Cubic critical

Percentage Accurate: 51.1% → 85.8%

Time: 13.5s

Alternatives: 13

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.1% 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: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+139}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+139)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 1.4e-92)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+139) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.4e-92) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d+139)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 1.4d-92) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+139) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.4e-92) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+139:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 1.4e-92:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+139)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 1.4e-92)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+139)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 1.4e-92)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+139], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4e-92], 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[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000003e139

   1. Initial program 31.4%

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

      1. sqr-neg 31.4%

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

      2. sqr-neg 31.4%

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

      3. associate-*l* 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in b around -inf 90.5%

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

      1. *-commutative 90.5%

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

   7. Simplified 90.5%

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

   if -5.0000000000000003e139 < b < 1.4e-92

   1. Initial program 88.5%

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

   if 1.4e-92 < b

   1. Initial program 18.7%

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

      1. sqr-neg 18.7%

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

      2. sqr-neg 18.7%

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

      3. associate-*l* 18.7%

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

   3. Simplified 18.7%

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

      1. frac-2neg 18.7%

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

      2. div-inv 18.7%

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

   6. Applied egg-rr 18.7%

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

   7. Taylor expanded in b around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

   9. Simplified 86.2%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+139}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+140}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+140)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 2.85e-92)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+140) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.85e-92) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d+140)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 2.85d-92) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+140) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.85e-92) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+140:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 2.85e-92:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+140)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2.85e-92)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+140)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 2.85e-92)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+140], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.85e-92], 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[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.00000000000000012e140

   1. Initial program 31.4%

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

      1. sqr-neg 31.4%

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

      2. sqr-neg 31.4%

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

      3. associate-*l* 31.4%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in b around -inf 90.5%

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

      1. *-commutative 90.5%

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

   7. Simplified 90.5%

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

   if -2.00000000000000012e140 < b < 2.85000000000000004e-92

   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. sqr-neg 88.5%

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

      2. sqr-neg 88.5%

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

      3. associate-*l* 88.3%

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

   3. Simplified 88.3%

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

   if 2.85000000000000004e-92 < b

   1. Initial program 18.7%

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

      1. sqr-neg 18.7%

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

      2. sqr-neg 18.7%

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

      3. associate-*l* 18.7%

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

   3. Simplified 18.7%

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

      1. frac-2neg 18.7%

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

      2. div-inv 18.7%

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

   6. Applied egg-rr 18.7%

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

   7. Taylor expanded in b around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

   9. Simplified 86.2%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+140}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-62}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-62)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 3e-92)
     (/ (- (sqrt (* c (* a -3.0))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-62) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 3e-92) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-62)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 3d-92) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-62) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 3e-92) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-62:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 3e-92:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-62)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 3e-92)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-62)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 3e-92)
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-62], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3e-92], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000002e-62

   1. Initial program 62.1%

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

      1. sqr-neg 62.1%

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

      2. sqr-neg 62.1%

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

      3. associate-*l* 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in b around -inf 85.1%

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

      1. *-commutative 85.1%

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

   7. Simplified 85.1%

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

   if -5.0000000000000002e-62 < b < 3.00000000000000013e-92

   1. Initial program 85.5%

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

      1. sqr-neg 85.5%

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

      2. sqr-neg 85.5%

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

      3. associate-*l* 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in b around 0 78.5%

