quadp (p42, positive)

Percentage Accurate: 52.2% → 86.2%

Time: 16.9s

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: 52.2% 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: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+120}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+120)
   (/ (- b) a)
   (if (<= b 1.55e-53)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+120) {
		tmp = -b / a;
	} else if (b <= 1.55e-53) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e+120)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.55e-53)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e+120], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.55e-53], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3e120

   1. Initial program 45.5%

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

   2. Taylor expanded in b around -inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. mul-1-neg 98.1%

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

   4. Simplified 98.1%

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

   if -3e120 < b < 1.55000000000000008e-53

   1. Initial program 87.5%

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

      1. remove-double-neg 87.5%

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

      2. distribute-frac-neg 87.5%

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

      3. distribute-neg-out 87.5%

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

      4. remove-double-neg 87.5%

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

      5. sub-neg 87.5%

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

      6. distribute-frac-neg 87.5%

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

      7. neg-mul-1 87.5%

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

   3. Simplified 87.5%

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

   if 1.55000000000000008e-53 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

   4. Simplified 85.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+120}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+122}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+122)
   (/ (- b) a)
   (if (<= b 3.1e-59)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+122) {
		tmp = -b / a;
	} else if (b <= 3.1e-59) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d+122)) then
        tmp = -b / a
    else if (b <= 3.1d-59) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+122) {
		tmp = -b / a;
	} else if (b <= 3.1e-59) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+122:
		tmp = -b / a
	elif b <= 3.1e-59:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+122)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.1e-59)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+122)
		tmp = -b / a;
	elseif (b <= 3.1e-59)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+122], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.1e-59], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.99999999999999989e122

   1. Initial program 45.5%

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

   2. Taylor expanded in b around -inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. mul-1-neg 98.1%

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

   4. Simplified 98.1%

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

   if -4.99999999999999989e122 < b < 3.09999999999999999e-59

   1. Initial program 87.5%

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

      1. remove-double-neg 87.5%

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

      2. distribute-frac-neg 87.5%

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

      3. distribute-neg-out 87.5%

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

      4. remove-double-neg 87.5%

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

      5. sub-neg 87.5%

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

      6. distribute-frac-neg 87.5%

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

      7. neg-mul-1 87.5%

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

   3. Simplified 87.5%

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

      1. *-commutative 87.5%

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

      2. associate-*r* 87.5%

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

      3. metadata-eval 87.5%

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

      4. distribute-lft-neg-in 87.5%

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

      5. fma-neg 87.5%

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

   5. Applied egg-rr 87.5%

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

   if 3.09999999999999999e-59 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

   4. Simplified 85.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+122}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-59}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.8e-71)
   (- (/ c b) (/ b a))
   (if (<= b 7e-59) (* (/ 0.5 a) (+ b (sqrt (* a (* c -4.0))))) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-71) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7e-59) {
		tmp = (0.5 / a) * (b + sqrt((a * (c * -4.0))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.8d-71)) then
        tmp = (c / b) - (b / a)
    else if (b <= 7d-59) then
        tmp = (0.5d0 / a) * (b + sqrt((a * (c * (-4.0d0)))))
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-71) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7e-59) {
		tmp = (0.5 / a) * (b + Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.8e-71:
		tmp = (c / b) - (b / a)
	elif b <= 7e-59:
		tmp = (0.5 / a) * (b + math.sqrt((a * (c * -4.0))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.8e-71)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7e-59)
		tmp = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.8e-71)
		tmp = (c / b) - (b / a);
	elseif (b <= 7e-59)
		tmp = (0.5 / a) * (b + sqrt((a * (c * -4.0))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.8e-71], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-59], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.7999999999999997e-71

   1. Initial program 70.4%

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

   2. Taylor expanded in b around -inf 87.0%

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

      1. +-commutative 87.0%

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

      2. mul-1-neg 87.0%

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

      3. unsub-neg 87.0%

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

   4. Simplified 87.0%

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

   if -5.7999999999999997e-71 < b < 7.0000000000000002e-59

   1. Initial program 81.4%

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

   2. Taylor expanded in b around 0 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-*r* 78.7%

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

   4. Simplified 78.7%

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

      1. expm1-log1p-u 54.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)\right)} \]

      2. expm1-udef 16.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1} \]

      3. add-sqr-sqrt 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      4. sqrt-unprod 16.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      5. sqr-neg 16.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{b \cdot b}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      6. sqrt-unprod 8.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      7. add-sqr-sqrt 16.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{b} + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      8. associate-*r* 16.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}{2 \cdot a}\right)} - 1 \]

      9. *-commutative 16.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}}{2 \cdot a}\right)} - 1 \]

      10. associate-*r* 16.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}{2 \cdot a}\right)} - 1 \]

      11. *-commutative 16.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{\color{blue}{a \cdot 2}}\right)} - 1 \]

   6. Applied egg-rr 16.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 54.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right)\right)} \]

      2. expm1-log1p 78.5%

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

      3. *-lft-identity 78.5%

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

      4. associate-*l/ 78.3%

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

      5. *-commutative 78.3%

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

      6. associate-/r* 78.3%

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

      7. metadata-eval 78.3%

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

      8. *-commutative 78.3%

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

      9. rem-square-sqrt 0.0%

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

      10. unpow2 0.0%

          \[\leadsto \frac{0.5}{a} \cdot \left(b + \sqrt{\left(a \cdot \color{blue}{{\left(\sqrt{-4}\right)}^{2}}\right) \cdot c}\right) \]

