quadm (p42, negative)

Percentage Accurate: 50.9% → 86.5%

Time: 12.1s

Alternatives: 8

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.9% 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: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\\ \mathbf{if}\;b \leq -205000:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-142}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{c \cdot \left(a \cdot 4\right)}{b - t_0}}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+123}:\\ \;\;\;\;-0.5 \cdot \frac{b + t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* a (* c 4.0))))))
   (if (<= b -205000.0)
     (/ (- c) b)
     (if (<= b -5e-142)
       (* -0.5 (/ (/ (* c (* a 4.0)) (- b t_0)) a))
       (if (<= b 2.9e+123) (* -0.5 (/ (+ b t_0) a)) (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (a * (c * 4.0))));
	double tmp;
	if (b <= -205000.0) {
		tmp = -c / b;
	} else if (b <= -5e-142) {
		tmp = -0.5 * (((c * (a * 4.0)) / (b - t_0)) / a);
	} else if (b <= 2.9e+123) {
		tmp = -0.5 * ((b + t_0) / a);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (a * (c * 4.0d0))))
    if (b <= (-205000.0d0)) then
        tmp = -c / b
    else if (b <= (-5d-142)) then
        tmp = (-0.5d0) * (((c * (a * 4.0d0)) / (b - t_0)) / a)
    else if (b <= 2.9d+123) then
        tmp = (-0.5d0) * ((b + t_0) / a)
    else
        tmp = -b / 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) - (a * (c * 4.0))));
	double tmp;
	if (b <= -205000.0) {
		tmp = -c / b;
	} else if (b <= -5e-142) {
		tmp = -0.5 * (((c * (a * 4.0)) / (b - t_0)) / a);
	} else if (b <= 2.9e+123) {
		tmp = -0.5 * ((b + t_0) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (a * (c * 4.0))))
	tmp = 0
	if b <= -205000.0:
		tmp = -c / b
	elif b <= -5e-142:
		tmp = -0.5 * (((c * (a * 4.0)) / (b - t_0)) / a)
	elif b <= 2.9e+123:
		tmp = -0.5 * ((b + t_0) / a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 4.0))))
	tmp = 0.0
	if (b <= -205000.0)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -5e-142)
		tmp = Float64(-0.5 * Float64(Float64(Float64(c * Float64(a * 4.0)) / Float64(b - t_0)) / a));
	elseif (b <= 2.9e+123)
		tmp = Float64(-0.5 * Float64(Float64(b + t_0) / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (a * (c * 4.0))));
	tmp = 0.0;
	if (b <= -205000.0)
		tmp = -c / b;
	elseif (b <= -5e-142)
		tmp = -0.5 * (((c * (a * 4.0)) / (b - t_0)) / a);
	elseif (b <= 2.9e+123)
		tmp = -0.5 * ((b + t_0) / a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -205000.0], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -5e-142], N[(-0.5 * N[(N[(N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision] / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e+123], N[(-0.5 * N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -205000

