quadp (p42, positive)

Percentage Accurate: 52.5% → 84.5%

Time: 11.6s

Alternatives: 8

Speedup: 19.1×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * 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) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * 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) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.7e-7)
   (/ (- b) a)
   (if (<= b 1.26e-88)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.7e-7) {
		tmp = -b / a;
	} else if (b <= 1.26e-88) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.7e-7)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.26e-88)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.7e-7], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.26e-88], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.7e-7

   1. Initial program 61.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 93.0%

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

      1. associate-*r/ 93.0%

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

      2. mul-1-neg 93.0%

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

   6. Simplified 93.0%

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

   if -4.7e-7 < b < 1.25999999999999992e-88

   1. Initial program 75.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. +-commutative 75.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]

      2. unsub-neg 75.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a} \]

      3. fma-neg 75.5%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} - b}{2 \cdot a} \]

      4. distribute-lft-neg-in 75.5%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)} - b}{2 \cdot a} \]

      5. *-commutative 75.5%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{2 \cdot a} \]

      6. *-commutative 75.5%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right)} \cdot \left(-4\right)\right)} - b}{2 \cdot a} \]

      7. associate-*l* 75.6%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(a \cdot \left(-4\right)\right)}\right)} - b}{2 \cdot a} \]

      8. metadata-eval 75.6%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{2 \cdot a} \]

      9. *-commutative 75.6%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{\color{blue}{a \cdot 2}} \]

   3. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]

   if 1.25999999999999992e-88 < b

   1. Initial program 10.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 10.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 10.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   6. Simplified 86.9%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-7)
   (/ (- b) a)
   (if (<= b 7.6e-91)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-7) {
		tmp = -b / a;
	} else if (b <= 7.6e-91) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -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 <= (-6d-7)) then
        tmp = -b / a
    else if (b <= 7.6d-91) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-7) {
		tmp = -b / a;
	} else if (b <= 7.6e-91) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-7:
		tmp = -b / a
	elif b <= 7.6e-91:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-7)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 7.6e-91)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e-7)
		tmp = -b / a;
	elseif (b <= 7.6e-91)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-7], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 7.6e-91], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.9999999999999997e-7

   1. Initial program 61.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 93.0%

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

      1. associate-*r/ 93.0%

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

      2. mul-1-neg 93.0%

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

   6. Simplified 93.0%

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

   if -5.9999999999999997e-7 < b < 7.59999999999999957e-91

   1. Initial program 75.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   if 7.59999999999999957e-91 < b

   1. Initial program 10.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 10.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 10.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   6. Simplified 86.9%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{-86}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-87}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.55e-86)
   (- (/ c b) (/ b a))
   (if (<= b 1.12e-87)
     (* 0.5 (* (sqrt (* a (* c -4.0))) (/ 1.0 a)))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.55e-86) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.12e-87) {
		tmp = 0.5 * (sqrt((a * (c * -4.0))) * (1.0 / a));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.55d-86)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.12d-87) then
        tmp = 0.5d0 * (sqrt((a * (c * (-4.0d0)))) * (1.0d0 / a))
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.55e-86) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.12e-87) {
		tmp = 0.5 * (Math.sqrt((a * (c * -4.0))) * (1.0 / a));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.55e-86:
		tmp = (c / b) - (b / a)
	elif b <= 1.12e-87:
		tmp = 0.5 * (math.sqrt((a * (c * -4.0))) * (1.0 / a))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.55e-86)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.12e-87)
		tmp = Float64(0.5 * Float64(sqrt(Float64(a * Float64(c * -4.0))) * Float64(1.0 / a)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.55e-86)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.12e-87)
		tmp = 0.5 * (sqrt((a * (c * -4.0))) * (1.0 / a));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.55e-86], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.12e-87], N[(0.5 * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.55000000000000003e-86

   1. Initial program 66.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 88.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   5. Step-by-step derivation

      1. +-commutative 88.8%

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

      2. mul-1-neg 88.8%

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

      3. unsub-neg 88.8%

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

   6. Simplified 88.8%

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

   if -2.55000000000000003e-86 < b < 1.1199999999999999e-87

   1. Initial program 72.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. +-commutative 72.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]

      2. unsub-neg 72.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a} \]

      3. fma-neg 72.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} - b}{2 \cdot a} \]

      4. distribute-lft-neg-in 72.3%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)} - b}{2 \cdot a} \]

      5. *-commutative 72.3%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{2 \cdot a} \]

