The quadratic formula (r1)

Percentage Accurate: 51.4% → 85.5%

Time: 13.7s

Alternatives: 7

Speedup: 12.9×

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: 51.4% 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: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+115)
   (/ b (- a))
   (if (<= b 1.16e-82)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+115) {
		tmp = b / -a;
	} else if (b <= 1.16e-82) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * 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 <= (-4.5d+115)) then
        tmp = b / -a
    else if (b <= 1.16d-82) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * 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 <= -4.5e+115) {
		tmp = b / -a;
	} else if (b <= 1.16e-82) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e+115:
		tmp = b / -a
	elif b <= 1.16e-82:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+115)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.16e-82)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * 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 <= -4.5e+115)
		tmp = b / -a;
	elseif (b <= 1.16e-82)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+115], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.16e-82], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.49999999999999963e115

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in b around -inf 94.8%

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

      1. associate-*r/ 94.8%

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

      2. mul-1-neg 94.8%

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

   7. Simplified 94.8%

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

   if -4.49999999999999963e115 < b < 1.16e-82

   1. Initial program 82.8%

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

   if 1.16e-82 < b

   1. Initial program 17.4%

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

      1. *-commutative 17.4%

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

   3. Simplified 17.5%

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

   5. Taylor expanded in a around 0 85.3%

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

      1. associate-*r/ 85.3%

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

      2. mul-1-neg 85.3%

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

   7. Simplified 85.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-46}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.8e-46)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b 1.55e-82)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.8e-46) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 1.55e-82) {
		tmp = (sqrt((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 <= (-4.8d-46)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= 1.55d-82) then
        tmp = (sqrt((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 <= -4.8e-46) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 1.55e-82) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.8e-46:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= 1.55e-82:
		tmp = (math.sqrt((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 <= -4.8e-46)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= 1.55e-82)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -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 <= -4.8e-46)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= 1.55e-82)
		tmp = (sqrt((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, -4.8e-46], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e-82], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.80000000000000027e-46

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in b around -inf 92.3%

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

      1. mul-1-neg 92.3%

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

      2. distribute-rgt-neg-in 92.3%

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

      3. +-commutative 92.3%

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

      4. mul-1-neg 92.3%

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

      5. unsub-neg 92.3%

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

   7. Simplified 92.3%

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

   if -4.80000000000000027e-46 < b < 1.55e-82

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in a around inf 69.6%

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

      1. associate-*r* 69.7%

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

      2. *-commutative 69.7%

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

   7. Simplified 69.7%

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

   if 1.55e-82 < b

   1. Initial program 17.4%

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

      1. *-commutative 17.4%

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

   3. Simplified 17.5%

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

   5. Taylor expanded in a around 0 85.3%

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

      1. associate-*r/ 85.3%

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

      2. mul-1-neg 85.3%

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

   7. Simplified 85.3%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-46}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-238}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-205}:\\ \;\;\;\;\sqrt{\frac{c \cdot -4}{a}} \cdot \left(--0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-238)
   (/ b (- a))
   (if (<= b 2e-205) (* (sqrt (/ (* c -4.0) a)) (- -0.5)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-238) {
		tmp = b / -a;
	} else if (b <= 2e-205) {
		tmp = sqrt(((c * -4.0) / a)) * -(-0.5);
	} 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-238)) then
        tmp = b / -a
    else if (b <= 2d-205) then
        tmp = sqrt(((c * (-4.0d0)) / a)) * -(-0.5d0)
    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-238) {
		tmp = b / -a;
	} else if (b <= 2e-205) {
		tmp = Math.sqrt(((c * -4.0) / a)) * -(-0.5);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-238:
		tmp = b / -a
	elif b <= 2e-205:
		tmp = math.sqrt(((c * -4.0) / a)) * -(-0.5)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-238)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2e-205)
		tmp = Float64(sqrt(Float64(Float64(c * -4.0) / a)) * Float64(-(-0.5)));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-238)
		tmp = b / -a;
	elseif (b <= 2e-205)
		tmp = sqrt(((c * -4.0) / a)) * -(-0.5);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-238], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2e-205], N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] / a), $MachinePrecision]], $MachinePrecision] * (--0.5)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2e-238

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in b around -inf 77.0%

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

      1. associate-*r/ 77.0%

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

      2. mul-1-neg 77.0%

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

   7. Simplified 77.0%

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

   if -2e-238 < b < 2e-205

   1. Initial program 80.9%

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

      1. *-commutative 80.9%

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

   3. Simplified 80.9%

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

      1. add-cube-cbrt 80.1%

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

      2. pow3 79.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot a}\right)}^{3}} \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 79.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot a}\right)}^{3}} \cdot c}}{a \cdot 2} \]

   7. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 49.6%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 49.7%