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

      1. +-commutative 78.5%

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

      2. unsub-neg 78.5%

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

      3. associate-*r* 78.8%

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

      4. *-commutative 78.8%

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

      5. *-commutative 78.8%

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

   7. Applied egg-rr 78.8%

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

   if 3.00000000000000013e-92 < b

   1. Initial program 18.7%

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

      1. sqr-neg 18.7%

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

      2. sqr-neg 18.7%

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

      3. associate-*l* 18.7%

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

   3. Simplified 18.7%

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

      1. frac-2neg 18.7%

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

      2. div-inv 18.7%

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

   6. Applied egg-rr 18.7%

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

   7. Taylor expanded in b around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

   9. Simplified 86.2%

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

Alternative 4: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-65}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-92}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e-65)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 1.4e-92)
     (* (/ -0.3333333333333333 a) (- b (sqrt (* c (* a -3.0)))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-65) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.4e-92) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt((c * (a * -3.0))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.95d-65)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 1.4d-92) then
        tmp = ((-0.3333333333333333d0) / a) * (b - sqrt((c * (a * (-3.0d0)))))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-65) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.4e-92) {
		tmp = (-0.3333333333333333 / a) * (b - Math.sqrt((c * (a * -3.0))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e-65:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 1.4e-92:
		tmp = (-0.3333333333333333 / a) * (b - math.sqrt((c * (a * -3.0))))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e-65)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 1.4e-92)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(Float64(c * Float64(a * -3.0)))));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.95e-65)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 1.4e-92)
		tmp = (-0.3333333333333333 / a) * (b - sqrt((c * (a * -3.0))));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e-65], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4e-92], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9500000000000002e-65

   1. Initial program 62.1%

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

      1. sqr-neg 62.1%

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

      2. sqr-neg 62.1%

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

      3. associate-*l* 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in b around -inf 85.1%

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

      1. *-commutative 85.1%

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

   7. Simplified 85.1%

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

   if -1.9500000000000002e-65 < b < 1.4e-92

   1. Initial program 85.5%

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

      1. sqr-neg 85.5%

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

      2. sqr-neg 85.5%

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

      3. associate-*l* 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in b around 0 78.5%

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

      1. frac-2neg 78.5%

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

      2. div-inv 78.4%

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

      3. distribute-neg-in 78.4%

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

      4. add-sqr-sqrt 31.1%

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

      5. sqrt-unprod 77.7%

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

      6. sqr-neg 77.7%

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

      7. sqrt-unprod 47.2%

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

      8. add-sqr-sqrt 76.4%

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

      9. sub-neg 76.4%

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

      10. add-sqr-sqrt 29.3%

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

      11. sqrt-unprod 76.7%

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

      12. sqr-neg 76.7%

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

      13. sqrt-unprod 47.3%

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

      14. add-sqr-sqrt 78.4%

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

      15. associate-*r* 78.8%

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

      16. *-commutative 78.8%

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

      17. *-commutative 78.8%

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

      18. distribute-lft-neg-in 78.8%

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

      19. metadata-eval 78.8%

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

      20. *-commutative 78.8%

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

      21. associate-/r* 78.5%

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

   7. Applied egg-rr 78.5%

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

      1. *-commutative 78.5%

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

      2. associate-*l/ 78.6%

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

      3. metadata-eval 78.6%

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

   9. Simplified 78.6%

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

   if 1.4e-92 < b

   1. Initial program 18.7%

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

      1. sqr-neg 18.7%

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

      2. sqr-neg 18.7%

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

      3. associate-*l* 18.7%

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

   3. Simplified 18.7%

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

      1. frac-2neg 18.7%

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

      2. div-inv 18.7%

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

   6. Applied egg-rr 18.7%

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

   7. Taylor expanded in b around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

   9. Simplified 86.2%

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

Alternative 5: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-120}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-95}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-120)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 2.4e-95)
     (* 0.3333333333333333 (/ (+ b (sqrt (* c (* a -3.0)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-120) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.4e-95) {
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3d-120)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 2.4d-95) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((c * (a * (-3.0d0))))) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-120) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.4e-95) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-120:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 2.4e-95:
		tmp = 0.3333333333333333 * ((b + math.sqrt((c * (a * -3.0)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-120)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2.4e-95)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e-120)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 2.4e-95)
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-120], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-95], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000011e-120

   1. Initial program 64.9%

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

      1. sqr-neg 64.9%

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

      2. sqr-neg 64.9%

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

      3. associate-*l* 64.9%

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

   3. Simplified 64.9%

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

   5. Taylor expanded in b around -inf 81.2%

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

      1. *-commutative 81.2%

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

   7. Simplified 81.2%

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

   if -3.00000000000000011e-120 < b < 2.4e-95

   1. Initial program 85.0%

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

      1. sqr-neg 85.0%

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

      2. sqr-neg 85.0%

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

      3. associate-*l* 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in b around 0 82.1%

