The quadratic formula (r1)

Percentage Accurate: 51.6% → 85.7%

Time: 15.8s

Alternatives: 9

Speedup: 19.1×

Specification

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.6% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+155)
   (- (/ c b) (/ b a))
   (if (<= b 5.2e-70)
     (/ (- (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 <= -2e+155) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.2e-70) {
		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 <= -2e+155)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5.2e-70)
		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, -2e+155], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-70], 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 < -2.00000000000000001e155

   1. Initial program 33.5%

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

      1. *-commutative 33.5%

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

   3. Simplified 33.5%

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

   4. Taylor expanded in b around -inf 95.5%

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

      1. +-commutative 95.5%

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

      2. mul-1-neg 95.5%

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

      3. unsub-neg 95.5%

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

   6. Simplified 95.5%

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

   if -2.00000000000000001e155 < b < 5.20000000000000004e-70

   1. Initial program 85.4%

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

   3. Simplified 85.5%

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

   if 5.20000000000000004e-70 < b

   1. Initial program 18.4%

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

      1. *-commutative 18.4%

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

   3. Simplified 18.4%

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

   4. Taylor expanded in b around inf 88.9%

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

      1. mul-1-neg 88.9%

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

      2. distribute-neg-frac 88.9%

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

   6. Simplified 88.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+153)
   (- (/ c b) (/ b a))
   (if (<= b 1.1e-69)
     (/ (- (sqrt (- (* 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 <= -1e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.1e-69) {
		tmp = (sqrt(((b * b) - (c * (a * 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 <= (-1d+153)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.1d-69) then
        tmp = (sqrt(((b * b) - (c * (a * 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 <= -1e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.1e-69) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+153:
		tmp = (c / b) - (b / a)
	elif b <= 1.1e-69:
		tmp = (math.sqrt(((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 <= -1e+153)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.1e-69)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 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 <= -1e+153)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.1e-69)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+153], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e-69], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - 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 < -1e153

   1. Initial program 33.5%

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

      1. *-commutative 33.5%

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

   3. Simplified 33.5%

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

   4. Taylor expanded in b around -inf 95.5%

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

      1. +-commutative 95.5%

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

      2. mul-1-neg 95.5%

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

      3. unsub-neg 95.5%

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

   6. Simplified 95.5%

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

   if -1e153 < b < 1.1e-69

   1. Initial program 85.4%

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

   if 1.1e-69 < b

   1. Initial program 18.4%

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

      1. *-commutative 18.4%

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

   3. Simplified 18.4%

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

   4. Taylor expanded in b around inf 88.9%

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

      1. mul-1-neg 88.9%

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

      2. distribute-neg-frac 88.9%

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

   6. Simplified 88.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{0.5}}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.1e-76)
   (- (/ c b) (/ b a))
   (if (<= b 1.6e-70)
     (/ 1.0 (/ (/ a 0.5) (- (sqrt (* a (* c -4.0))) b)))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.1e-76) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.6e-70) {
		tmp = 1.0 / ((a / 0.5) / (sqrt((a * (c * -4.0))) - b));
	} 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 <= (-5.1d-76)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.6d-70) then
        tmp = 1.0d0 / ((a / 0.5d0) / (sqrt((a * (c * (-4.0d0)))) - b))
    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 <= -5.1e-76) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.6e-70) {
		tmp = 1.0 / ((a / 0.5) / (Math.sqrt((a * (c * -4.0))) - b));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.1e-76:
		tmp = (c / b) - (b / a)
	elif b <= 1.6e-70:
		tmp = 1.0 / ((a / 0.5) / (math.sqrt((a * (c * -4.0))) - b))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.1e-76)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.6e-70)
		tmp = Float64(1.0 / Float64(Float64(a / 0.5) / Float64(sqrt(Float64(a * Float64(c * -4.0))) - b)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.1e-76)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.6e-70)
		tmp = 1.0 / ((a / 0.5) / (sqrt((a * (c * -4.0))) - b));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.1e-76], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e-70], N[(1.0 / N[(N[(a / 0.5), $MachinePrecision] / N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.09999999999999986e-76

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in b around -inf 86.2%

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

      1. +-commutative 86.2%

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

      2. mul-1-neg 86.2%

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

      3. unsub-neg 86.2%

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

   6. Simplified 86.2%

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

   if -5.09999999999999986e-76 < b < 1.5999999999999999e-70

   1. Initial program 76.7%

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

      1. *-commutative 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in b around 0 76.7%

