jeff quadratic root 2

Percentage Accurate: 72.8% → 90.7%

Time: 16.7s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (+ (- 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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-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(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 72.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (+ (- 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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-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(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b} - \frac{b}{a}\\ t_1 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{+43}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ c b) (/ b a))) (t_1 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -1e+137)
     (if (>= b 0.0) (* -2.0 (* 0.5 (/ b a))) t_0)
     (if (<= b 1.02e+43)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_1)) (/ (- t_1 b) (* a 2.0)))
       (if (>= b 0.0) (/ (* -2.0 (* 0.5 c)) b) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1e+137) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (0.5 * (b / a));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.02e+43) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_1);
		} else {
			tmp_3 = (t_1 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * (0.5 * c)) / b;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

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) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = (c / b) - (b / a)
    t_1 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b <= (-1d+137)) then
        if (b >= 0.0d0) then
            tmp_2 = (-2.0d0) * (0.5d0 * (b / a))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= 1.02d+43) then
        if (b >= 0.0d0) then
            tmp_3 = (c * 2.0d0) / (-b - t_1)
        else
            tmp_3 = (t_1 - b) / (a * 2.0d0)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = ((-2.0d0) * (0.5d0 * c)) / b
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1e+137) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (0.5 * (b / a));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.02e+43) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_1);
		} else {
			tmp_3 = (t_1 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * (0.5 * c)) / b;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c / b) - (b / a)
	t_1 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -1e+137:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -2.0 * (0.5 * (b / a))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= 1.02e+43:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (c * 2.0) / (-b - t_1)
		else:
			tmp_3 = (t_1 - b) / (a * 2.0)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (-2.0 * (0.5 * c)) / b
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / b) - Float64(b / a))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -1e+137)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(0.5 * Float64(b / a)));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.02e+43)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_1));
		else
			tmp_3 = Float64(Float64(t_1 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * Float64(0.5 * c)) / b);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (c / b) - (b / a);
	t_1 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp_2 = 0.0;
	if (b <= -1e+137)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -2.0 * (0.5 * (b / a));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.02e+43)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (c * 2.0) / (-b - t_1);
		else
			tmp_4 = (t_1 - b) / (a * 2.0);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * (0.5 * c)) / b;
	else
		tmp_2 = t_0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1e+137], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[LessEqual[b, 1.02e+43], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * N[(0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e137

   1. Initial program 45.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. mul-1-neg 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]

      2. unsub-neg 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   6. Simplified 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   7. Taylor expanded in b around -inf 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\left(0.5 \cdot \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   if -1e137 < b < 1.02e43

   1. Initial program 87.8%

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

   if 1.02e43 < b

   1. Initial program 65.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. mul-1-neg 65.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]

      2. unsub-neg 65.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   6. Simplified 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   7. Taylor expanded in c around 0 93.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\left(0.5 \cdot \frac{c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-*r/ 93.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   9. Simplified 93.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 93.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

       2. *-commutative 93.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{\left(c \cdot 0.5\right)}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   11. Applied egg-rr 93.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-2 \cdot \left(c \cdot 0.5\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{+43}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b} - \frac{b}{a}\\ t_1 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{+43}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ c b) (/ b a))) (t_1 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -2e+137)
     (if (>= b 0.0) (* -2.0 (* 0.5 (/ b a))) t_0)
     (if (<= b 1.02e+43)
       (if (>= b 0.0) (* c (/ 2.0 (- (- b) t_1))) (/ (- t_1 b) (* a 2.0)))
       (if (>= b 0.0) (/ (* -2.0 (* 0.5 c)) b) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -2e+137) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (0.5 * (b / a));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.02e+43) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (2.0 / (-b - t_1));
		} else {
			tmp_3 = (t_1 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * (0.5 * c)) / b;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

