The quadratic formula (r2)

Percentage Accurate: 51.9% → 85.0%

Time: 13.1s

Alternatives: 9

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: 51.9% 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: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.55e-90)
   (/ (- c) b)
   (if (<= b 1.5e+124)
     (/ (- (- b) (sqrt (- (* b b) (* (* c 4.0) a)))) (* a 2.0))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-90) {
		tmp = -c / b;
	} else if (b <= 1.5e+124) {
		tmp = (-b - sqrt(((b * b) - ((c * 4.0) * a)))) / (a * 2.0);
	} else {
		tmp = -b / 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 <= (-1.55d-90)) then
        tmp = -c / b
    else if (b <= 1.5d+124) then
        tmp = (-b - sqrt(((b * b) - ((c * 4.0d0) * a)))) / (a * 2.0d0)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-90) {
		tmp = -c / b;
	} else if (b <= 1.5e+124) {
		tmp = (-b - Math.sqrt(((b * b) - ((c * 4.0) * a)))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.55e-90:
		tmp = -c / b
	elif b <= 1.5e+124:
		tmp = (-b - math.sqrt(((b * b) - ((c * 4.0) * a)))) / (a * 2.0)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.55e-90)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.5e+124)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(c * 4.0) * a)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.55e-90)
		tmp = -c / b;
	elseif (b <= 1.5e+124)
		tmp = (-b - sqrt(((b * b) - ((c * 4.0) * a)))) / (a * 2.0);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.55e-90], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.5e+124], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5500000000000001e-90

   1. Initial program 16.4%

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

      1. *-commutative 16.4%

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

      2. sqr-neg 16.4%

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

      3. *-commutative 16.4%

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

      4. sqr-neg 16.4%

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

      5. *-commutative 16.4%

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

      6. associate-*r* 16.4%

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

      7. *-commutative 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in b around -inf 90.5%

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

      1. mul-1-neg 90.5%

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

      2. distribute-neg-frac 90.5%

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

   6. Simplified 90.5%

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

   if -1.5500000000000001e-90 < b < 1.5e124

   1. Initial program 84.1%

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

      1. *-commutative 84.1%

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

      2. sqr-neg 84.1%

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

      3. *-commutative 84.1%

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

      4. sqr-neg 84.1%

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

      5. *-commutative 84.1%

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

      6. associate-*r* 84.1%

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

      7. *-commutative 84.1%

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

   3. Simplified 84.1%

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

   if 1.5e124 < b

   1. Initial program 59.6%

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

      1. *-commutative 59.6%

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

      2. sqr-neg 59.6%

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

      3. *-commutative 59.6%

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

      4. sqr-neg 59.6%

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

      5. *-commutative 59.6%

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

      6. associate-*r* 59.6%

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

      7. *-commutative 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.55e-90)
   (/ (- c) b)
   (if (<= b 3.2e+123)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-90) {
		tmp = -c / b;
	} else if (b <= 3.2e+123) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = -b / 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 <= (-1.55d-90)) then
        tmp = -c / b
    else if (b <= 3.2d+123) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-90) {
		tmp = -c / b;
	} else if (b <= 3.2e+123) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.55e-90:
		tmp = -c / b
	elif b <= 3.2e+123:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.55e-90)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 3.2e+123)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.55e-90)
		tmp = -c / b;
	elseif (b <= 3.2e+123)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.55e-90], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 3.2e+123], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5500000000000001e-90

   1. Initial program 16.4%

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

      1. *-commutative 16.4%

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

      2. sqr-neg 16.4%

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

      3. *-commutative 16.4%

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

      4. sqr-neg 16.4%

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

      5. *-commutative 16.4%

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

      6. associate-*r* 16.4%

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

      7. *-commutative 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in b around -inf 90.5%

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

      1. mul-1-neg 90.5%

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

      2. distribute-neg-frac 90.5%

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

   6. Simplified 90.5%

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

   if -1.5500000000000001e-90 < b < 3.20000000000000005e123

   1. Initial program 84.1%

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

   if 3.20000000000000005e123 < b

   1. Initial program 59.6%

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

      1. *-commutative 59.6%

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

      2. sqr-neg 59.6%

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

      3. *-commutative 59.6%

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

      4. sqr-neg 59.6%

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

      5. *-commutative 59.6%

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

      6. associate-*r* 59.6%

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

      7. *-commutative 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-98}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.55e-90)
   (/ (- c) b)
   (if (<= b 1.35e-98)
     (/ (- (- b) (sqrt (* c (* a -4.0)))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-90) {
		tmp = -c / b;
	} else if (b <= 1.35e-98) {
		tmp = (-b - sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / 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 <= (-1.55d-90)) then
        tmp = -c / b
    else if (b <= 1.35d-98) then
        tmp = (-b - sqrt((c * (a * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-90) {
		tmp = -c / b;
	} else if (b <= 1.35e-98) {
		tmp = (-b - Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.55e-90:
		tmp = -c / b
	elif b <= 1.35e-98:
		tmp = (-b - math.sqrt((c * (a * -4.0)))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.55e-90)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.35e-98)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.55e-90)
		tmp = -c / b;
	elseif (b <= 1.35e-98)
		tmp = (-b - sqrt((c * (a * -4.0)))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.55e-90], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.35e-98], N[(N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5500000000000001e-90