      6. *-commutative 72.3%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right)} \cdot \left(-4\right)\right)} - b}{2 \cdot a} \]

      7. associate-*l* 72.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(a \cdot \left(-4\right)\right)}\right)} - b}{2 \cdot a} \]

      8. metadata-eval 72.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{2 \cdot a} \]

      9. *-commutative 72.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{\color{blue}{a \cdot 2}} \]

   3. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]

   5. Applied egg-rr 72.1%

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

      1. associate--r- 72.1%

         \[\leadsto \frac{\sqrt{\color{blue}{\left({b}^{2} - a \cdot \left(-4 \cdot c\right)\right) + \left(a \cdot \left(-4 \cdot c\right) + a \cdot \left(-4 \cdot c\right)\right)}} - b}{a \cdot 2} \]

      2. *-commutative 72.1%

         \[\leadsto \frac{\sqrt{\left({b}^{2} - a \cdot \color{blue}{\left(c \cdot -4\right)}\right) + \left(a \cdot \left(-4 \cdot c\right) + a \cdot \left(-4 \cdot c\right)\right)} - b}{a \cdot 2} \]

      3. count-2 72.1%

         \[\leadsto \frac{\sqrt{\left({b}^{2} - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{2 \cdot \left(a \cdot \left(-4 \cdot c\right)\right)}} - b}{a \cdot 2} \]

      4. *-commutative 72.1%

         \[\leadsto \frac{\sqrt{\left({b}^{2} - a \cdot \left(c \cdot -4\right)\right) + 2 \cdot \left(a \cdot \color{blue}{\left(c \cdot -4\right)}\right)} - b}{a \cdot 2} \]

   7. Simplified 72.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left({b}^{2} - a \cdot \left(c \cdot -4\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot -4\right)\right)}} - b}{a \cdot 2} \]

   8. Taylor expanded in b around 0 66.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{a} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 66.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)} \cdot \frac{1}{a}\right)} \]

      2. distribute-rgt-out-- 67.1%

         \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 - -4\right)}} \cdot \frac{1}{a}\right) \]

      3. metadata-eval 67.1%

         \[\leadsto 0.5 \cdot \left(\sqrt{\left(a \cdot c\right) \cdot \color{blue}{-4}} \cdot \frac{1}{a}\right) \]

      4. associate-*r* 67.1%

         \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}} \cdot \frac{1}{a}\right) \]

   10. Simplified 67.1%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \frac{1}{a}\right)} \]

   if 1.1199999999999999e-87 < b

   1. Initial program 10.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 10.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 10.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   6. Simplified 86.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{-86}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-87}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-91)
   (- (/ c b) (/ b a))
   (if (<= b 1.85e-95)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-91) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.85e-95) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -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 <= (-8.5d-91)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.85d-95) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-91) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.85e-95) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e-91:
		tmp = (c / b) - (b / a)
	elif b <= 1.85e-95:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-91)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.85e-95)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e-91)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.85e-95)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-91], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.85e-95], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.49999999999999985e-91

   1. Initial program 66.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 88.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   5. Step-by-step derivation

      1. +-commutative 88.8%

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

      2. mul-1-neg 88.8%

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

      3. unsub-neg 88.8%

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

   6. Simplified 88.8%

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

   if -8.49999999999999985e-91 < b < 1.84999999999999997e-95

   1. Initial program 72.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. +-commutative 72.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]

      2. unsub-neg 72.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a} \]

      3. fma-neg 72.9%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} - b}{2 \cdot a} \]

      4. distribute-lft-neg-in 72.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)} - b}{2 \cdot a} \]

      5. *-commutative 72.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{2 \cdot a} \]

      6. *-commutative 72.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right)} \cdot \left(-4\right)\right)} - b}{2 \cdot a} \]

      7. associate-*l* 72.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(a \cdot \left(-4\right)\right)}\right)} - b}{2 \cdot a} \]

      8. metadata-eval 72.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{2 \cdot a} \]

      9. *-commutative 72.9%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{\color{blue}{a \cdot 2}} \]

   3. Simplified 72.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]

   5. Applied egg-rr 72.6%

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

      1. associate--r- 72.6%

         \[\leadsto \frac{\sqrt{\color{blue}{\left({b}^{2} - a \cdot \left(-4 \cdot c\right)\right) + \left(a \cdot \left(-4 \cdot c\right) + a \cdot \left(-4 \cdot c\right)\right)}} - b}{a \cdot 2} \]

