quadp (p42, positive)

Percentage Accurate: 52.5% → 86.1%

Time: 12.7s

Alternatives: 7

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+90}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e+90)
   (/ b (- a))
   (if (<= b 6.2e-93)
     (/ (- (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 <= -6e+90) {
		tmp = b / -a;
	} else if (b <= 6.2e-93) {
		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 <= -6e+90)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 6.2e-93)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e+90], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 6.2e-93], 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 < -5.99999999999999957e90

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in b around -inf 96.9%

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

      1. associate-*r/ 96.9%

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

      2. mul-1-neg 96.9%

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

   7. Simplified 96.9%

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

   if -5.99999999999999957e90 < b < 6.19999999999999999e-93

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. +-commutative 84.2%

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

      3. unsub-neg 84.2%

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

      4. fma-neg 84.2%

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

      5. *-commutative 84.2%

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

      6. associate-*r* 84.3%

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

      7. distribute-lft-neg-in 84.3%

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

      8. *-commutative 84.3%

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

      9. distribute-rgt-neg-in 84.3%

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

      10. associate-*r* 84.3%

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

      11. metadata-eval 84.3%

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

   3. Simplified 84.3%

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

   if 6.19999999999999999e-93 < b

   1. Initial program 11.3%

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

      1. *-commutative 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in b around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. neg-mul-1 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+90}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+89}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.7e+89)
   (/ b (- a))
   (if (<= b 2.6e-89)
     (/ (- (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 <= -1.7e+89) {
		tmp = b / -a;
	} else if (b <= 2.6e-89) {
		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 <= (-1.7d+89)) then
        tmp = b / -a
    else if (b <= 2.6d-89) 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 <= -1.7e+89) {
		tmp = b / -a;
	} else if (b <= 2.6e-89) {
		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 <= -1.7e+89:
		tmp = b / -a
	elif b <= 2.6e-89:
		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 <= -1.7e+89)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2.6e-89)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.7e+89)
		tmp = b / -a;
	elseif (b <= 2.6e-89)
		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, -1.7e+89], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2.6e-89], 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 < -1.7000000000000001e89

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in b around -inf 96.9%

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

      1. associate-*r/ 96.9%

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

      2. mul-1-neg 96.9%

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

   7. Simplified 96.9%

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

   if -1.7000000000000001e89 < b < 2.5999999999999999e-89

   1. Initial program 84.2%

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

   if 2.5999999999999999e-89 < b

   1. Initial program 11.3%

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

      1. *-commutative 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in b around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. neg-mul-1 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+89}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{-9}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.8e-9)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b 4.5e-87)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.8e-9) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 4.5e-87) {
		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 <= (-9.8d-9)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= 4.5d-87) 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 <= -9.8e-9) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 4.5e-87) {
		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 <= -9.8e-9:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= 4.5e-87:
		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 <= -9.8e-9)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= 4.5e-87)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.8e-9)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= 4.5e-87)
		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, -9.8e-9], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-87], 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 < -9.80000000000000007e-9

   1. Initial program 61.2%

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

      1. *-commutative 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in b around -inf 90.3%

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

      1. mul-1-neg 90.3%

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

      2. *-commutative 90.3%

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

      3. distribute-rgt-neg-in 90.3%

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

      4. +-commutative 90.3%

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

      5. mul-1-neg 90.3%

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

      6. unsub-neg 90.3%

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

   7. Simplified 90.3%

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

   if -9.80000000000000007e-9 < b < 4.49999999999999958e-87

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in b around 0 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-*r* 70.9%

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

   7. Simplified 70.9%

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

   if 4.49999999999999958e-87 < b

   1. Initial program 11.3%

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

      1. *-commutative 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in b around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. neg-mul-1 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{-9}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.2% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = b / -a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 71.7%

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

      1. *-commutative 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in b around -inf 63.2%

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

      1. associate-*r/ 63.2%

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

      2. mul-1-neg 63.2%

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

   7. Simplified 63.2%

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

   if -1.999999999999994e-310 < b

   1. Initial program 27.4%

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

      1. *-commutative 27.4%

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

   3. Simplified 27.4%

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

   5. Taylor expanded in b around inf 71.8%

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

      1. associate-*r/ 71.8%

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

      2. neg-mul-1 71.8%

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

   7. Simplified 71.8%

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

4. Final simplification 67.0%

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

Alternative 5: 44.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-48}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 2.9e-48) (/ b (- a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e-48) {
		tmp = b / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.9d-48) then
        tmp = b / -a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e-48) {
		tmp = b / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.9e-48:
		tmp = b / -a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.9e-48)
		tmp = Float64(b / Float64(-a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.9e-48)
		tmp = b / -a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.9e-48], N[(b / (-a)), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 2.9000000000000003e-48