   1. Initial program 16.4%

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

      1. *-commutative 16.4%

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

      2. sqr-neg 16.4%

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

      3. *-commutative 16.4%

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

      4. sqr-neg 16.4%

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

      5. *-commutative 16.4%

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

      6. associate-*r* 16.4%

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

      7. *-commutative 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in b around -inf 90.5%

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

      1. mul-1-neg 90.5%

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

      2. distribute-neg-frac 90.5%

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

   6. Simplified 90.5%

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

   if -1.5500000000000001e-90 < b < 1.3499999999999999e-98

   1. Initial program 71.4%

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

      1. *-commutative 71.4%

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

      2. sqr-neg 71.4%

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

      3. *-commutative 71.4%

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

      4. sqr-neg 71.4%

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

      5. *-commutative 71.4%

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

      6. associate-*r* 71.5%

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

      7. *-commutative 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in b around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. *-commutative 71.4%

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

      3. associate-*l* 71.5%

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

   6. Simplified 71.5%

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

   if 1.3499999999999999e-98 < b

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. sqr-neg 79.8%

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

      3. *-commutative 79.8%

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

      4. sqr-neg 79.8%

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

      5. *-commutative 79.8%

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

      6. associate-*r* 79.8%

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

      7. *-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in b around inf 93.8%

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

      1. +-commutative 93.8%

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

      2. mul-1-neg 93.8%

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

      3. unsub-neg 93.8%

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

   6. Simplified 93.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-98}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e-90)
   (/ (- c) b)
   (if (<= b 6.8e-99)
     (* (/ 0.5 a) (- b (sqrt (* c (* a -4.0)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-90) {
		tmp = -c / b;
	} else if (b <= 6.8e-99) {
		tmp = (0.5 / a) * (b - sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / 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 <= (-1.25d-90)) then
        tmp = -c / b
    else if (b <= 6.8d-99) then
        tmp = (0.5d0 / a) * (b - sqrt((c * (a * (-4.0d0)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-90) {
		tmp = -c / b;
	} else if (b <= 6.8e-99) {
		tmp = (0.5 / a) * (b - Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.25e-90:
		tmp = -c / b
	elif b <= 6.8e-99:
		tmp = (0.5 / a) * (b - math.sqrt((c * (a * -4.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.25e-90)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 6.8e-99)
		tmp = Float64(Float64(0.5 / a) * Float64(b - sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.25e-90)
		tmp = -c / b;
	elseif (b <= 6.8e-99)
		tmp = (0.5 / a) * (b - sqrt((c * (a * -4.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e-90], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 6.8e-99], N[(N[(0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25000000000000005e-90

   1. Initial program 16.4%

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

      1. *-commutative 16.4%

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

      2. sqr-neg 16.4%

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

      3. *-commutative 16.4%

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

      4. sqr-neg 16.4%

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

      5. *-commutative 16.4%

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

      6. associate-*r* 16.4%

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

      7. *-commutative 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in b around -inf 90.5%

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

      1. mul-1-neg 90.5%

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

      2. distribute-neg-frac 90.5%

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

   6. Simplified 90.5%

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

   if -1.25000000000000005e-90 < b < 6.80000000000000014e-99

   1. Initial program 71.4%

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

      1. *-commutative 71.4%

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

      2. sqr-neg 71.4%

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

      3. *-commutative 71.4%

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

      4. sqr-neg 71.4%

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

      5. *-commutative 71.4%

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

      6. associate-*r* 71.5%

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

      7. *-commutative 71.5%

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

   3. Simplified 71.5%

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

      1. clear-num 71.6%

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

      2. associate-/r/ 71.5%

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

      3. *-commutative 71.5%

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

      4. associate-/r* 71.5%

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

      5. metadata-eval 71.5%

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

      6. add-sqr-sqrt 43.7%

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

      7. sqrt-unprod 71.1%

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

      8. sqr-neg 71.1%

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

      9. sqrt-prod 27.3%

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

      10. add-sqr-sqrt 71.1%

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

      11. fma-neg 71.1%

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

      12. *-commutative 71.1%

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

      13. distribute-rgt-neg-in 71.1%

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

      14. *-commutative 71.1%

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

      15. distribute-rgt-neg-in 71.1%

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

      16. metadata-eval 71.1%

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

   5. Applied egg-rr 71.1%

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

   6. Taylor expanded in b around 0 71.0%

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

      1. associate-*r* 71.1%

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

      2. *-commutative 71.1%

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

   8. Simplified 71.1%

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

   if 6.80000000000000014e-99 < b

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. sqr-neg 79.8%

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

      3. *-commutative 79.8%

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

      4. sqr-neg 79.8%

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

      5. *-commutative 79.8%

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

      6. associate-*r* 79.8%

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

      7. *-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in b around inf 93.8%

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

      1. +-commutative 93.8%

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

      2. mul-1-neg 93.8%

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

      3. unsub-neg 93.8%

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

   6. Simplified 93.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-90}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 67.6% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-312) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / 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 <= (-5d-312)) then
        tmp = -c / b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-312:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-312)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-312)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-312], N[((-c) / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.0000000000022e-312