   1. Initial program 12.2%

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

   2. Taylor expanded in b around -inf 88.7%

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

      1. associate-*r/ 88.7%

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

      2. neg-mul-1 88.7%

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

   4. Simplified 88.7%

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

   if -205000 < b < -5.0000000000000002e-142

   1. Initial program 58.2%

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

   3. Simplified 58.2%

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

      1. fma-udef 58.2%

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

      2. associate-*r* 58.2%

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

      3. metadata-eval 58.2%

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

      4. distribute-rgt-neg-in 58.2%

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

      5. *-commutative 58.2%

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

      6. +-commutative 58.2%

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

      7. sub-neg 58.2%

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

      8. *-commutative 58.2%

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

      9. associate-*l* 58.2%

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

   5. Applied egg-rr 58.2%

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

      1. flip-+ 57.9%

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

      2. add-sqr-sqrt 58.1%

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

   7. Applied egg-rr 58.1%

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

   8. Taylor expanded in b around 0 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*l* 84.8%

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

   10. Simplified 84.8%

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

   if -5.0000000000000002e-142 < b < 2.9000000000000001e123

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

      1. fma-udef 74.5%

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

      2. associate-*r* 74.5%

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

      3. metadata-eval 74.5%

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

      4. distribute-rgt-neg-in 74.5%

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

      5. *-commutative 74.5%

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

      6. +-commutative 74.5%

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

      7. sub-neg 74.5%

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

      8. *-commutative 74.5%

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

      9. associate-*l* 74.5%

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

   5. Applied egg-rr 74.5%

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

   if 2.9000000000000001e123 < b

   1. Initial program 59.3%

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

   2. Taylor expanded in b around inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. mul-1-neg 97.8%

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

   4. Simplified 97.8%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -205000:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-142}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{c \cdot \left(a \cdot 4\right)}{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+123}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-59}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+121}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-59)
   (/ (- c) b)
   (if (<= b 4e+121)
     (* -0.5 (/ (+ b (sqrt (- (* b b) (* a (* c 4.0))))) a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-59) {
		tmp = -c / b;
	} else if (b <= 4e+121) {
		tmp = -0.5 * ((b + sqrt(((b * b) - (a * (c * 4.0))))) / a);
	} 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-59)) then
        tmp = -c / b
    else if (b <= 4d+121) then
        tmp = (-0.5d0) * ((b + sqrt(((b * b) - (a * (c * 4.0d0))))) / a)
    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-59) {
		tmp = -c / b;
	} else if (b <= 4e+121) {
		tmp = -0.5 * ((b + Math.sqrt(((b * b) - (a * (c * 4.0))))) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-59:
		tmp = -c / b
	elif b <= 4e+121:
		tmp = -0.5 * ((b + math.sqrt(((b * b) - (a * (c * 4.0))))) / a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-59)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 4e+121)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 4.0))))) / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-59)
		tmp = -c / b;
	elseif (b <= 4e+121)
		tmp = -0.5 * ((b + sqrt(((b * b) - (a * (c * 4.0))))) / a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-59], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 4e+121], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000001e-59

   1. Initial program 17.1%

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

   2. Taylor expanded in b around -inf 82.4%

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

      1. associate-*r/ 82.4%

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

      2. neg-mul-1 82.4%

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

   4. Simplified 82.4%

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

   if -5.0000000000000001e-59 < b < 4.00000000000000015e121

   1. Initial program 75.2%

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

   3. Simplified 75.2%

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

      1. fma-udef 75.2%

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

      2. associate-*r* 75.2%

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

      3. metadata-eval 75.2%

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

      4. distribute-rgt-neg-in 75.2%

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

      5. *-commutative 75.2%

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

      6. +-commutative 75.2%

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

      7. sub-neg 75.2%

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

      8. *-commutative 75.2%

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

      9. associate-*l* 75.2%

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

   5. Applied egg-rr 75.2%

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

   if 4.00000000000000015e121 < b

   1. Initial program 59.3%

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

   2. Taylor expanded in b around inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. mul-1-neg 97.8%

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

   4. Simplified 97.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-59}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+121}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-68}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-68)
   (/ (- c) b)
   (if (<= b 6.8e-90)
     (* -0.5 (/ (+ b (sqrt (* (* c a) -4.0))) a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-68) {
		tmp = -c / b;
	} else if (b <= 6.8e-90) {
		tmp = -0.5 * ((b + sqrt(((c * a) * -4.0))) / a);
	} 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-68)) then
        tmp = -c / b
    else if (b <= 6.8d-90) then
        tmp = (-0.5d0) * ((b + sqrt(((c * a) * (-4.0d0)))) / a)
    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-68) {
		tmp = -c / b;
	} else if (b <= 6.8e-90) {
		tmp = -0.5 * ((b + Math.sqrt(((c * a) * -4.0))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-68:
		tmp = -c / b
	elif b <= 6.8e-90:
		tmp = -0.5 * ((b + math.sqrt(((c * a) * -4.0))) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-68)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 6.8e-90)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(c * a) * -4.0))) / a));
	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-68)
		tmp = -c / b;
	elseif (b <= 6.8e-90)
		tmp = -0.5 * ((b + sqrt(((c * a) * -4.0))) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-68], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 6.8e-90], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.99999999999999971e-68