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) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = (c / b) - (b / a)
    t_1 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b <= (-2d+137)) then
        if (b >= 0.0d0) then
            tmp_2 = (-2.0d0) * (0.5d0 * (b / a))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= 1.02d+43) then
        if (b >= 0.0d0) then
            tmp_3 = c * (2.0d0 / (-b - t_1))
        else
            tmp_3 = (t_1 - b) / (a * 2.0d0)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = ((-2.0d0) * (0.5d0 * c)) / b
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double t_1 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -2e+137) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (0.5 * (b / a));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.02e+43) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (2.0 / (-b - t_1));
		} else {
			tmp_3 = (t_1 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * (0.5 * c)) / b;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c / b) - (b / a)
	t_1 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -2e+137:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -2.0 * (0.5 * (b / a))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= 1.02e+43:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = c * (2.0 / (-b - t_1))
		else:
			tmp_3 = (t_1 - b) / (a * 2.0)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (-2.0 * (0.5 * c)) / b
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / b) - Float64(b / a))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -2e+137)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(0.5 * Float64(b / a)));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.02e+43)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(2.0 / Float64(Float64(-b) - t_1)));
		else
			tmp_3 = Float64(Float64(t_1 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * Float64(0.5 * c)) / b);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (c / b) - (b / a);
	t_1 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp_2 = 0.0;
	if (b <= -2e+137)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -2.0 * (0.5 * (b / a));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.02e+43)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = c * (2.0 / (-b - t_1));
		else
			tmp_4 = (t_1 - b) / (a * 2.0);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * (0.5 * c)) / b;
	else
		tmp_2 = t_0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2e+137], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[LessEqual[b, 1.02e+43], If[GreaterEqual[b, 0.0], N[(c * N[(2.0 / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * N[(0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e137

   1. Initial program 45.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. mul-1-neg 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]

      2. unsub-neg 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   6. Simplified 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   7. Taylor expanded in b around -inf 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\left(0.5 \cdot \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   if -2.0000000000000001e137 < b < 1.02e43

   1. Initial program 87.8%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 78.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. expm1-udef 61.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      3. associate-/l* 61.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      4. *-commutative 61.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}}{c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      5. *-commutative 61.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}}}{c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   3. Applied egg-rr 61.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{c}}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   4. Step-by-step derivation

      1. expm1-def 78.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{c}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. expm1-log1p 87.7%

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

      3. associate-/r/ 87.7%

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

   5. Simplified 87.7%

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

   if 1.02e43 < b

   1. Initial program 65.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. mul-1-neg 65.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]

      2. unsub-neg 65.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   6. Simplified 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   7. Taylor expanded in c around 0 93.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\left(0.5 \cdot \frac{c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-*r/ 93.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   9. Simplified 93.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 93.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

       2. *-commutative 93.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{\left(c \cdot 0.5\right)}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   11. Applied egg-rr 93.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-2 \cdot \left(c \cdot 0.5\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{+43}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b} - \frac{b}{a}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-290}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+25}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ c b) (/ b a))))
   (if (<= b -1e+137)
     (if (>= b 0.0) (* -2.0 (* 0.5 (/ b a))) t_0)
     (if (<= b -4e-290)
       (if (>= b 0.0)
         (* c (/ 2.0 (- (- b) b)))
         (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)))
       (if (<= b 1.3e+25)
         (if (>= b 0.0)
           (* -2.0 (/ c (+ b (sqrt (* a (* c -4.0))))))
           (* (/ -0.5 a) (+ b b)))
         (if (>= b 0.0) (/ (* -2.0 (* 0.5 c)) b) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double tmp_1;
	if (b <= -1e+137) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (0.5 * (b / a));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -4e-290) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (2.0 / (-b - b));
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.3e+25) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -2.0 * (c / (b + sqrt((a * (c * -4.0)))));
		} else {
			tmp_4 = (-0.5 / a) * (b + b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * (0.5 * c)) / b;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