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

      1. *-un-lft-identity 82.1%

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

      2. times-frac 82.1%

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

      3. metadata-eval 82.1%

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

      4. add-sqr-sqrt 27.9%

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

      5. sqrt-unprod 81.3%

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

      6. sqr-neg 81.3%

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

      7. sqrt-unprod 54.0%

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

      8. add-sqr-sqrt 80.9%

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

      9. associate-*r* 81.0%

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

      10. *-commutative 81.0%

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

      11. *-commutative 81.0%

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

   7. Applied egg-rr 81.0%

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

   if 2.4e-95 < b

   1. Initial program 18.7%

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

      1. sqr-neg 18.7%

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

      2. sqr-neg 18.7%

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

      3. associate-*l* 18.7%

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

   3. Simplified 18.7%

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

      1. frac-2neg 18.7%

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

      2. div-inv 18.7%

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

   6. Applied egg-rr 18.7%

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

   7. Taylor expanded in b around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

   9. Simplified 86.2%

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

Alternative 6: 68.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2e-309) (/ (* b -2.0) (* 3.0 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-309) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2d-309) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-309) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2e-309:
		tmp = (b * -2.0) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2e-309)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2e-309)
		tmp = (b * -2.0) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2e-309], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9999999999999988e-309

   1. Initial program 68.2%

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

      1. sqr-neg 68.2%

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

      2. sqr-neg 68.2%

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

      3. associate-*l* 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in b around -inf 67.2%

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

      1. *-commutative 67.2%

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

   7. Simplified 67.2%

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

   if 1.9999999999999988e-309 < b

   1. Initial program 37.3%

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

      1. sqr-neg 37.3%

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

      2. sqr-neg 37.3%

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

      3. associate-*l* 37.2%

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

   3. Simplified 37.2%

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

      1. frac-2neg 37.2%

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

      2. div-inv 37.2%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in b around inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. *-commutative 65.1%

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

   9. Simplified 65.1%

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

Alternative 7: 68.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left(b \cdot \frac{1}{a}\right) \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (* (* b (/ 1.0 a)) -0.6666666666666666) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (b * (1.0 / a)) * -0.6666666666666666;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-311)) then
        tmp = (b * (1.0d0 / a)) * (-0.6666666666666666d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (b * (1.0 / a)) * -0.6666666666666666;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = (b * (1.0 / a)) * -0.6666666666666666
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(Float64(b * Float64(1.0 / a)) * -0.6666666666666666);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = (b * (1.0 / a)) * -0.6666666666666666;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(N[(b * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 68.2%

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

      1. sqr-neg 68.2%

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

      2. sqr-neg 68.2%

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

      3. associate-*l* 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. *-commutative 67.1%

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

   7. Simplified 67.1%

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

      1. div-inv 67.1%

         \[\leadsto \color{blue}{\left(b \cdot \frac{1}{a}\right)} \cdot -0.6666666666666666 \]

   9. Applied egg-rr 67.1%

      \[\leadsto \color{blue}{\left(b \cdot \frac{1}{a}\right)} \cdot -0.6666666666666666 \]

   if -5.00000000000023e-311 < b

   1. Initial program 37.3%

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

      1. sqr-neg 37.3%

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

      2. sqr-neg 37.3%

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

      3. associate-*l* 37.2%

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

   3. Simplified 37.2%

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

      1. frac-2neg 37.2%

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

      2. div-inv 37.2%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in b around inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. *-commutative 65.1%

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

   9. Simplified 65.1%

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

Alternative 8: 68.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-311)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 68.2%

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

      1. sqr-neg 68.2%

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

      2. sqr-neg 68.2%

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

      3. associate-*l* 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. *-commutative 67.1%

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

   7. Simplified 67.1%

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

      1. associate-*l/ 67.1%

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

   9. Applied egg-rr 67.1%

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

   if -5.00000000000023e-311 < b

   1. Initial program 37.3%

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

      1. sqr-neg 37.3%

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

      2. sqr-neg 37.3%

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

      3. associate-*l* 37.2%

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

   3. Simplified 37.2%

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

      1. frac-2neg 37.2%

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

      2. div-inv 37.2%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in b around inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. *-commutative 65.1%