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in b around -inf 51.4%

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

      1. associate-*r/ 51.4%

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

      2. mul-1-neg 51.4%

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

   7. Simplified 51.4%

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

   if 2.9000000000000003e-48 < b

   1. Initial program 10.7%

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

      1. *-commutative 10.7%

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

   3. Simplified 10.7%

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

   6. Applied egg-rr 10.5%

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

   7. Taylor expanded in b around -inf 2.4%

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

   9. Simplified 2.4%

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

       1. *-un-lft-identity 2.4%

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

       2. associate-*r/ 2.4%

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

       3. associate-*r/ 2.4%

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

       4. sub-div 2.4%

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

       5. add-sqr-sqrt 0.0%

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

       6. sqrt-unprod 7.2%

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

       7. sqr-neg 7.2%

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

       8. sqrt-unprod 13.2%

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

       9. add-sqr-sqrt 22.9%

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

   11. Applied egg-rr 22.9%

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

       1. *-lft-identity 22.9%

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

       2. div-sub 17.1%

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

       3. *-commutative 17.1%

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

       4. +-inverses 22.9%

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

   13. Simplified 22.9%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-48}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 13.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-86}:\\ \;\;\;\;\frac{-0.5}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.05e-86) (/ -0.5 (- a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-86) {
		tmp = -0.5 / -a;
	} else {
		tmp = 0.0;
	}
	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 <= 1.05d-86) then
        tmp = (-0.5d0) / -a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e-86) {
		tmp = -0.5 / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.05e-86:
		tmp = -0.5 / -a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.05e-86)
		tmp = Float64(-0.5 / Float64(-a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.05e-86)
		tmp = -0.5 / -a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.05e-86], N[(-0.5 / (-a)), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 1.05e-86

   1. Initial program 72.5%

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

      1. *-commutative 72.5%

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

   3. Simplified 72.5%

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

   6. Applied egg-rr 38.4%

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

      1. *-commutative 38.4%

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

      2. flip-+ 37.8%

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

      3. frac-2neg 37.8%

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

      4. frac-times 28.7%

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

   8. Applied egg-rr 28.7%

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

   10. Simplified 6.0%

       \[\leadsto \color{blue}{\frac{-0.5}{-a}} \]

   if 1.05e-86 < b

   1. Initial program 11.3%

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

      1. *-commutative 11.3%

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

   3. Simplified 11.3%

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

   6. Applied egg-rr 10.9%

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

   7. Taylor expanded in b around -inf 2.6%

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

   9. Simplified 2.6%

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

       1. *-un-lft-identity 2.6%

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

       2. associate-*r/ 2.6%

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

       3. associate-*r/ 2.6%

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

       4. sub-div 2.6%

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

       5. add-sqr-sqrt 0.0%

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

       6. sqrt-unprod 6.9%

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

       7. sqr-neg 6.9%

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

       8. sqrt-unprod 12.5%

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

       9. add-sqr-sqrt 21.4%

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

   11. Applied egg-rr 21.4%

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

       1. *-lft-identity 21.4%

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

       2. div-sub 16.1%

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

       3. *-commutative 16.1%

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

       4. +-inverses 21.4%

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

   13. Simplified 21.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 11.1% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 52.0%

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

   1. *-commutative 52.0%

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

3. Simplified 52.0%

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

6. Applied egg-rr 45.9%

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

7. Taylor expanded in b around -inf 36.2%

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

9. Simplified 36.2%

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

    1. *-un-lft-identity 36.2%

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

    2. associate-*r/ 36.3%

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

    3. associate-*r/ 36.3%

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

    4. sub-div 36.3%

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

    5. add-sqr-sqrt 34.9%

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

    6. sqrt-unprod 26.2%

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

    7. sqr-neg 26.2%

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

    8. sqrt-unprod 4.7%

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

    9. add-sqr-sqrt 9.1%

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

11. Applied egg-rr 9.1%

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

    1. *-lft-identity 9.1%

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

    2. div-sub 7.1%

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

    3. *-commutative 7.1%

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

    4. +-inverses 9.1%

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

13. Simplified 9.1%

    \[\leadsto \color{blue}{0} \]
14. Add Preprocessing

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024107 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

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

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