quadp (p42, positive)

Percentage Accurate: 53.1% → 85.4%

Time: 12.0s

Alternatives: 5

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: 53.1% 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: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e+134)
   (/ (- b) a)
   (if (<= b 1.8e-116)
     (/ (- (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 <= -7.8e+134) {
		tmp = -b / a;
	} else if (b <= 1.8e-116) {
		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 <= (-7.8d+134)) then
        tmp = -b / a
    else if (b <= 1.8d-116) 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 <= -7.8e+134) {
		tmp = -b / a;
	} else if (b <= 1.8e-116) {
		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 <= -7.8e+134:
		tmp = -b / a
	elif b <= 1.8e-116:
		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 <= -7.8e+134)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.8e-116)
		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 <= -7.8e+134)
		tmp = -b / a;
	elseif (b <= 1.8e-116)
		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, -7.8e+134], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.8e-116], 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 < -7.79999999999999967e134

   1. Initial program 52.4%

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

      1. *-commutative 52.4%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in b around -inf 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 -7.79999999999999967e134 < b < 1.79999999999999988e-116

   1. Initial program 89.1%

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

   if 1.79999999999999988e-116 < b

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

   3. Simplified 17.6%

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

   4. Taylor expanded in b around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. neg-mul-1 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.52 \cdot 10^{-53}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.52e-53)
   (/ (- b) a)
   (if (<= b 3.5e-116)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.52e-53) {
		tmp = -b / a;
	} else if (b <= 3.5e-116) {
		tmp = (sqrt((a * (c * -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 <= (-1.52d-53)) then
        tmp = -b / a
    else if (b <= 3.5d-116) then
        tmp = (sqrt((a * (c * (-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 <= -1.52e-53) {
		tmp = -b / a;
	} else if (b <= 3.5e-116) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.52e-53:
		tmp = -b / a
	elif b <= 3.5e-116:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.52e-53)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.5e-116)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -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 <= -1.52e-53)
		tmp = -b / a;
	elseif (b <= 3.5e-116)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.52e-53], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.5e-116], N[(N[(N[Sqrt[N[(a * N[(c * -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 < -1.5200000000000001e-53

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in b around -inf 90.5%

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

      1. associate-*r/ 90.5%

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

      2. mul-1-neg 90.5%

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

   6. Simplified 90.5%

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

   if -1.5200000000000001e-53 < b < 3.49999999999999984e-116

   1. Initial program 80.8%

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

      1. *-commutative 80.8%

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

   3. Simplified 80.8%

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

   5. Applied egg-rr 79.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right) + \mathsf{fma}\left(-\sqrt[3]{a \cdot \left(-4 \cdot c\right)}, {\left(\sqrt[3]{a \cdot \left(-4 \cdot c\right)}\right)}^{2}, a \cdot \left(-4 \cdot c\right)\right)}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. +-commutative 79.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(-\sqrt[3]{a \cdot \left(-4 \cdot c\right)}, {\left(\sqrt[3]{a \cdot \left(-4 \cdot c\right)}\right)}^{2}, a \cdot \left(-4 \cdot c\right)\right) + \mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)}}}{a \cdot 2} \]

      2. fma-udef 79.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(\left(-\sqrt[3]{a \cdot \left(-4 \cdot c\right)}\right) \cdot {\left(\sqrt[3]{a \cdot \left(-4 \cdot c\right)}\right)}^{2} + a \cdot \left(-4 \cdot c\right)\right)} + \mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)}}{a \cdot 2} \]

      3. associate-+l+ 79.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-\sqrt[3]{a \cdot \left(-4 \cdot c\right)}\right) \cdot {\left(\sqrt[3]{a \cdot \left(-4 \cdot c\right)}\right)}^{2} + \left(a \cdot \left(-4 \cdot c\right) + \mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}}}{a \cdot 2} \]

      4. *-commutative 79.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(-\sqrt[3]{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right) \cdot {\left(\sqrt[3]{a \cdot \left(-4 \cdot c\right)}\right)}^{2} + \left(a \cdot \left(-4 \cdot c\right) + \mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}}{a \cdot 2} \]