      11. associate-*r* 0.0%

          \[\leadsto \frac{0.5}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left({\left(\sqrt{-4}\right)}^{2} \cdot c\right)}}\right) \]

      12. unpow2 0.0%

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

      13. rem-square-sqrt 78.3%

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

      14. *-commutative 78.3%

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

   8. Simplified 78.3%

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

   if 7.0000000000000002e-59 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

   4. Simplified 85.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-59}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-69}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.14e-69)
   (- (/ c b) (/ b a))
   (if (<= b 1.6e-58)
     (/ (+ b (sqrt (* c (* a -4.0)))) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.14e-69) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.6e-58) {
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.14d-69)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.6d-58) then
        tmp = (b + sqrt((c * (a * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.14e-69) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.6e-58) {
		tmp = (b + Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.14e-69:
		tmp = (c / b) - (b / a)
	elif b <= 1.6e-58:
		tmp = (b + math.sqrt((c * (a * -4.0)))) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.14e-69)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.6e-58)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.14e-69)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.6e-58)
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.14e-69], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e-58], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14000000000000006e-69

   1. Initial program 70.4%

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

   2. Taylor expanded in b around -inf 87.0%

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

      1. +-commutative 87.0%

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

      2. mul-1-neg 87.0%

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

      3. unsub-neg 87.0%

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

   4. Simplified 87.0%

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

   if -1.14000000000000006e-69 < b < 1.6e-58

   1. Initial program 81.4%

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

   2. Taylor expanded in b around 0 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-*r* 78.7%

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

   4. Simplified 78.7%

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

      1. expm1-log1p-u 54.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)\right)} \]

      2. expm1-udef 16.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1} \]

      3. add-sqr-sqrt 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      4. sqrt-unprod 16.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      5. sqr-neg 16.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{b \cdot b}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      6. sqrt-unprod 8.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      7. add-sqr-sqrt 16.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{b} + \sqrt{a \cdot \left(c \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      8. associate-*r* 16.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}{2 \cdot a}\right)} - 1 \]

      9. *-commutative 16.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}}{2 \cdot a}\right)} - 1 \]

      10. associate-*r* 16.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}{2 \cdot a}\right)} - 1 \]

      11. *-commutative 16.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{\color{blue}{a \cdot 2}}\right)} - 1 \]

   6. Applied egg-rr 16.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 54.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right)\right)} \]

      2. expm1-log1p 78.5%

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

   8. Simplified 78.5%

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

   if 1.6e-58 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

   4. Simplified 85.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.14 \cdot 10^{-69}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.24 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e-41)
   (- (/ c b) (/ b a))
   (if (<= b 1.24e-54)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-41) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.24e-54) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.4d-41)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.24d-54) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-41) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.24e-54) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.4e-41:
		tmp = (c / b) - (b / a)
	elif b <= 1.24e-54:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e-41)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.24e-54)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.4e-41)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.24e-54)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e-41], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.24e-54], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.4e-41

   1. Initial program 69.2%

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

   2. Taylor expanded in b around -inf 88.5%

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

      1. +-commutative 88.5%

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

      2. mul-1-neg 88.5%

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

      3. unsub-neg 88.5%

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

   4. Simplified 88.5%

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

   if -4.4e-41 < b < 1.23999999999999999e-54

   1. Initial program 82.4%

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

   2. Taylor expanded in b around 0 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-*r* 77.6%

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

   4. Simplified 77.6%

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

   if 1.23999999999999999e-54 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

   4. Simplified 85.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.24 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 68.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-311) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-311) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-311)) then
        tmp = (c / b) - (b / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-311) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-311:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-311)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-311)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-311], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.99999999999948e-312

   1. Initial program 73.7%

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

   2. Taylor expanded in b around -inf 69.7%

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

      1. +-commutative 69.7%

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

      2. mul-1-neg 69.7%

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

      3. unsub-neg 69.7%

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

   4. Simplified 69.7%

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

   if -9.99999999999948e-312 < b

   1. Initial program 33.5%

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

   2. Taylor expanded in b around inf 65.2%

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

      1. associate-*r/ 65.2%

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

      2. neg-mul-1 65.2%

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

   4. Simplified 65.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 68.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-311) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-311) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-311)) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-311) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-311:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-311)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-311)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-311], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.99999999999948e-312

   1. Initial program 73.7%

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

   2. Taylor expanded in b around -inf 69.5%

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

      1. associate-*r/ 69.5%

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

      2. mul-1-neg 69.5%

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

   4. Simplified 69.5%

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

   if -9.99999999999948e-312 < b

   1. Initial program 33.5%

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

   2. Taylor expanded in b around inf 65.2%

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

      1. associate-*r/ 65.2%

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

      2. neg-mul-1 65.2%

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

   4. Simplified 65.2%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 35.6% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

2. Taylor expanded in b around -inf 35.6%

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

   1. associate-*r/ 35.6%

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

   2. mul-1-neg 35.6%

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

4. Simplified 35.6%

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

5. Final simplification 35.6%

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

Alternative 9: 2.5% 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 53.3%

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

3. Applied egg-rr 28.2%

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

4. Taylor expanded in a around 0 2.5%

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

5. Final simplification 2.5%

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

Developer target: 70.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* 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 = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-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 = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-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 = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-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 = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-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(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(a * 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 = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-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[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023283 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

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

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