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. sqr-neg 29.8%

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

      3. *-commutative 29.8%

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

      4. sqr-neg 29.8%

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

      5. *-commutative 29.8%

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

      6. associate-*r* 29.8%

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

      7. *-commutative 29.8%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in b around -inf 71.6%

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

      1. mul-1-neg 71.6%

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

      2. distribute-neg-frac 71.6%

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

   6. Simplified 71.6%

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

   if -5.0000000000022e-312 < b

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

      2. sqr-neg 77.9%

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

      3. *-commutative 77.9%

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

      4. sqr-neg 77.9%

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

      5. *-commutative 77.9%

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

      6. associate-*r* 77.9%

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

      7. *-commutative 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in b around inf 77.8%

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

      1. +-commutative 77.8%

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

      2. mul-1-neg 77.8%

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

      3. unsub-neg 77.8%

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

   6. Simplified 77.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6: 42.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -4.5e-47) (/ c b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-47) {
		tmp = c / b;
	} else {
		tmp = -b / 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 <= (-4.5d-47)) then
        tmp = c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-47) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e-47:
		tmp = c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-47)
		tmp = Float64(c / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e-47)
		tmp = c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-47], N[(c / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5e-47

   1. Initial program 14.8%

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

      1. *-commutative 14.8%

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

      2. sqr-neg 14.8%

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

      3. *-commutative 14.8%

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

      4. sqr-neg 14.8%

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

      5. *-commutative 14.8%

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

      6. associate-*r* 14.8%

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

      7. *-commutative 14.8%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in b around inf 2.6%

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

   5. Taylor expanded in b around 0 27.1%

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

   if -4.5e-47 < b

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

      2. sqr-neg 74.0%

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

      3. *-commutative 74.0%

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

      4. sqr-neg 74.0%

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

      5. *-commutative 74.0%

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

      6. associate-*r* 74.0%

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

      7. *-commutative 74.0%

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

   3. Simplified 74.0%

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

   4. Taylor expanded in b around inf 57.3%

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

      1. associate-*r/ 57.3%

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

      2. mul-1-neg 57.3%

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

   6. Simplified 57.3%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7: 67.5% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-312) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = -c / b;
	} else {
		tmp = -b / 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 <= (-5d-312)) then
        tmp = -c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-312:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-312)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-312)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-312], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.0000000000022e-312

   1. Initial program 29.8%

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

      1. *-commutative 29.8%

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

      2. sqr-neg 29.8%

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

      3. *-commutative 29.8%

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

      4. sqr-neg 29.8%

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

      5. *-commutative 29.8%

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

      6. associate-*r* 29.8%

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

      7. *-commutative 29.8%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in b around -inf 71.6%

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

      1. mul-1-neg 71.6%

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

      2. distribute-neg-frac 71.6%

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

   6. Simplified 71.6%

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

   if -5.0000000000022e-312 < b

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

      2. sqr-neg 77.9%

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

      3. *-commutative 77.9%

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

      4. sqr-neg 77.9%

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

      5. *-commutative 77.9%

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

      6. associate-*r* 77.9%

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

      7. *-commutative 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in b around inf 77.4%

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

      1. associate-*r/ 77.4%

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

      2. mul-1-neg 77.4%

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

   6. Simplified 77.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8: 2.6% accurate, 38.7× 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(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 51.8%

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

   1. *-commutative 51.8%

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

   2. sqr-neg 51.8%

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

   3. *-commutative 51.8%

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

   4. sqr-neg 51.8%

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

   5. *-commutative 51.8%

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

   6. associate-*r* 51.8%

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

   7. *-commutative 51.8%

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

3. Simplified 51.8%

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

   1. clear-num 51.7%

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

   2. inv-pow 51.7%

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

5. Applied egg-rr 29.2%

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

6. Taylor expanded in b around -inf 2.7%

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

7. Final simplification 2.7%

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

Alternative 9: 11.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

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

   1. *-commutative 51.8%

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

   2. sqr-neg 51.8%

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

   3. *-commutative 51.8%

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

   4. sqr-neg 51.8%

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

   5. *-commutative 51.8%

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

   6. associate-*r* 51.8%

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

   7. *-commutative 51.8%

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

3. Simplified 51.8%

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

4. Taylor expanded in b around inf 35.5%

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

5. Taylor expanded in b around 0 11.9%

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

6. Final simplification 11.9%

   \[\leadsto \frac{c}{b} \]

Developer target: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
     (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp = 0.0;
	if (b < 0.0)
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-b - t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023306 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

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