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

      1. associate-*r* 76.7%

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

      2. fma-def 76.7%

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

      3. *-commutative 76.7%

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

   6. Simplified 76.7%

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

      1. clear-num 76.7%

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

      2. inv-pow 76.7%

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

      3. neg-mul-1 76.7%

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

      4. fma-def 76.7%

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

   8. Applied egg-rr 76.7%

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

      1. unpow-1 76.7%

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

      2. associate-/l* 76.7%

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

   10. Simplified 76.7%

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

       1. *-un-lft-identity 76.7%

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

       2. div-inv 76.7%

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

       3. metadata-eval 76.7%

          \[\leadsto \frac{1}{\frac{1 \cdot a}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot -4, c, {b}^{2}\right)}\right) \cdot \color{blue}{0.5}}} \]

       4. times-frac 76.4%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot -4, c, {b}^{2}\right)}\right)} \cdot \frac{a}{0.5}}} \]

       5. *-un-lft-identity 76.4%

          \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{1 \cdot \mathsf{fma}\left(a \cdot -4, c, {b}^{2}\right)}}\right)} \cdot \frac{a}{0.5}} \]

       6. fma-udef 76.4%

          \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(-1, b, \sqrt{1 \cdot \color{blue}{\left(\left(a \cdot -4\right) \cdot c + {b}^{2}\right)}}\right)} \cdot \frac{a}{0.5}} \]

       7. *-un-lft-identity 76.4%

          \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c + {b}^{2}}}\right)} \cdot \frac{a}{0.5}} \]

       8. associate-*r* 76.4%

          \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{a \cdot \left(-4 \cdot c\right)} + {b}^{2}}\right)} \cdot \frac{a}{0.5}} \]

       9. fma-udef 76.4%

          \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)}}\right)} \cdot \frac{a}{0.5}} \]

   12. Applied egg-rr 76.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)}\right)} \cdot \frac{a}{0.5}}} \]
   13. Step-by-step derivation

       1. associate-*l/ 76.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot \frac{a}{0.5}}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)}\right)}}} \]

       2. *-lft-identity 76.7%

          \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a}{0.5}}}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)}\right)}} \]

       3. fma-udef 76.7%

          \[\leadsto \frac{1}{\frac{\frac{a}{0.5}}{\color{blue}{-1 \cdot b + \sqrt{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)}}}} \]

       4. neg-mul-1 76.7%

          \[\leadsto \frac{1}{\frac{\frac{a}{0.5}}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)}}} \]

       5. +-commutative 76.7%

          \[\leadsto \frac{1}{\frac{\frac{a}{0.5}}{\color{blue}{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)} + \left(-b\right)}}} \]

       6. sub-neg 76.7%

          \[\leadsto \frac{1}{\frac{\frac{a}{0.5}}{\color{blue}{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)} - b}}} \]

       7. *-commutative 76.7%

          \[\leadsto \frac{1}{\frac{\frac{a}{0.5}}{\sqrt{\mathsf{fma}\left(a, \color{blue}{c \cdot -4}, {b}^{2}\right)} - b}} \]

   14. Simplified 76.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{a}{0.5}}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b}}} \]

   15. Taylor expanded in a around inf 70.2%

       \[\leadsto \frac{1}{\frac{\frac{a}{0.5}}{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}} - b}} \]
   16. Step-by-step derivation

       1. *-commutative 70.1%

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

       2. associate-*r* 70.1%

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

   17. Simplified 70.2%

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

   if 1.5999999999999999e-70 < b

   1. Initial program 18.4%

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

      1. *-commutative 18.4%

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

   3. Simplified 18.4%

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

   4. Taylor expanded in b around inf 88.9%

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

      1. mul-1-neg 88.9%

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

      2. distribute-neg-frac 88.9%

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

   6. Simplified 88.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{0.5}}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-76}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-70}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.45e-76)
   (- (/ c b) (/ b a))
   (if (<= b 3.7e-70)
     (* (- (sqrt (* a (* c -4.0))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e-76) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.7e-70) {
		tmp = (sqrt((a * (c * -4.0))) - b) * (0.5 / 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 <= (-1.45d-76)) then
        tmp = (c / b) - (b / a)
    else if (b <= 3.7d-70) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) * (0.5d0 / 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 <= -1.45e-76) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.7e-70) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.45e-76:
		tmp = (c / b) - (b / a)
	elif b <= 3.7e-70:
		tmp = (math.sqrt((a * (c * -4.0))) - b) * (0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.45e-76)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 3.7e-70)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.45e-76)
		tmp = (c / b) - (b / a);
	elseif (b <= 3.7e-70)
		tmp = (sqrt((a * (c * -4.0))) - b) * (0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.45e-76], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.7e-70], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.4500000000000001e-76