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
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = (c / b) - (b / a)
    if (b <= (-1d+137)) then
        if (b >= 0.0d0) then
            tmp_2 = (-2.0d0) * (0.5d0 * (b / a))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= (-4d-290)) then
        if (b >= 0.0d0) then
            tmp_3 = c * (2.0d0 / (-b - b))
        else
            tmp_3 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
        end if
        tmp_1 = tmp_3
    else if (b <= 1.3d+25) then
        if (b >= 0.0d0) then
            tmp_4 = (-2.0d0) * (c / (b + sqrt((a * (c * (-4.0d0))))))
        else
            tmp_4 = ((-0.5d0) / a) * (b + b)
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = ((-2.0d0) * (0.5d0 * c)) / b
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c / b) - (b / a);
	double tmp_1;
	if (b <= -1e+137) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (0.5 * (b / a));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -4e-290) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (2.0 / (-b - b));
		} else {
			tmp_3 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.3e+25) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -2.0 * (c / (b + Math.sqrt((a * (c * -4.0)))));
		} else {
			tmp_4 = (-0.5 / a) * (b + b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * (0.5 * c)) / b;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c / b) - (b / a)
	tmp_1 = 0
	if b <= -1e+137:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -2.0 * (0.5 * (b / a))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= -4e-290:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = c * (2.0 / (-b - b))
		else:
			tmp_3 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
		tmp_1 = tmp_3
	elif b <= 1.3e+25:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = -2.0 * (c / (b + math.sqrt((a * (c * -4.0)))))
		else:
			tmp_4 = (-0.5 / a) * (b + b)
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (-2.0 * (0.5 * c)) / b
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / b) - Float64(b / a))
	tmp_1 = 0.0
	if (b <= -1e+137)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(0.5 * Float64(b / a)));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -4e-290)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(2.0 / Float64(Float64(-b) - b)));
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.3e+25)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-2.0 * Float64(c / Float64(b + sqrt(Float64(a * Float64(c * -4.0))))));
		else
			tmp_4 = Float64(Float64(-0.5 / a) * Float64(b + b));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * Float64(0.5 * c)) / b);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = (c / b) - (b / a);
	tmp_2 = 0.0;
	if (b <= -1e+137)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -2.0 * (0.5 * (b / a));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= -4e-290)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = c * (2.0 / (-b - b));
		else
			tmp_4 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.3e+25)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = -2.0 * (c / (b + sqrt((a * (c * -4.0)))));
		else
			tmp_5 = (-0.5 / a) * (b + b);
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * (0.5 * c)) / b;
	else
		tmp_2 = t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1e+137], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[LessEqual[b, -4e-290], If[GreaterEqual[b, 0.0], N[(c * N[(2.0 / N[((-b) - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]], If[LessEqual[b, 1.3e+25], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * N[(0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1e137

   1. Initial program 45.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. mul-1-neg 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]

      2. unsub-neg 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   6. Simplified 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   7. Taylor expanded in b around -inf 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\left(0.5 \cdot \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   if -1e137 < b < -4.0000000000000003e-290

   1. Initial program 88.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 88.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. expm1-udef 88.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      3. associate-/l* 88.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      4. *-commutative 88.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}}{c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      5. *-commutative 88.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}}}{c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   3. Applied egg-rr 88.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{c}}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   4. Step-by-step derivation

      1. expm1-def 88.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{c}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. expm1-log1p 88.0%

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

      3. associate-/r/ 88.0%

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

   5. Simplified 88.0%

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

   6. Taylor expanded in b around inf 88.0%

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

   if -4.0000000000000003e-290 < b < 1.2999999999999999e25

   1. Initial program 87.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified 87.4%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 87.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   5. Taylor expanded in b around 0 69.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r* 69.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\color{blue}{\left(-4 \cdot c\right) \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      2. *-commutative 69.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\color{blue}{a \cdot \left(-4 \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      3. *-commutative 69.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   7. Simplified 69.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   if 1.2999999999999999e25 < b

   1. Initial program 65.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. mul-1-neg 65.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]

      2. unsub-neg 65.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   6. Simplified 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   7. Taylor expanded in c around 0 93.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\left(0.5 \cdot \frac{c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-*r/ 93.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   9. Simplified 93.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 93.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

       2. *-commutative 93.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{\left(c \cdot 0.5\right)}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   11. Applied egg-rr 93.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-2 \cdot \left(c \cdot 0.5\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-290}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+25}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+25}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.3e+25)
   (if (>= b 0.0)
     (* -2.0 (/ c (+ b (sqrt (* a (* c -4.0))))))
     (* (/ -0.5 a) (+ b b)))
   (if (>= b 0.0) (/ (* -2.0 (* 0.5 c)) b) (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 1.3e+25) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + sqrt((a * (c * -4.0)))));
		} else {
			tmp_2 = (-0.5 / a) * (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * (0.5 * c)) / b;
	} else {
		tmp_1 = (c / b) - (b / a);
	}
	return tmp_1;
}