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

   9. Simplified 49.7%

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

   if 2e-205 < b

   1. Initial program 27.3%

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

      1. *-commutative 27.3%

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

   3. Simplified 27.3%

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

   5. Taylor expanded in a around 0 73.3%

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

      1. associate-*r/ 73.3%

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

      2. mul-1-neg 73.3%

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

   7. Simplified 73.3%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-238}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-205}:\\ \;\;\;\;\sqrt{\frac{c \cdot -4}{a}} \cdot \left(--0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.05 \cdot 10^{-289}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.05e-289) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.05e-289) {
		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 <= 3.05d-289) 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 <= 3.05e-289) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.05e-289:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.05e-289)
		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 <= 3.05e-289)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.05e-289], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.0499999999999999e-289

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in b around -inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. mul-1-neg 67.3%

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

   7. Simplified 67.3%

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

   if 3.0499999999999999e-289 < b

   1. Initial program 35.1%

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

      1. *-commutative 35.1%

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

   3. Simplified 35.1%

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

   5. Taylor expanded in a around 0 65.4%

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

      1. associate-*r/ 65.4%

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

      2. mul-1-neg 65.4%

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

   7. Simplified 65.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.05 \cdot 10^{-289}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 6.2e-51) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.2e-51) {
		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 <= 6.2d-51) 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 <= 6.2e-51) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.2e-51:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.2e-51)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6.2e-51)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.2e-51], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.1999999999999995e-51

   1. Initial program 72.5%

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

      1. *-commutative 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in b around -inf 51.9%

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

      1. associate-*r/ 51.9%

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

      2. mul-1-neg 51.9%

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

   7. Simplified 51.9%

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

   if 6.1999999999999995e-51 < b

   1. Initial program 16.0%

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

      1. *-commutative 16.0%

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

   3. Simplified 16.0%

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

      1. add-exp-log 7.2%

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

   6. Applied egg-rr 7.2%

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

      1. rem-exp-log 16.0%

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

      2. clear-num 16.0%

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

      3. inv-pow 16.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 8.0%

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

      6. sqr-neg 8.0%

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

      7. sqrt-prod 8.0%

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

      8. add-sqr-sqrt 8.0%

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

      9. pow2 8.0%

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

      10. *-commutative 8.0%

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

      11. *-commutative 8.0%

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

   8. Applied egg-rr 8.0%

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

      1. unpow-1 8.0%

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

      2. unpow2 8.0%

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

      3. fma-neg 8.0%

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

      4. associate-*r* 8.0%

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

      5. *-commutative 8.0%

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

      6. distribute-rgt-neg-in 8.0%

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

      7. *-commutative 8.0%

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

      8. metadata-eval 8.0%

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

   10. Simplified 8.0%

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

   11. Taylor expanded in b around -inf 21.0%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 10.9% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

   1. *-commutative 55.5%

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

3. Simplified 55.5%

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

   1. add-exp-log 27.4%

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

6. Applied egg-rr 27.4%

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

   1. rem-exp-log 55.5%

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

   2. clear-num 55.4%

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

   3. inv-pow 55.4%

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

   4. add-sqr-sqrt 37.8%

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

   5. sqrt-unprod 52.7%

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

   6. sqr-neg 52.7%

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

   7. sqrt-prod 15.2%

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

   8. add-sqr-sqrt 33.2%

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

   9. pow2 33.2%

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

   10. *-commutative 33.2%

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

   11. *-commutative 33.2%

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

8. Applied egg-rr 33.2%

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

   1. unpow-1 33.2%

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

   2. unpow2 33.2%

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

   3. fma-neg 33.2%

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

   4. associate-*r* 33.2%

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

   5. *-commutative 33.2%

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

   6. distribute-rgt-neg-in 33.2%

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

   7. *-commutative 33.2%

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

   8. metadata-eval 33.2%

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

10. Simplified 33.2%

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

11. Taylor expanded in b around -inf 8.4%

    \[\leadsto \color{blue}{\frac{c}{b}} \]
12. Add Preprocessing

Alternative 7: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

   1. *-commutative 55.5%

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

3. Simplified 55.5%

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

5. Taylor expanded in b around -inf 37.1%

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

   1. associate-*r/ 37.1%

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

   2. mul-1-neg 37.1%

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

7. Simplified 37.1%

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

   1. div-inv 37.0%

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

   2. add-sqr-sqrt 35.5%

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

   3. sqrt-unprod 27.5%

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

   4. sqr-neg 27.5%

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

   5. sqrt-prod 1.9%

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

   6. add-sqr-sqrt 2.7%

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

9. Applied egg-rr 2.7%

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

    1. associate-*r/ 2.7%

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

    2. *-rgt-identity 2.7%

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

11. Simplified 2.7%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
12. Add Preprocessing

Developer Target 1: 69.9% accurate, 0.9× 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 2024125 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (- (* b b) (* (* 4 a) c)))) (let ((r1 (/ (+ (- b) (sqrt d)) (* 2 a)))) (let ((r2 (/ (- (- b) (sqrt d)) (* 2 a)))) (if (< b 0) r1 (/ c (* a r2)))))))

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