-47) {
		tmp = ((sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.95d+105)) then
        tmp = ((b * (-2.0d0)) + abs(((a / b) * (c * 1.5d0)))) / (3.0d0 * a)
    else if (b <= 1.75d-47) then
        tmp = ((sqrt(((b * b) - (a * (3.0d0 * c)))) - b) / a) * 0.3333333333333333d0
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e+105) {
		tmp = ((b * -2.0) + Math.abs(((a / b) * (c * 1.5)))) / (3.0 * a);
	} else if (b <= 1.75e-47) {
		tmp = ((Math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e+105:
		tmp = ((b * -2.0) + math.fabs(((a / b) * (c * 1.5)))) / (3.0 * a)
	elif b <= 1.75e-47:
		tmp = ((math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e+105)
		tmp = Float64(Float64(Float64(b * -2.0) + abs(Float64(Float64(a / b) * Float64(c * 1.5)))) / Float64(3.0 * a));
	elseif (b <= 1.75e-47)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(3.0 * c)))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.95e+105)
		tmp = ((b * -2.0) + abs(((a / b) * (c * 1.5)))) / (3.0 * a);
	elseif (b <= 1.75e-47)
		tmp = ((sqrt(((b * b) - (a * (3.0 * c)))) - b) / a) * 0.3333333333333333;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e+105], N[(N[(N[(b * -2.0), $MachinePrecision] + N[Abs[N[(N[(a / b), $MachinePrecision] * N[(c * 1.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.75e-47], N[(N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.94999999999999989e105

   1. Initial program 52.2%

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

      1. sqr-neg 52.2%

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

      2. sqr-neg 52.2%

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

      3. associate-*l* 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in b around -inf 95.7%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 68.6%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\sqrt{1.5 \cdot \frac{a \cdot c}{b}} \cdot \sqrt{1.5 \cdot \frac{a \cdot c}{b}}}}{3 \cdot a} \]

      2. sqrt-unprod 95.9%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\sqrt{\left(1.5 \cdot \frac{a \cdot c}{b}\right) \cdot \left(1.5 \cdot \frac{a \cdot c}{b}\right)}}}{3 \cdot a} \]

      3. pow2 95.9%

         \[\leadsto \frac{-2 \cdot b + \sqrt{\color{blue}{{\left(1.5 \cdot \frac{a \cdot c}{b}\right)}^{2}}}}{3 \cdot a} \]

      4. *-commutative 95.9%

         \[\leadsto \frac{-2 \cdot b + \sqrt{{\color{blue}{\left(\frac{a \cdot c}{b} \cdot 1.5\right)}}^{2}}}{3 \cdot a} \]

      5. associate-/l* 99.8%

         \[\leadsto \frac{-2 \cdot b + \sqrt{{\left(\color{blue}{\frac{a}{\frac{b}{c}}} \cdot 1.5\right)}^{2}}}{3 \cdot a} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\sqrt{{\left(\frac{a}{\frac{b}{c}} \cdot 1.5\right)}^{2}}}}{3 \cdot a} \]
   7. Step-by-step derivation

      1. *-commutative 99.8%

         \[\leadsto \frac{-2 \cdot b + \sqrt{{\color{blue}{\left(1.5 \cdot \frac{a}{\frac{b}{c}}\right)}}^{2}}}{3 \cdot a} \]

      2. associate-/l* 95.9%

         \[\leadsto \frac{-2 \cdot b + \sqrt{{\left(1.5 \cdot \color{blue}{\frac{a \cdot c}{b}}\right)}^{2}}}{3 \cdot a} \]

      3. unpow2 95.9%

         \[\leadsto \frac{-2 \cdot b + \sqrt{\color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b}\right) \cdot \left(1.5 \cdot \frac{a \cdot c}{b}\right)}}}{3 \cdot a} \]

      4. rem-sqrt-square 95.9%

         \[\leadsto \frac{-2 \cdot b + \color{blue}{\left|1.5 \cdot \frac{a \cdot c}{b}\right|}}{3 \cdot a} \]

      5. associate-/l* 99.8%

         \[\leadsto \frac{-2 \cdot b + \left|1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right|}{3 \cdot a} \]

      6. *-commutative 99.8%

         \[\leadsto \frac{-2 \cdot b + \left|\color{blue}{\frac{a}{\frac{b}{c}} \cdot 1.5}\right|}{3 \cdot a} \]

      7. associate-/r/ 99.8%

         \[\leadsto \frac{-2 \cdot b + \left|\color{blue}{\left(\frac{a}{b} \cdot c\right)} \cdot 1.5\right|}{3 \cdot a} \]

      8. associate-*l* 99.8%

         \[\leadsto \frac{-2 \cdot b + \left|\color{blue}{\frac{a}{b} \cdot \left(c \cdot 1.5\right)}\right|}{3 \cdot a} \]