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

   9. Simplified 65.1%

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

Alternative 9: 68.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-309}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.5e-309) (/ (* b -0.6666666666666666) a) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.5e-309) {
		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 <= 6.5d-309) 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 <= 6.5e-309) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.5e-309:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.5e-309)
		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 <= 6.5e-309)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.5e-309], 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 < 6.4999999999999999e-309

   1. Initial program 68.2%

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

      1. sqr-neg 68.2%

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

      2. sqr-neg 68.2%

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

      3. associate-*l* 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. *-commutative 67.1%

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

   7. Simplified 67.1%

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

      1. associate-*l/ 67.1%

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

   9. Applied egg-rr 67.1%

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

   if 6.4999999999999999e-309 < b

   1. Initial program 37.3%

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

      1. sqr-neg 37.3%

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

      2. sqr-neg 37.3%

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

      3. associate-*l* 37.2%

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

   3. Simplified 37.2%

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

   5. Taylor expanded in b around inf 65.1%

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

      1. *-commutative 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-309}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-309}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.5e-309) (* -0.6666666666666666 (/ b a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.5e-309) {
		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 <= 4.5d-309) 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 <= 4.5e-309) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.5e-309:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.5e-309)
		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 <= 4.5e-309)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.5e-309], 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 < 4.5000000000000011e-309

   1. Initial program 68.2%

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

      1. sqr-neg 68.2%

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

      2. sqr-neg 68.2%

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

      3. associate-*l* 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. *-commutative 67.1%

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

   7. Simplified 67.1%

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

   if 4.5000000000000011e-309 < b

   1. Initial program 37.3%

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

      1. sqr-neg 37.3%

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

      2. sqr-neg 37.3%

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

      3. associate-*l* 37.2%

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

   3. Simplified 37.2%

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

   5. Taylor expanded in b around inf 65.1%

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

      1. *-commutative 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-309}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-299}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e-299) (* -0.6666666666666666 (/ b a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-299) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.5d-299)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-299) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.5e-299:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e-299)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.5e-299)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e-299], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -2.49999999999999978e-299

   1. Initial program 67.9%

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

      1. sqr-neg 67.9%

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

      2. sqr-neg 67.9%

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

      3. associate-*l* 67.8%

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

   3. Simplified 67.8%

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

   5. Taylor expanded in b around -inf 67.6%

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

      1. *-commutative 67.6%

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

   7. Simplified 67.6%

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

   if -2.49999999999999978e-299 < b

   1. Initial program 37.7%

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

      1. sqr-neg 37.7%

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

      2. sqr-neg 37.7%

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

      3. associate-*l* 37.6%

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

   3. Simplified 37.6%

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

      1. frac-2neg 37.6%

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

      2. div-inv 37.6%

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

   6. Applied egg-rr 37.7%

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

      1. *-commutative 37.7%

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

      2. sub-neg 37.7%

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

      3. distribute-lft-in 37.4%

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

      4. associate-/r* 35.5%

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

      5. div-inv 34.6%

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

      6. metadata-eval 34.6%

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

      7. associate-/r* 35.1%

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

      8. div-inv 37.3%

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

      9. metadata-eval 37.3%

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

      10. fma-undefine 37.3%

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

      11. *-commutative 37.3%

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

      12. *-commutative 37.3%

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

      13. associate-*r* 37.3%

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

      14. add-sqr-sqrt 33.6%

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

      15. hypot-define 35.4%

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

      16. associate-*r* 35.4%

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

      17. *-commutative 35.4%

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

      18. *-commutative 35.4%

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

   8. Applied egg-rr 35.4%

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

   9. Taylor expanded in c around 0 11.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b}{a} + 0.3333333333333333 \cdot \frac{b}{a}} \]
   10. Step-by-step derivation