      2. *-commutative 72.6%

         \[\leadsto \frac{\sqrt{\left({b}^{2} - a \cdot \color{blue}{\left(c \cdot -4\right)}\right) + \left(a \cdot \left(-4 \cdot c\right) + a \cdot \left(-4 \cdot c\right)\right)} - b}{a \cdot 2} \]

      3. count-2 72.6%

         \[\leadsto \frac{\sqrt{\left({b}^{2} - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{2 \cdot \left(a \cdot \left(-4 \cdot c\right)\right)}} - b}{a \cdot 2} \]

      4. *-commutative 72.6%

         \[\leadsto \frac{\sqrt{\left({b}^{2} - a \cdot \left(c \cdot -4\right)\right) + 2 \cdot \left(a \cdot \color{blue}{\left(c \cdot -4\right)}\right)} - b}{a \cdot 2} \]

   7. Simplified 72.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left({b}^{2} - a \cdot \left(c \cdot -4\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot -4\right)\right)}} - b}{a \cdot 2} \]

   8. Taylor expanded in b around 0 68.4%

      \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]
   9. Step-by-step derivation

      1. distribute-rgt-out-- 68.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 - -4\right)}} - b}{a \cdot 2} \]

      2. metadata-eval 68.8%

         \[\leadsto \frac{\sqrt{\left(a \cdot c\right) \cdot \color{blue}{-4}} - b}{a \cdot 2} \]

      3. *-commutative 68.8%

         \[\leadsto \frac{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]

      4. associate-*r* 68.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}} - b}{a \cdot 2} \]

      5. *-commutative 68.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot -4\right)} \cdot c} - b}{a \cdot 2} \]

      6. metadata-eval 68.8%

         \[\leadsto \frac{\sqrt{\left(a \cdot \color{blue}{\left(-8 - -4\right)}\right) \cdot c} - b}{a \cdot 2} \]

      7. distribute-rgt-out-- 68.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot a - -4 \cdot a\right)} \cdot c} - b}{a \cdot 2} \]

      8. *-commutative 68.8%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-8 \cdot a - -4 \cdot a\right)}} - b}{a \cdot 2} \]

      9. distribute-rgt-out-- 68.8%

         \[\leadsto \frac{\sqrt{c \cdot \color{blue}{\left(a \cdot \left(-8 - -4\right)\right)}} - b}{a \cdot 2} \]

      10. metadata-eval 68.8%

          \[\leadsto \frac{\sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)} - b}{a \cdot 2} \]

   10. Simplified 68.8%

       \[\leadsto \frac{\color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)}} - b}{a \cdot 2} \]

   if 1.84999999999999997e-95 < b

   1. Initial program 11.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 11.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 11.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e-100)
   (- (/ c b) (/ b a))
   (if (<= b 1.08e-91) (* 0.5 (/ (sqrt (* c (* a -4.0))) a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-100) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.08e-91) {
		tmp = 0.5 * (sqrt((c * (a * -4.0))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.6d-100)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.08d-91) then
        tmp = 0.5d0 * (sqrt((c * (a * (-4.0d0)))) / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-100) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.08e-91) {
		tmp = 0.5 * (Math.sqrt((c * (a * -4.0))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e-100:
		tmp = (c / b) - (b / a)
	elif b <= 1.08e-91:
		tmp = 0.5 * (math.sqrt((c * (a * -4.0))) / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e-100)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.08e-91)
		tmp = Float64(0.5 * Float64(sqrt(Float64(c * Float64(a * -4.0))) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.6e-100)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.08e-91)
		tmp = 0.5 * (sqrt((c * (a * -4.0))) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e-100], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.08e-91], N[(0.5 * N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.60000000000000008e-100

   1. Initial program 66.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 88.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   5. Step-by-step derivation

      1. +-commutative 88.8%

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

      2. mul-1-neg 88.8%

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

      3. unsub-neg 88.8%

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

   6. Simplified 88.8%

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

   if -1.60000000000000008e-100 < b < 1.07999999999999998e-91

   1. Initial program 72.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. +-commutative 72.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}}{2 \cdot a} \]

      2. unsub-neg 72.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a} \]

      3. fma-neg 72.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} - b}{2 \cdot a} \]

      4. distribute-lft-neg-in 72.3%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)} - b}{2 \cdot a} \]

      5. *-commutative 72.3%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{2 \cdot a} \]

      6. *-commutative 72.3%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right)} \cdot \left(-4\right)\right)} - b}{2 \cdot a} \]

      7. associate-*l* 72.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(a \cdot \left(-4\right)\right)}\right)} - b}{2 \cdot a} \]