   8. Simplified 99.8%

      \[\leadsto \frac{-2 \cdot b + \color{blue}{\left|\frac{a}{b} \cdot \left(c \cdot 1.5\right)\right|}}{3 \cdot a} \]

   if -1.94999999999999989e105 < b < 1.7499999999999999e-47

   1. Initial program 87.8%

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

      1. neg-sub0 87.8%

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

      2. sqr-neg 87.8%

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

      3. associate-+l- 87.8%

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

      4. sub0-neg 87.8%

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

      5. neg-mul-1 87.8%

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

   3. Simplified 87.6%

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

      1. associate-*r* 87.6%

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

      2. metadata-eval 87.6%

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

      3. distribute-rgt-neg-in 87.6%

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

      4. *-commutative 87.6%

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

      5. fma-neg 87.6%

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

      6. associate-*r* 87.6%

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

      7. *-commutative 87.6%

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

      8. associate-*l* 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. *-commutative 87.6%

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

   7. Simplified 87.6%

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

      1. associate-/r/ 87.7%

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

   9. Applied egg-rr 87.7%

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

   if 1.7499999999999999e-47 < b

   1. Initial program 18.6%

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

      1. sqr-neg 18.6%

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

      2. sqr-neg 18.6%

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

      3. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in b around inf 85.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{+105}:\\ \;\;\;\;\frac{b \cdot -2 + \left|\frac{a}{b} \cdot \left(c \cdot 1.5\right)\right|}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+154)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 2.4e-46)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+154) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.4e-46) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d+154)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 2.4d-46) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+154) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.4e-46) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+154:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 2.4e-46:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+154)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2.4e-46)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+154)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 2.4e-46)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+154], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-46], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.00000000000000007e154

   1. Initial program 39.7%

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

      1. sqr-neg 39.7%

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

      2. sqr-neg 39.7%

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

      3. associate-*l* 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in b around -inf 99.8%

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

      1. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if -2.00000000000000007e154 < b < 2.40000000000000013e-46

   1. Initial program 88.7%

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

      1. sqr-neg 88.7%

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

      2. sqr-neg 88.7%

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

      3. associate-*l* 88.7%

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

   3. Simplified 88.7%

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

   if 2.40000000000000013e-46 < b

   1. Initial program 18.6%

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

      1. sqr-neg 18.6%

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

      2. sqr-neg 18.6%

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

      3. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in b around inf 85.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.5:\\ \;\;\;\;\frac{\left(\frac{a}{b} \cdot \left(c \cdot 1.5\right) - b\right) - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-46}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -0.5)
   (/ (- (- (* (/ a b) (* c 1.5)) b) b) (/ a 0.3333333333333333))
   (if (<= b 1.12e-46)
     (* (/ 0.3333333333333333 a) (- (sqrt (* a (* c -3.0))) b))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -0.5) {
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333);
	} else if (b <= 1.12e-46) {
		tmp = (0.3333333333333333 / a) * (sqrt((a * (c * -3.0))) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-0.5d0)) then
        tmp = ((((a / b) * (c * 1.5d0)) - b) - b) / (a / 0.3333333333333333d0)
    else if (b <= 1.12d-46) then
        tmp = (0.3333333333333333d0 / a) * (sqrt((a * (c * (-3.0d0)))) - b)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -0.5) {
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333);
	} else if (b <= 1.12e-46) {
		tmp = (0.3333333333333333 / a) * (Math.sqrt((a * (c * -3.0))) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -0.5:
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333)
	elif b <= 1.12e-46:
		tmp = (0.3333333333333333 / a) * (math.sqrt((a * (c * -3.0))) - b)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -0.5)
		tmp = Float64(Float64(Float64(Float64(Float64(a / b) * Float64(c * 1.5)) - b) - b) / Float64(a / 0.3333333333333333));
	elseif (b <= 1.12e-46)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(a * Float64(c * -3.0))) - b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -0.5)
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333);
	elseif (b <= 1.12e-46)
		tmp = (0.3333333333333333 / a) * (sqrt((a * (c * -3.0))) - b);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -0.5], N[(N[(N[(N[(N[(a / b), $MachinePrecision] * N[(c * 1.5), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.12e-46], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.5

   1. Initial program 66.7%

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

      1. neg-sub0 66.7%

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

      2. sqr-neg 66.7%

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

      3. associate-+l- 66.7%

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

      4. sub0-neg 66.7%

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

      5. neg-mul-1 66.7%

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

   3. Simplified 66.6%

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

      1. associate-*r* 66.6%

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

      2. metadata-eval 66.6%

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

      3. distribute-rgt-neg-in 66.6%

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

      4. *-commutative 66.6%

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

      5. fma-neg 66.6%

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

      6. associate-*r* 66.6%

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

      7. *-commutative 66.6%

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

      8. associate-*l* 66.6%

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

   5. Applied egg-rr 66.6%

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

      1. *-commutative 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in b around -inf 93.0%