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in b around -inf 86.2%

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

      1. +-commutative 86.2%

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

      2. mul-1-neg 86.2%

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

      3. unsub-neg 86.2%

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

   6. Simplified 86.2%

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

   if -1.4500000000000001e-76 < b < 3.7e-70

   1. Initial program 76.7%

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

      1. *-commutative 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in b around 0 76.7%

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

      1. associate-*r* 76.7%

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

      2. fma-def 76.7%

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

      3. *-commutative 76.7%

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

   6. Simplified 76.7%

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

      1. +-commutative 76.7%

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

      2. *-un-lft-identity 76.7%

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

      3. fma-def 76.7%

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

   8. Applied egg-rr 76.7%

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

      1. fma-udef 76.7%

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

      2. *-lft-identity 76.7%

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

      3. unsub-neg 76.7%

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

      4. fma-def 76.7%

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

      5. associate-*l* 76.7%

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

      6. fma-def 76.7%

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

   10. Simplified 76.7%

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

       1. div-sub 76.6%

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

       2. sub-neg 76.6%

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

       3. *-un-lft-identity 76.6%

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

       4. *-commutative 76.6%

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

       5. times-frac 76.6%

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

       6. metadata-eval 76.6%

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

   12. Applied egg-rr 76.6%

       \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)}}{a \cdot 2} + \left(-0.5 \cdot \frac{b}{a}\right)} \]
   13. Step-by-step derivation

       1. sub-neg 76.6%

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

       2. *-commutative 76.6%

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

       3. metadata-eval 76.6%

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

       4. times-frac 76.6%

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

       5. *-rgt-identity 76.6%

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

       6. div-sub 76.7%

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

       7. *-lft-identity 76.7%

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

       8. *-lft-identity 76.7%

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

       9. sub-neg 76.7%

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

       10. +-commutative 76.7%

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

       11. neg-mul-1 76.7%

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

       12. fma-udef 76.7%

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

       13. *-lft-identity 76.7%

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

       14. *-commutative 76.7%

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

       15. times-frac 76.7%

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

   14. Simplified 76.6%

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

   15. Taylor expanded in a around inf 70.1%

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

       1. *-commutative 70.1%

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

       2. associate-*r* 70.1%

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

   17. Simplified 70.1%

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

   if 3.7e-70 < b

   1. Initial program 18.4%

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

      1. *-commutative 18.4%

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

   3. Simplified 18.4%

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

   4. Taylor expanded in b around inf 88.9%

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

      1. mul-1-neg 88.9%

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

      2. distribute-neg-frac 88.9%

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

   6. Simplified 88.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-76}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-70}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-76}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e-76)
   (- (/ c b) (/ b a))
   (if (<= b 1.08e-69)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-76) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.08e-69) {
		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 <= (-8d-76)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.08d-69) 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 <= -8e-76) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.08e-69) {
		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 <= -8e-76:
		tmp = (c / b) - (b / a)
	elif b <= 1.08e-69:
		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 <= -8e-76)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.08e-69)
		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 <= -8e-76)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.08e-69)
		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, -8e-76], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.08e-69], 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 < -7.99999999999999942e-76

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in b around -inf 86.2%

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

      1. +-commutative 86.2%

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

      2. mul-1-neg 86.2%

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

      3. unsub-neg 86.2%

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

   6. Simplified 86.2%

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

   if -7.99999999999999942e-76 < b < 1.0800000000000001e-69

   1. Initial program 76.7%

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

      1. *-commutative 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in b around 0 76.7%