      8. metadata-eval 72.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{2 \cdot a} \]

      9. *-commutative 72.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{\color{blue}{a \cdot 2}} \]

   3. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]

   5. Applied egg-rr 72.1%

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

      1. associate--r- 72.1%

         \[\leadsto \frac{\sqrt{\color{blue}{\left({b}^{2} - a \cdot \left(-4 \cdot c\right)\right) + \left(a \cdot \left(-4 \cdot c\right) + a \cdot \left(-4 \cdot c\right)\right)}} - b}{a \cdot 2} \]

      2. *-commutative 72.1%

         \[\leadsto \frac{\sqrt{\left({b}^{2} - a \cdot \color{blue}{\left(c \cdot -4\right)}\right) + \left(a \cdot \left(-4 \cdot c\right) + a \cdot \left(-4 \cdot c\right)\right)} - b}{a \cdot 2} \]

      3. count-2 72.1%

         \[\leadsto \frac{\sqrt{\left({b}^{2} - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{2 \cdot \left(a \cdot \left(-4 \cdot c\right)\right)}} - b}{a \cdot 2} \]

      4. *-commutative 72.1%

         \[\leadsto \frac{\sqrt{\left({b}^{2} - a \cdot \left(c \cdot -4\right)\right) + 2 \cdot \left(a \cdot \color{blue}{\left(c \cdot -4\right)}\right)} - b}{a \cdot 2} \]

   7. Simplified 72.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\left({b}^{2} - a \cdot \left(c \cdot -4\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot -4\right)\right)}} - b}{a \cdot 2} \]

   8. Taylor expanded in b around 0 66.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{a} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 66.8%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}}{a}} \]

      2. *-lft-identity 66.8%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}}}{a} \]

      3. distribute-rgt-out-- 67.1%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 - -4\right)}}}{a} \]

      4. metadata-eval 67.1%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot \color{blue}{-4}}}{a} \]

      5. *-commutative 67.1%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a} \]

      6. associate-*r* 67.1%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a} \]

      7. *-commutative 67.1%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot -4\right)} \cdot c}}{a} \]

      8. metadata-eval 67.1%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\left(a \cdot \color{blue}{\left(-8 - -4\right)}\right) \cdot c}}{a} \]

      9. distribute-rgt-out-- 67.1%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{\left(-8 \cdot a - -4 \cdot a\right)} \cdot c}}{a} \]

      10. *-commutative 67.1%

          \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{c \cdot \left(-8 \cdot a - -4 \cdot a\right)}}}{a} \]

      11. distribute-rgt-out-- 67.1%

          \[\leadsto 0.5 \cdot \frac{\sqrt{c \cdot \color{blue}{\left(a \cdot \left(-8 - -4\right)\right)}}}{a} \]

      12. metadata-eval 67.1%

          \[\leadsto 0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)}}{a} \]

   10. Simplified 67.1%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}} \]

   if 1.07999999999999998e-91 < b

   1. Initial program 10.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 10.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 10.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 86.9%

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

      1. associate-*r/ 86.9%

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

      2. neg-mul-1 86.9%

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

   6. Simplified 86.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-91}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 68.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -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 <= (-4d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 69.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 69.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 69.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 67.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   5. Step-by-step derivation

      1. +-commutative 67.3%

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

      2. mul-1-neg 67.3%

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

      3. unsub-neg 67.3%

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

   6. Simplified 67.3%

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

   if -3.999999999999988e-310 < b

   1. Initial program 27.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 27.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 27.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. neg-mul-1 69.2%

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

   6. Simplified 69.2%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 68.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -b / a;
	} else {
		tmp = -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 <= (-4d-310)) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 69.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 69.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 69.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around -inf 67.0%

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

      1. associate-*r/ 67.0%

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

      2. mul-1-neg 67.0%

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

   6. Simplified 67.0%

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

   if -3.999999999999988e-310 < b

   1. Initial program 27.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 27.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 27.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

   4. Taylor expanded in b around inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. neg-mul-1 69.2%

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

   6. Simplified 69.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 35.9% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 49.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

3. Simplified 49.5%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]

4. Taylor expanded in b around -inf 36.3%

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

   1. associate-*r/ 36.3%

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

   2. mul-1-neg 36.3%

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

6. Simplified 36.3%

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

7. Final simplification 36.3%

   \[\leadsto \frac{-b}{a} \]

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t_0 - t_1} \cdot \sqrt{t_0 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2023334 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))