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{\frac{a}{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. neg-mul-1 93.0%

         \[\leadsto \frac{\left(\color{blue}{\left(-b\right)} + 1.5 \cdot \frac{a \cdot c}{b}\right) - b}{\frac{a}{0.3333333333333333}} \]

      2. +-commutative 93.0%

         \[\leadsto \frac{\color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + \left(-b\right)\right)} - b}{\frac{a}{0.3333333333333333}} \]

      3. unsub-neg 93.0%

         \[\leadsto \frac{\color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)} - b}{\frac{a}{0.3333333333333333}} \]

      4. associate-/l* 95.7%

         \[\leadsto \frac{\left(1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} - b\right) - b}{\frac{a}{0.3333333333333333}} \]

      5. *-commutative 95.7%

         \[\leadsto \frac{\left(\color{blue}{\frac{a}{\frac{b}{c}} \cdot 1.5} - b\right) - b}{\frac{a}{0.3333333333333333}} \]

      6. associate-/r/ 95.7%

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{a}{b} \cdot c\right)} \cdot 1.5 - b\right) - b}{\frac{a}{0.3333333333333333}} \]

      7. associate-*l* 95.7%

         \[\leadsto \frac{\left(\color{blue}{\frac{a}{b} \cdot \left(c \cdot 1.5\right)} - b\right) - b}{\frac{a}{0.3333333333333333}} \]

   10. Simplified 95.7%

       \[\leadsto \frac{\color{blue}{\left(\frac{a}{b} \cdot \left(c \cdot 1.5\right) - b\right)} - b}{\frac{a}{0.3333333333333333}} \]

   if -0.5 < b < 1.11999999999999997e-46

   1. Initial program 85.6%

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

      1. neg-sub0 85.6%

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

      2. sqr-neg 85.6%

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

      3. associate-+l- 85.6%

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

      4. sub0-neg 85.6%

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

      5. neg-mul-1 85.6%

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

   3. Simplified 85.4%

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

      1. associate-*r* 85.4%

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

      2. metadata-eval 85.4%

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

      3. distribute-rgt-neg-in 85.4%

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

      4. *-commutative 85.4%

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

      5. fma-neg 85.4%

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

      6. associate-*r* 85.4%

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

      7. *-commutative 85.4%

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

      8. associate-*l* 85.4%

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

   5. Applied egg-rr 85.4%

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

      1. *-commutative 85.4%

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

   7. Simplified 85.4%

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

      1. div-inv 85.3%

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

      2. clear-num 85.3%

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

   9. Applied egg-rr 85.3%

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

   10. Taylor expanded in b around 0 71.4%

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

       1. *-commutative 71.4%

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

       2. associate-*r* 71.3%

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

   12. Simplified 71.3%

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

   if 1.11999999999999997e-46 < b

   1. Initial program 18.6%

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

      1. sqr-neg 18.6%

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

      2. sqr-neg 18.6%

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

      3. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in b around inf 85.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.5:\\ \;\;\;\;\frac{\left(\frac{a}{b} \cdot \left(c \cdot 1.5\right) - b\right) - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-46}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4:\\ \;\;\;\;\frac{\left(\frac{a}{b} \cdot \left(c \cdot 1.5\right) - b\right) - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-46}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.4)
   (/ (- (- (* (/ a b) (* c 1.5)) b) b) (/ a 0.3333333333333333))
   (if (<= b 3.8e-46)
     (* 0.3333333333333333 (/ (- (sqrt (* (* a c) -3.0)) b) a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4) {
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333);
	} else if (b <= 3.8e-46) {
		tmp = 0.3333333333333333 * ((sqrt(((a * c) * -3.0)) - b) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.4d0)) then
        tmp = ((((a / b) * (c * 1.5d0)) - b) - b) / (a / 0.3333333333333333d0)
    else if (b <= 3.8d-46) then
        tmp = 0.3333333333333333d0 * ((sqrt(((a * c) * (-3.0d0))) - b) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4) {
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333);
	} else if (b <= 3.8e-46) {
		tmp = 0.3333333333333333 * ((Math.sqrt(((a * c) * -3.0)) - b) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.4:
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333)
	elif b <= 3.8e-46:
		tmp = 0.3333333333333333 * ((math.sqrt(((a * c) * -3.0)) - b) / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.4)
		tmp = Float64(Float64(Float64(Float64(Float64(a / b) * Float64(c * 1.5)) - b) - b) / Float64(a / 0.3333333333333333));
	elseif (b <= 3.8e-46)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(Float64(a * c) * -3.0)) - b) / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.4)
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333);
	elseif (b <= 3.8e-46)
		tmp = 0.3333333333333333 * ((sqrt(((a * c) * -3.0)) - b) / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.4], N[(N[(N[(N[(N[(a / b), $MachinePrecision] * N[(c * 1.5), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e-46], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.39999999999999991