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

      1. associate-*r* 76.7%

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

      2. fma-def 76.7%

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

      3. *-commutative 76.7%

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

   6. Simplified 76.7%

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

      1. +-commutative 76.7%

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

      2. *-un-lft-identity 76.7%

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

      3. fma-def 76.7%

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

   8. Applied egg-rr 76.7%

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

      1. fma-udef 76.7%

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

      2. *-lft-identity 76.7%

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

      3. unsub-neg 76.7%

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

      4. fma-def 76.7%

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

      5. associate-*l* 76.7%

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

      6. fma-def 76.7%

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

   10. Simplified 76.7%

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

   11. Taylor expanded in a around inf 70.2%

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

       1. associate-*r* 70.2%

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

       2. *-commutative 70.2%

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

       3. associate-*l* 70.2%

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

       4. *-commutative 70.2%

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

   13. Simplified 70.2%

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

   if 1.0800000000000001e-69 < b

   1. Initial program 18.4%

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

      1. *-commutative 18.4%

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

   3. Simplified 18.4%

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

   4. Taylor expanded in b around inf 88.9%

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

      1. mul-1-neg 88.9%

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

      2. distribute-neg-frac 88.9%

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

   6. Simplified 88.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-76}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 68.2% accurate, 12.8× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		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 <= -5e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in b around -inf 66.4%

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

      1. +-commutative 66.4%

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

      2. mul-1-neg 66.4%

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

      3. unsub-neg 66.4%

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

   6. Simplified 66.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 35.3%

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

      1. *-commutative 35.3%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in b around inf 66.9%

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

      1. mul-1-neg 66.9%

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

      2. distribute-neg-frac 66.9%

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

   6. Simplified 66.9%

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

4. Final simplification 66.7%

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

Alternative 7: 44.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e-303) (/ (- b) a) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e-303) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.2d-303)) then
        tmp = -b / a
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e-303) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.2e-303:
		tmp = -b / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e-303)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.2e-303)
		tmp = -b / a;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e-303], N[((-b) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.2e-303

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in b around -inf 67.8%

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

      1. associate-*r/ 67.8%

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

      2. mul-1-neg 67.8%

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

   6. Simplified 67.8%

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

   if -1.2e-303 < b

   1. Initial program 36.6%

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

      1. *-commutative 36.6%

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

   3. Simplified 36.6%

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

   4. Taylor expanded in b around 0 36.6%

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

      1. associate-*r* 36.6%

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

      2. fma-def 36.6%

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

      3. *-commutative 36.6%

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

   6. Simplified 36.6%

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

      1. clear-num 36.6%

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

      2. inv-pow 36.6%

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

      3. neg-mul-1 36.6%

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

      4. fma-def 36.6%

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

   8. Applied egg-rr 36.6%

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

      1. unpow-1 36.6%

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

      2. associate-/l* 36.6%

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

   10. Simplified 36.6%

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

   11. Taylor expanded in a around 0 18.8%

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

       1. associate-*r/ 18.8%

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

       2. distribute-rgt1-in 18.8%

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

       3. metadata-eval 18.8%

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

       4. mul0-lft 18.8%

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

       5. metadata-eval 18.8%

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

   13. Simplified 18.8%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 8: 68.0% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-256}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-256) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-256) {
		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 <= 4.2d-256) 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 <= 4.2e-256) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.2e-256:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-256)
		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 <= 4.2e-256)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-256], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.20000000000000005e-256

   1. Initial program 73.3%

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

      1. *-commutative 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in b around -inf 62.7%

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

      1. associate-*r/ 62.7%

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

      2. mul-1-neg 62.7%

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

   6. Simplified 62.7%

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

   if 4.20000000000000005e-256 < b

   1. Initial program 33.2%

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

      1. *-commutative 33.2%

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

   3. Simplified 33.2%

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

   4. Taylor expanded in b around inf 70.6%

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

      1. mul-1-neg 70.6%

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

      2. distribute-neg-frac 70.6%

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

   6. Simplified 70.6%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-256}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 9: 11.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

   1. *-commutative 54.5%

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

3. Simplified 54.5%

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

4. Taylor expanded in b around 0 54.5%

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

   1. associate-*r* 54.5%

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

   2. fma-def 54.6%

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

   3. *-commutative 54.6%

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

6. Simplified 54.6%

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

   1. clear-num 54.5%

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

   2. inv-pow 54.5%

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

   3. neg-mul-1 54.5%

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

   4. fma-def 54.5%

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

8. Applied egg-rr 54.5%

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

   1. unpow-1 54.5%

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

   2. associate-/l* 54.5%

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

10. Simplified 54.5%

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

11. Taylor expanded in a around 0 10.9%

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

    1. associate-*r/ 10.9%

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

    2. distribute-rgt1-in 10.9%

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

    3. metadata-eval 10.9%

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

    4. mul0-lft 10.9%

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

    5. metadata-eval 10.9%

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

13. Simplified 10.9%

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

14. Final simplification 10.9%

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

Developer target: 69.7% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $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 2023314 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))