   1. Initial program 18.3%

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

   2. Taylor expanded in b around -inf 81.7%

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

      1. associate-*r/ 81.7%

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

      2. neg-mul-1 81.7%

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

   4. Simplified 81.7%

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

   if -4.99999999999999971e-68 < b < 6.79999999999999988e-90

   1. Initial program 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in a around inf 67.8%

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

      1. *-commutative 67.8%

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

   6. Simplified 67.8%

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

   if 6.79999999999999988e-90 < b

   1. Initial program 72.0%

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

   2. Taylor expanded in b around inf 81.8%

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

      1. mul-1-neg 81.8%

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

      2. unsub-neg 81.8%

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

   4. Simplified 81.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-68}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4: 67.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		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 <= (-2d-310)) 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 <= -2e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		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 <= -2e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[((-c) / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 33.7%

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

   2. Taylor expanded in b around -inf 63.3%

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

      1. associate-*r/ 63.3%

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

      2. neg-mul-1 63.3%

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

   4. Simplified 63.3%

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

   if -1.999999999999994e-310 < b

   1. Initial program 70.8%

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

   2. Taylor expanded in b around inf 58.0%

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

      1. mul-1-neg 58.0%

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

      2. unsub-neg 58.0%

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

   4. Simplified 58.0%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 42.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -112:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -112.0) (/ c b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -112.0) {
		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 <= (-112.0d0)) 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 <= -112.0) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -112.0:
		tmp = c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -112.0)
		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 <= -112.0)
		tmp = c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -112.0], N[(c / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -112

   1. Initial program 12.2%

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

   2. Taylor expanded in b around inf 2.1%

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

   3. Taylor expanded in c around inf 27.2%

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

   if -112 < b

   1. Initial program 68.3%

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

   2. Taylor expanded in b around inf 40.6%

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

      1. associate-*r/ 40.6%

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

      2. mul-1-neg 40.6%

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

   4. Simplified 40.6%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -112:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6: 67.5% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		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 <= (-2d-310)) 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 <= -2e-310) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		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 <= -2e-310)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 33.7%

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

   2. Taylor expanded in b around -inf 63.3%

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

      1. associate-*r/ 63.3%

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

      2. neg-mul-1 63.3%

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

   4. Simplified 63.3%

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

   if -1.999999999999994e-310 < b

   1. Initial program 70.8%

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

   2. Taylor expanded in b around inf 57.7%

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

      1. associate-*r/ 57.7%

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

      2. mul-1-neg 57.7%

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

   4. Simplified 57.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7: 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 51.7%

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

3. Simplified 51.7%

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

   1. clear-num 51.6%

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

   2. inv-pow 51.6%

      \[\leadsto -0.5 \cdot \color{blue}{{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{-1}} \]

5. Applied egg-rr 51.6%

   \[\leadsto -0.5 \cdot \color{blue}{{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 51.6%

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

   2. fma-udef 51.6%

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

   3. *-commutative 51.6%

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

   4. associate-*l* 51.6%

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

   5. *-commutative 51.6%

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

   6. fma-def 51.6%

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

7. Simplified 51.6%

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

8. Taylor expanded in b around -inf 33.9%

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

9. Taylor expanded in b around 0 2.9%

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

10. Final simplification 2.9%

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

Alternative 8: 10.5% 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 51.7%

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

2. Taylor expanded in b around inf 27.8%

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

3. Taylor expanded in c around inf 10.3%

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

4. Final simplification 10.3%

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

Developer target: 69.7% 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 2023192 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))