   1. Initial program 66.7%

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

      1. neg-sub0 66.7%

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

      2. sqr-neg 66.7%

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

      3. associate-+l- 66.7%

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

      4. sub0-neg 66.7%

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

      5. neg-mul-1 66.7%

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

   3. Simplified 66.6%

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

      1. associate-*r* 66.6%

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

      2. metadata-eval 66.6%

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

      3. distribute-rgt-neg-in 66.6%

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

      4. *-commutative 66.6%

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

      5. fma-neg 66.6%

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

      6. associate-*r* 66.6%

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

      7. *-commutative 66.6%

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

      8. associate-*l* 66.6%

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

   5. Applied egg-rr 66.6%

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

      1. *-commutative 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in b around -inf 93.0%

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{\frac{a}{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. neg-mul-1 93.0%

         \[\leadsto \frac{\left(\color{blue}{\left(-b\right)} + 1.5 \cdot \frac{a \cdot c}{b}\right) - b}{\frac{a}{0.3333333333333333}} \]

      2. +-commutative 93.0%

         \[\leadsto \frac{\color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + \left(-b\right)\right)} - b}{\frac{a}{0.3333333333333333}} \]

      3. unsub-neg 93.0%

         \[\leadsto \frac{\color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)} - b}{\frac{a}{0.3333333333333333}} \]

      4. associate-/l* 95.7%

         \[\leadsto \frac{\left(1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} - b\right) - b}{\frac{a}{0.3333333333333333}} \]

      5. *-commutative 95.7%

         \[\leadsto \frac{\left(\color{blue}{\frac{a}{\frac{b}{c}} \cdot 1.5} - b\right) - b}{\frac{a}{0.3333333333333333}} \]

      6. associate-/r/ 95.7%

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{a}{b} \cdot c\right)} \cdot 1.5 - b\right) - b}{\frac{a}{0.3333333333333333}} \]

      7. associate-*l* 95.7%

         \[\leadsto \frac{\left(\color{blue}{\frac{a}{b} \cdot \left(c \cdot 1.5\right)} - b\right) - b}{\frac{a}{0.3333333333333333}} \]

   10. Simplified 95.7%

       \[\leadsto \frac{\color{blue}{\left(\frac{a}{b} \cdot \left(c \cdot 1.5\right) - b\right)} - b}{\frac{a}{0.3333333333333333}} \]

   if -3.39999999999999991 < b < 3.7999999999999997e-46

   1. Initial program 85.6%

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

      1. neg-sub0 85.6%

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

      2. sqr-neg 85.6%

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

      3. associate-+l- 85.6%

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

      4. sub0-neg 85.6%

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

      5. neg-mul-1 85.6%

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

   3. Simplified 85.4%

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

      1. associate-*r* 85.4%

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

      2. metadata-eval 85.4%

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

      3. distribute-rgt-neg-in 85.4%

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

      4. *-commutative 85.4%

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

      5. fma-neg 85.4%

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

      6. associate-*r* 85.4%

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

      7. *-commutative 85.4%

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

      8. associate-*l* 85.4%

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

   5. Applied egg-rr 85.4%

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

      1. *-commutative 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in b around 0 71.4%