      5. *-commutative 79.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(-\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right) \cdot {\left(\sqrt[3]{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right)}^{2} + \left(a \cdot \left(-4 \cdot c\right) + \mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}}{a \cdot 2} \]

      6. +-commutative 79.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(-\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right) \cdot {\left(\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right)}^{2} + \color{blue}{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right) + a \cdot \left(-4 \cdot c\right)\right)}}}{a \cdot 2} \]

      7. fma-udef 79.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(-\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right) \cdot {\left(\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right)}^{2} + \left(\color{blue}{\left(b \cdot b + a \cdot \left(-4 \cdot c\right)\right)} + a \cdot \left(-4 \cdot c\right)\right)}}{a \cdot 2} \]

      8. unpow2 79.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(-\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right) \cdot {\left(\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right)}^{2} + \left(\left(\color{blue}{{b}^{2}} + a \cdot \left(-4 \cdot c\right)\right) + a \cdot \left(-4 \cdot c\right)\right)}}{a \cdot 2} \]

      9. associate-+l+ 79.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left(-\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right) \cdot {\left(\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right)}^{2} + \color{blue}{\left({b}^{2} + \left(a \cdot \left(-4 \cdot c\right) + a \cdot \left(-4 \cdot c\right)\right)\right)}}}{a \cdot 2} \]

      10. distribute-lft-out 79.5%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left(-\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right) \cdot {\left(\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right)}^{2} + \left({b}^{2} + \color{blue}{a \cdot \left(-4 \cdot c + -4 \cdot c\right)}\right)}}{a \cdot 2} \]

   7. Simplified 79.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right) \cdot {\left(\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right)}^{2} + \left({b}^{2} + a \cdot \left(c \cdot -8\right)\right)}}}{a \cdot 2} \]

   8. Taylor expanded in b around 0 77.2%

      \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. neg-mul-1 77.2%

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

      2. unsub-neg 77.2%

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

      3. distribute-rgt-out 77.5%

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

      4. metadata-eval 77.5%

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

      5. associate-*r* 77.5%

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

   10. Simplified 77.5%

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

   if 3.49999999999999984e-116 < b

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

   3. Simplified 17.6%

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

   4. Taylor expanded in b around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. neg-mul-1 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.52 \cdot 10^{-53}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 43.7% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 20000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 20000.0) (/ (- b) a) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 20000.0) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	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 <= 20000.0d0) then
        tmp = -b / a
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 20000.0) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 20000.0:
		tmp = -b / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 20000.0)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 20000.0)
		tmp = -b / a;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 20000.0], N[((-b) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2e4

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in b around -inf 48.4%

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

      1. associate-*r/ 48.4%

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

      2. mul-1-neg 48.4%

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

   6. Simplified 48.4%

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

   if 2e4 < b

   1. Initial program 11.4%

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

      1. *-commutative 11.4%

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

   3. Simplified 11.4%

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

      1. fma-neg 11.4%

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

      2. *-commutative 11.4%

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

      3. distribute-rgt-neg-in 11.4%

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

      4. *-commutative 11.4%

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

      5. metadata-eval 11.4%

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

      6. associate-*r* 11.4%

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

      7. add-cbrt-cube 5.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]

      8. pow3 5.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{3}}}}}{a \cdot 2} \]

      9. *-commutative 5.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot -4\right) \cdot c}\right)\right)}^{3}}}}{a \cdot 2} \]

      10. associate-*l* 5.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-4 \cdot c\right)}\right)\right)}^{3}}}}{a \cdot 2} \]

   5. Applied egg-rr 5.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}^{3}}}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. clear-num 5.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}^{3}}}}}} \]

      2. inv-pow 5.7%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}^{3}}}}\right)}^{-1}} \]

      3. neg-mul-1 5.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}^{3}}}}\right)}^{-1} \]

      4. fma-def 5.7%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}^{3}}}\right)}}\right)}^{-1} \]

      5. unpow3 5.6%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{\color{blue}{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right) \cdot \mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)}}}\right)}\right)}^{-1} \]

      6. add-cbrt-cube 11.4%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)}}\right)}\right)}^{-1} \]