       1. distribute-rgt-out 11.4%

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

       2. metadata-eval 11.4%

          \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

       3. mul0-rgt 16.2%

          \[\leadsto \color{blue}{0} \]

   11. Simplified 16.2%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-299}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-299}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e-299) (* b (/ -0.6666666666666666 a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-299) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.5d-299)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-299) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.5e-299:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e-299)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.5e-299)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e-299], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -2.49999999999999978e-299

   1. Initial program 67.9%

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

      1. sqr-neg 67.9%

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

      2. sqr-neg 67.9%

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

      3. associate-*l* 67.8%

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

   3. Simplified 67.8%

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

   5. Taylor expanded in b around -inf 67.6%

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

      1. *-commutative 67.6%

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

   7. Simplified 67.6%

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

      1. *-commutative 67.6%

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

      2. clear-num 67.6%

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

      3. un-div-inv 67.6%

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

   9. Applied egg-rr 67.6%

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

       1. associate-/r/ 67.6%

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

   11. Simplified 67.6%

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

   if -2.49999999999999978e-299 < b

   1. Initial program 37.7%

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

      1. sqr-neg 37.7%

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

      2. sqr-neg 37.7%

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

      3. associate-*l* 37.6%

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

   3. Simplified 37.6%

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

      1. frac-2neg 37.6%

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

      2. div-inv 37.6%

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

   6. Applied egg-rr 37.7%

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

      1. *-commutative 37.7%

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

      2. sub-neg 37.7%

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

      3. distribute-lft-in 37.4%

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

      4. associate-/r* 35.5%

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

      5. div-inv 34.6%

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

      6. metadata-eval 34.6%

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

      7. associate-/r* 35.1%

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

      8. div-inv 37.3%

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

      9. metadata-eval 37.3%

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

      10. fma-undefine 37.3%

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

      11. *-commutative 37.3%

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

      12. *-commutative 37.3%

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

      13. associate-*r* 37.3%

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

      14. add-sqr-sqrt 33.6%

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

      15. hypot-define 35.4%

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

      16. associate-*r* 35.4%

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

      17. *-commutative 35.4%

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

      18. *-commutative 35.4%

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

   8. Applied egg-rr 35.4%

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

   9. Taylor expanded in c around 0 11.4%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b}{a} + 0.3333333333333333 \cdot \frac{b}{a}} \]
   10. Step-by-step derivation

       1. distribute-rgt-out 11.4%

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

       2. metadata-eval 11.4%

          \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

       3. mul0-rgt 16.2%

          \[\leadsto \color{blue}{0} \]

   11. Simplified 16.2%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-299}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 11.1% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 50.3%

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

   1. sqr-neg 50.3%

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

   2. sqr-neg 50.3%

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

   3. associate-*l* 50.2%

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

3. Simplified 50.2%

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

   1. frac-2neg 50.2%

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

   2. div-inv 50.2%

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

6. Applied egg-rr 50.3%

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

   1. *-commutative 50.3%

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

   2. sub-neg 50.3%

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

   3. distribute-lft-in 50.2%

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

   4. associate-/r* 49.0%

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

   5. div-inv 48.5%

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

   6. metadata-eval 48.5%

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

   7. associate-/r* 48.8%

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

   8. div-inv 50.0%

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

   9. metadata-eval 50.0%

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

   10. fma-undefine 50.0%

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

   11. *-commutative 50.0%

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

   12. *-commutative 50.0%

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

   13. associate-*r* 50.0%

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

   14. add-sqr-sqrt 43.1%

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

   15. hypot-define 50.1%

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

   16. associate-*r* 50.1%

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

   17. *-commutative 50.1%

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

   18. *-commutative 50.1%

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

8. Applied egg-rr 50.1%

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

9. Taylor expanded in c around 0 7.7%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b}{a} + 0.3333333333333333 \cdot \frac{b}{a}} \]
10. Step-by-step derivation

    1. distribute-rgt-out 7.7%

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

    2. metadata-eval 7.7%

       \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

    3. mul0-rgt 10.6%

       \[\leadsto \color{blue}{0} \]

11. Simplified 10.6%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))