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

      1. associate-/r/ 71.4%

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

   10. Applied egg-rr 71.4%

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

   if 3.7999999999999997e-46 < b

   1. Initial program 18.6%

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

      1. sqr-neg 18.6%

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

      2. sqr-neg 18.6%

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

      3. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in b around inf 85.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4:\\ \;\;\;\;\frac{\left(\frac{a}{b} \cdot \left(c \cdot 1.5\right) - b\right) - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-46}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.15:\\ \;\;\;\;\frac{\left(\frac{a}{b} \cdot \left(c \cdot 1.5\right) - b\right) - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-47}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -0.15)
   (/ (- (- (* (/ a b) (* c 1.5)) b) b) (/ a 0.3333333333333333))
   (if (<= b 8.5e-47)
     (* 0.3333333333333333 (/ (- (sqrt (* a (* c -3.0))) b) a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -0.15) {
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333);
	} else if (b <= 8.5e-47) {
		tmp = 0.3333333333333333 * ((sqrt((a * (c * -3.0))) - b) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-0.15d0)) then
        tmp = ((((a / b) * (c * 1.5d0)) - b) - b) / (a / 0.3333333333333333d0)
    else if (b <= 8.5d-47) then
        tmp = 0.3333333333333333d0 * ((sqrt((a * (c * (-3.0d0)))) - b) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -0.15) {
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333);
	} else if (b <= 8.5e-47) {
		tmp = 0.3333333333333333 * ((Math.sqrt((a * (c * -3.0))) - b) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -0.15:
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333)
	elif b <= 8.5e-47:
		tmp = 0.3333333333333333 * ((math.sqrt((a * (c * -3.0))) - b) / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -0.15)
		tmp = Float64(Float64(Float64(Float64(Float64(a / b) * Float64(c * 1.5)) - b) - b) / Float64(a / 0.3333333333333333));
	elseif (b <= 8.5e-47)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -0.15)
		tmp = ((((a / b) * (c * 1.5)) - b) - b) / (a / 0.3333333333333333);
	elseif (b <= 8.5e-47)
		tmp = 0.3333333333333333 * ((sqrt((a * (c * -3.0))) - b) / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -0.15], N[(N[(N[(N[(N[(a / b), $MachinePrecision] * N[(c * 1.5), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e-47], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.149999999999999994

   1. Initial program 66.7%

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

      1. neg-sub0 66.7%

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

      2. sqr-neg 66.7%

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

      3. associate-+l- 66.7%

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

      4. sub0-neg 66.7%

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

      5. neg-mul-1 66.7%

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

   3. Simplified 66.6%

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

      1. associate-*r* 66.6%

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

      2. metadata-eval 66.6%

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

      3. distribute-rgt-neg-in 66.6%

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

      4. *-commutative 66.6%

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

      5. fma-neg 66.6%

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

      6. associate-*r* 66.6%

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

      7. *-commutative 66.6%

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

      8. associate-*l* 66.6%

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

   5. Applied egg-rr 66.6%

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

      1. *-commutative 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in b around -inf 93.0%

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{\frac{a}{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. neg-mul-1 93.0%

         \[\leadsto \frac{\left(\color{blue}{\left(-b\right)} + 1.5 \cdot \frac{a \cdot c}{b}\right) - b}{\frac{a}{0.3333333333333333}} \]

      2. +-commutative 93.0%

         \[\leadsto \frac{\color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + \left(-b\right)\right)} - b}{\frac{a}{0.3333333333333333}} \]

      3. unsub-neg 93.0%

         \[\leadsto \frac{\color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)} - b}{\frac{a}{0.3333333333333333}} \]

      4. associate-/l* 95.7%

         \[\leadsto \frac{\left(1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} - b\right) - b}{\frac{a}{0.3333333333333333}} \]

      5. *-commutative 95.7%

         \[\leadsto \frac{\left(\color{blue}{\frac{a}{\frac{b}{c}} \cdot 1.5} - b\right) - b}{\frac{a}{0.3333333333333333}} \]

      6. associate-/r/ 95.7%

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{a}{b} \cdot c\right)} \cdot 1.5 - b\right) - b}{\frac{a}{0.3333333333333333}} \]

      7. associate-*l* 95.7%

         \[\leadsto \frac{\left(\color{blue}{\frac{a}{b} \cdot \left(c \cdot 1.5\right)} - b\right) - b}{\frac{a}{0.3333333333333333}} \]

   10. Simplified 95.7%

       \[\leadsto \frac{\color{blue}{\left(\frac{a}{b} \cdot \left(c \cdot 1.5\right) - b\right)} - b}{\frac{a}{0.3333333333333333}} \]