      7. fma-udef 11.3%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{b \cdot b + a \cdot \left(-4 \cdot c\right)}}\right)}\right)}^{-1} \]

      8. add-sqr-sqrt 9.0%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(-4 \cdot c\right)} \cdot \sqrt{a \cdot \left(-4 \cdot c\right)}}}\right)}\right)}^{-1} \]

      9. hypot-def 23.8%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)}\right)}\right)}^{-1} \]

      10. *-commutative 23.8%

          \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right)\right)}\right)}^{-1} \]

   7. Applied egg-rr 23.8%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 23.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}}} \]

   9. Simplified 23.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}}} \]

   10. Taylor expanded in a around 0 25.0%

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

       1. associate-*r/ 25.0%

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

       2. distribute-rgt1-in 25.0%

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

       3. metadata-eval 25.0%

          \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

       4. mul0-lft 25.0%

          \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

       5. metadata-eval 25.0%

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

   12. Simplified 25.0%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 20000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 4: 67.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-287}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 6e-287) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6e-287) {
		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 <= 6d-287) 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 <= 6e-287) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6e-287:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6e-287)
		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 <= 6e-287)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6e-287], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.99999999999999984e-287

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in b around -inf 66.4%

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

      1. associate-*r/ 66.4%

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

      2. mul-1-neg 66.4%

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

   6. Simplified 66.4%

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

   if 5.99999999999999984e-287 < b

   1. Initial program 28.0%

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

      1. *-commutative 28.0%

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

   3. Simplified 28.0%

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

   4. Taylor expanded in b around inf 71.7%

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

      1. associate-*r/ 71.7%

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

      2. neg-mul-1 71.7%

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

   6. Simplified 71.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-287}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 11.4% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. *-commutative 54.0%

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

3. Simplified 54.0%

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

   1. fma-neg 54.1%

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

   2. *-commutative 54.1%

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

   3. distribute-rgt-neg-in 54.1%

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

   4. *-commutative 54.1%

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

   5. metadata-eval 54.1%

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

   6. associate-*r* 54.1%

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

   7. add-cbrt-cube 31.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]

   8. pow3 31.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{3}}}}}{a \cdot 2} \]

   9. *-commutative 31.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot -4\right) \cdot c}\right)\right)}^{3}}}}{a \cdot 2} \]

   10. associate-*l* 31.3%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-4 \cdot c\right)}\right)\right)}^{3}}}}{a \cdot 2} \]

5. Applied egg-rr 31.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}^{3}}}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. clear-num 31.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}^{3}}}}}} \]

   2. inv-pow 31.3%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}^{3}}}}\right)}^{-1}} \]

   3. neg-mul-1 31.3%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}^{3}}}}\right)}^{-1} \]

   4. fma-def 31.3%

      \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right)}^{3}}}\right)}}\right)}^{-1} \]

   5. unpow3 31.3%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt[3]{\color{blue}{\left(\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)\right) \cdot \mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)}}}\right)}\right)}^{-1} \]

   6. add-cbrt-cube 53.9%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)}}\right)}\right)}^{-1} \]

   7. fma-udef 53.9%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{b \cdot b + a \cdot \left(-4 \cdot c\right)}}\right)}\right)}^{-1} \]

   8. add-sqr-sqrt 45.2%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(-4 \cdot c\right)} \cdot \sqrt{a \cdot \left(-4 \cdot c\right)}}}\right)}\right)}^{-1} \]

   9. hypot-def 52.2%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(-4 \cdot c\right)}\right)}\right)}\right)}^{-1} \]

   10. *-commutative 52.2%

       \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right)\right)}\right)}^{-1} \]

7. Applied egg-rr 52.2%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 52.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}}} \]

9. Simplified 52.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}}} \]

10. Taylor expanded in a around 0 9.6%

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

    1. associate-*r/ 9.6%

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

    2. distribute-rgt1-in 9.6%

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

    3. metadata-eval 9.6%

       \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 9.6%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 9.6%

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

12. Simplified 9.6%

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

13. Final simplification 9.6%

    \[\leadsto \frac{0}{a} \]

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 2023321 
(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)))