   if -0.149999999999999994 < b < 8.4999999999999999e-47

   1. Initial program 85.6%

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

      1. neg-sub0 85.6%

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

      2. sqr-neg 85.6%

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

      3. associate-+l- 85.6%

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

      4. sub0-neg 85.6%

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

      5. neg-mul-1 85.6%

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

   3. Simplified 85.4%

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

      1. associate-*r* 85.4%

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

      2. metadata-eval 85.4%

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

      3. distribute-rgt-neg-in 85.4%

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

      4. *-commutative 85.4%

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

      5. fma-neg 85.4%

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

      6. associate-*r* 85.4%

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

      7. *-commutative 85.4%

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

      8. associate-*l* 85.4%

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

   5. Applied egg-rr 85.4%

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

      1. *-commutative 85.4%

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

   7. Simplified 85.4%

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

      1. associate-/r/ 85.6%

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

   9. Applied egg-rr 85.6%

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

   10. Taylor expanded in b around 0 71.4%

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

       1. *-commutative 71.4%

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

       2. associate-*r* 71.3%

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

   12. Simplified 71.5%

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

   if 8.4999999999999999e-47 < b

   1. Initial program 18.6%

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

      1. sqr-neg 18.6%

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

      2. sqr-neg 18.6%

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

      3. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   4. Taylor expanded in b around inf 85.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.15:\\ \;\;\;\;\frac{\left(\frac{a}{b} \cdot \left(c \cdot 1.5\right) - b\right) - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-47}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9: 68.4% accurate, 8.9× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (+ (* (/ c b) 0.5) (* -0.6666666666666666 (/ b a)))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((c / b) * 0.5) + (-0.6666666666666666 * (b / a));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((c / b) * 0.5d0) + ((-0.6666666666666666d0) * (b / a))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((c / b) * 0.5) + (-0.6666666666666666 * (b / a));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = ((c / b) * 0.5) + (-0.6666666666666666 * (b / a))
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(Float64(c / b) * 0.5) + Float64(-0.6666666666666666 * Float64(b / a)));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = ((c / b) * 0.5) + (-0.6666666666666666 * (b / a));
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.9%

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

      1. sqr-neg 74.9%

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

      2. sqr-neg 74.9%

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

      3. associate-*l* 74.9%

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

   3. Simplified 74.9%

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

   4. Taylor expanded in b around -inf 69.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 36.9%

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

      1. sqr-neg 36.9%

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

      2. sqr-neg 36.9%

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

      3. associate-*l* 36.8%

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

   3. Simplified 36.8%

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

   4. Taylor expanded in b around inf 66.1%

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

4. Final simplification 67.8%

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

Alternative 10: 68.4% accurate, 8.9× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (+ (/ (* b -0.6666666666666666) a) (* (/ c b) 0.5))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((b * (-0.6666666666666666d0)) / a) + ((c / b) * 0.5d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(Float64(b * -0.6666666666666666) / a) + Float64(Float64(c / b) * 0.5));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = ((b * -0.6666666666666666) / a) + ((c / b) * 0.5);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.9%

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

      1. sqr-neg 74.9%

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

      2. sqr-neg 74.9%

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

      3. associate-*l* 74.9%

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

   3. Simplified 74.9%

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

   4. Taylor expanded in b around -inf 69.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. associate-*r/ 69.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   6. Applied egg-rr 69.7%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + 0.5 \cdot \frac{c}{b} \]

   if -4.999999999999985e-310 < b

   1. Initial program 36.9%

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

      1. sqr-neg 36.9%

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

      2. sqr-neg 36.9%

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

      3. associate-*l* 36.8%

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

   3. Simplified 36.8%

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

   4. Taylor expanded in b around inf 66.1%

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

4. Final simplification 67.8%

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

Alternative 11: 68.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-277) (/ 1.0 (* (/ a b) -1.5)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-277) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.2d-277) then
        tmp = 1.0d0 / ((a / b) * (-1.5d0))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-277) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-277:
		tmp = 1.0 / ((a / b) * -1.5)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-277)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.2e-277)
		tmp = 1.0 / ((a / b) * -1.5);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-277], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.1999999999999999e-277

   1. Initial program 75.3%

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

      1. neg-sub0 75.3%

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

      2. sqr-neg 75.3%

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

      3. associate-+l- 75.3%

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

      4. sub0-neg 75.3%

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

      5. neg-mul-1 75.3%

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

   3. Simplified 75.2%

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

      1. associate-*r* 75.2%

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

      2. metadata-eval 75.2%

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

      3. distribute-rgt-neg-in 75.2%

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

      4. *-commutative 75.2%

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

      5. fma-neg 75.2%

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

      6. associate-*r* 75.2%

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

      7. *-commutative 75.2%

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

      8. associate-*l* 75.2%

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

   5. Applied egg-rr 75.2%

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

      1. *-commutative 75.2%

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

   7. Simplified 75.2%

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

      1. clear-num 75.2%

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

      2. inv-pow 75.2%

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

      3. div-inv 75.2%

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

      4. metadata-eval 75.2%

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

   9. Applied egg-rr 75.2%

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

       1. unpow-1 75.2%

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

       2. *-commutative 75.2%

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

       3. *-lft-identity 75.2%

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

       4. times-frac 75.3%

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

       5. metadata-eval 75.3%

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

       6. sub-neg 75.3%

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

       7. +-commutative 75.3%

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

       8. distribute-rgt-neg-in 75.3%

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

       9. fma-def 75.4%

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

       10. distribute-rgt-neg-in 75.4%

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

       11. metadata-eval 75.4%

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

   11. Simplified 75.4%

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

   12. Taylor expanded in b around -inf 68.4%

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

       1. associate-*r/ 68.4%

          \[\leadsto \frac{1}{3 \cdot \color{blue}{\frac{-0.5 \cdot a}{b}}} \]

   14. Simplified 68.4%

       \[\leadsto \frac{1}{3 \cdot \color{blue}{\frac{-0.5 \cdot a}{b}}} \]

   15. Taylor expanded in a around 0 68.4%

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

   if 4.1999999999999999e-277 < b

   1. Initial program 35.9%

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

      1. sqr-neg 35.9%

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

      2. sqr-neg 35.9%

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

      3. associate-*l* 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in b around inf 67.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12: 68.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-277) (/ (* b -2.0) (* 3.0 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-277) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.2d-277) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-277) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-277:
		tmp = (b * -2.0) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-277)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.2e-277)
		tmp = (b * -2.0) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-277], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.1999999999999999e-277

   1. Initial program 75.3%

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

      1. sqr-neg 75.3%

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

      2. sqr-neg 75.3%

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

      3. associate-*l* 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in b around -inf 68.4%

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

      1. *-commutative 68.4%

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

   6. Simplified 68.4%

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

   if 4.1999999999999999e-277 < b

   1. Initial program 35.9%

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

      1. sqr-neg 35.9%

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

      2. sqr-neg 35.9%

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

      3. associate-*l* 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in b around inf 67.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 13: 68.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-277) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-277) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.2d-277) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-277) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-277:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-277)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.2e-277)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-277], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.1999999999999999e-277

   1. Initial program 75.3%

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

      1. neg-sub0 75.3%

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

      2. sqr-neg 75.3%

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

      3. associate-+l- 75.3%

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

      4. sub0-neg 75.3%

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

      5. neg-mul-1 75.3%

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

   3. Simplified 75.2%

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

      1. associate-*r* 75.2%

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

      2. metadata-eval 75.2%

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

      3. distribute-rgt-neg-in 75.2%

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

      4. *-commutative 75.2%

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

      5. fma-neg 75.2%

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

      6. associate-*r* 75.2%

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

      7. *-commutative 75.2%

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

      8. associate-*l* 75.2%

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

   5. Applied egg-rr 75.2%

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

      1. *-commutative 75.2%

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

   7. Simplified 75.2%

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

      1. clear-num 75.2%

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

      2. inv-pow 75.2%

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

      3. div-inv 75.2%

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

      4. metadata-eval 75.2%

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

   9. Applied egg-rr 75.2%

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

       1. unpow-1 75.2%

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

       2. *-commutative 75.2%

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

       3. *-lft-identity 75.2%

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

       4. times-frac 75.3%

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

       5. metadata-eval 75.3%

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

       6. sub-neg 75.3%

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

       7. +-commutative 75.3%

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

       8. distribute-rgt-neg-in 75.3%

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

       9. fma-def 75.4%

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

       10. distribute-rgt-neg-in 75.4%

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

       11. metadata-eval 75.4%

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

   11. Simplified 75.4%

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

   12. Taylor expanded in b around -inf 68.4%

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

       1. associate-*r/ 68.4%

          \[\leadsto \frac{1}{3 \cdot \color{blue}{\frac{-0.5 \cdot a}{b}}} \]

   14. Simplified 68.4%

       \[\leadsto \frac{1}{3 \cdot \color{blue}{\frac{-0.5 \cdot a}{b}}} \]
   15. Step-by-step derivation

       1. expm1-log1p-u 36.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{3 \cdot \frac{-0.5 \cdot a}{b}}\right)\right)} \]

       2. expm1-udef 26.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{3 \cdot \frac{-0.5 \cdot a}{b}}\right)} - 1} \]

       3. associate-/r* 26.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{3}}{\frac{-0.5 \cdot a}{b}}}\right)} - 1 \]

       4. metadata-eval 26.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.3333333333333333}}{\frac{-0.5 \cdot a}{b}}\right)} - 1 \]

       5. *-commutative 26.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{0.3333333333333333}{\frac{\color{blue}{a \cdot -0.5}}{b}}\right)} - 1 \]

   16. Applied egg-rr 26.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.3333333333333333}{\frac{a \cdot -0.5}{b}}\right)} - 1} \]
   17. Step-by-step derivation

       1. expm1-def 36.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{\frac{a \cdot -0.5}{b}}\right)\right)} \]

       2. expm1-log1p 68.2%

          \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{a \cdot -0.5}{b}}} \]

       3. associate-/r/ 68.2%

          \[\leadsto \color{blue}{\frac{0.3333333333333333}{a \cdot -0.5} \cdot b} \]

       4. *-commutative 68.2%

          \[\leadsto \color{blue}{b \cdot \frac{0.3333333333333333}{a \cdot -0.5}} \]

       5. *-commutative 68.2%

          \[\leadsto b \cdot \frac{0.3333333333333333}{\color{blue}{-0.5 \cdot a}} \]

       6. associate-/r* 68.2%

          \[\leadsto b \cdot \color{blue}{\frac{\frac{0.3333333333333333}{-0.5}}{a}} \]

       7. metadata-eval 68.2%

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

   18. Simplified 68.2%

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

   if 4.1999999999999999e-277 < b

   1. Initial program 35.9%

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

      1. sqr-neg 35.9%

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

      2. sqr-neg 35.9%

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

      3. associate-*l* 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in b around inf 67.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 14: 68.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-277) (* -0.6666666666666666 (/ b a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-277) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.2d-277) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-277) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-277:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-277)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.2e-277)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-277], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.1999999999999999e-277

   1. Initial program 75.3%

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

      1. sqr-neg 75.3%

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

      2. sqr-neg 75.3%

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

      3. associate-*l* 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in b around -inf 68.2%

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

      1. *-commutative 68.2%

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

   6. Simplified 68.2%

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

   if 4.1999999999999999e-277 < b

   1. Initial program 35.9%

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

      1. sqr-neg 35.9%

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

      2. sqr-neg 35.9%

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

      3. associate-*l* 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in b around inf 67.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 15: 68.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-277}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2e-277) (/ -0.6666666666666666 (/ a b)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-277) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.2d-277) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-277) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2e-277:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2e-277)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.2e-277)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2e-277], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2e-277

   1. Initial program 75.3%

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

      1. neg-sub0 75.3%

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

      2. sqr-neg 75.3%

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

      3. associate-+l- 75.3%

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

      4. sub0-neg 75.3%

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

      5. neg-mul-1 75.3%

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

   3. Simplified 75.2%

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

      1. associate-*r* 75.2%

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

      2. metadata-eval 75.2%

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

      3. distribute-rgt-neg-in 75.2%

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

      4. *-commutative 75.2%

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

      5. fma-neg 75.2%

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

      6. associate-*r* 75.2%

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

      7. *-commutative 75.2%

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

      8. associate-*l* 75.2%

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

   5. Applied egg-rr 75.2%

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

      1. *-commutative 75.2%

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

   7. Simplified 75.2%

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

   8. Taylor expanded in b around -inf 68.2%

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

      1. associate-*r/ 68.3%

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

      2. associate-/l* 68.2%

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

   10. Simplified 68.2%

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

   if 5.2e-277 < b

   1. Initial program 35.9%

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

      1. sqr-neg 35.9%

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

      2. sqr-neg 35.9%

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

      3. associate-*l* 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in b around inf 67.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-277}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 16: 68.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-277) (/ (* b -0.6666666666666666) a) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-277) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.2d-277) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-277) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-277:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-277)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.2e-277)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-277], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.1999999999999999e-277

   1. Initial program 75.3%

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

      1. neg-sub0 75.3%

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

      2. sqr-neg 75.3%

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

      3. associate-+l- 75.3%

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

      4. sub0-neg 75.3%

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

      5. neg-mul-1 75.3%

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

   3. Simplified 75.2%

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

      1. associate-*r* 75.2%

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

      2. metadata-eval 75.2%

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

      3. distribute-rgt-neg-in 75.2%

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

      4. *-commutative 75.2%

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

      5. fma-neg 75.2%

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

      6. associate-*r* 75.2%

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

      7. *-commutative 75.2%

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

      8. associate-*l* 75.2%

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

   5. Applied egg-rr 75.2%

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

      1. *-commutative 75.2%

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

   7. Simplified 75.2%

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

      1. clear-num 75.2%

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

      2. inv-pow 75.2%

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

      3. div-inv 75.2%

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

      4. metadata-eval 75.2%

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

   9. Applied egg-rr 75.2%

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

       1. unpow-1 75.2%

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

       2. *-commutative 75.2%

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

       3. *-lft-identity 75.2%

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

       4. times-frac 75.3%

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

       5. metadata-eval 75.3%

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

       6. sub-neg 75.3%

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

       7. +-commutative 75.3%

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

       8. distribute-rgt-neg-in 75.3%

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

       9. fma-def 75.4%

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

       10. distribute-rgt-neg-in 75.4%

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

       11. metadata-eval 75.4%

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

   11. Simplified 75.4%

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

   12. Taylor expanded in b around -inf 68.2%

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

       1. associate-*r/ 68.3%

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

   14. Simplified 68.3%

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

   if 4.1999999999999999e-277 < b

   1. Initial program 35.9%

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

      1. sqr-neg 35.9%

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

      2. sqr-neg 35.9%

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

      3. associate-*l* 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in b around inf 67.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-277}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 17: 35.8% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

   1. sqr-neg 55.2%

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

   2. sqr-neg 55.2%

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

   3. associate-*l* 55.1%

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

3. Simplified 55.1%

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

4. Taylor expanded in b around inf 35.3%

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

5. Final simplification 35.3%

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

Reproduce

herbie shell --seed 2023272 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))