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
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= 1.3d+25) then
        if (b >= 0.0d0) then
            tmp_2 = (-2.0d0) * (c / (b + sqrt((a * (c * (-4.0d0))))))
        else
            tmp_2 = ((-0.5d0) / a) * (b + b)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = ((-2.0d0) * (0.5d0 * c)) / b
    else
        tmp_1 = (c / b) - (b / a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 1.3e+25) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + Math.sqrt((a * (c * -4.0)))));
		} else {
			tmp_2 = (-0.5 / a) * (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * (0.5 * c)) / b;
	} else {
		tmp_1 = (c / b) - (b / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= 1.3e+25:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -2.0 * (c / (b + math.sqrt((a * (c * -4.0)))))
		else:
			tmp_2 = (-0.5 / a) * (b + b)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (-2.0 * (0.5 * c)) / b
	else:
		tmp_1 = (c / b) - (b / a)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 1.3e+25)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(c / Float64(b + sqrt(Float64(a * Float64(c * -4.0))))));
		else
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * Float64(0.5 * c)) / b);
	else
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= 1.3e+25)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -2.0 * (c / (b + sqrt((a * (c * -4.0)))));
		else
			tmp_3 = (-0.5 / a) * (b + b);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * (0.5 * c)) / b;
	else
		tmp_2 = (c / b) - (b / a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.3e+25], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * N[(0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2999999999999999e25

   1. Initial program 78.1%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   5. Taylor expanded in b around 0 66.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r* 66.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\color{blue}{\left(-4 \cdot c\right) \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      2. *-commutative 66.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\color{blue}{a \cdot \left(-4 \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      3. *-commutative 66.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   7. Simplified 66.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   if 1.2999999999999999e25 < b

   1. Initial program 65.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. mul-1-neg 65.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]

      2. unsub-neg 65.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   6. Simplified 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   7. Taylor expanded in c around 0 93.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\left(0.5 \cdot \frac{c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-*r/ 93.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   9. Simplified 93.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 93.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

       2. *-commutative 93.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{\left(c \cdot 0.5\right)}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   11. Applied egg-rr 93.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-2 \cdot \left(c \cdot 0.5\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+25}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 35.2% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* -2.0 (* 0.5 (/ b a))) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * (0.5 * (b / a));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-2.0d0) * (0.5d0 * (b / a))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * (0.5 * (b / a));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -2.0 * (0.5 * (b / a))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-2.0 * Float64(0.5 * Float64(b / a)));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -2.0 * (0.5 * (b / a));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-2.0 * N[(0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
2. Step-by-step derivation

3. Simplified 74.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

4. Taylor expanded in b around -inf 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
5. Step-by-step derivation

   1. mul-1-neg 70.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]

   2. unsub-neg 70.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

6. Simplified 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

7. Taylor expanded in b around -inf 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\left(0.5 \cdot \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

8. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6: 68.0% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* -2.0 (/ c (+ b b))) (* (/ -0.5 a) (* b 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * (c / (b + b));
	} else {
		tmp = (-0.5 / a) * (b * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-2.0d0) * (c / (b + b))
    else
        tmp = ((-0.5d0) / a) * (b * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * (c / (b + b));
	} else {
		tmp = (-0.5 / a) * (b * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -2.0 * (c / (b + b))
	else:
		tmp = (-0.5 / a) * (b * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-2.0 * Float64(c / Float64(b + b)));
	else
		tmp = Float64(Float64(-0.5 / a) * Float64(b * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -2.0 * (c / (b + b));
	else
		tmp = (-0.5 / a) * (b * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
2. Step-by-step derivation

3. Simplified 74.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

4. Taylor expanded in b around inf 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

5. Taylor expanded in b around -inf 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
6. Step-by-step derivation

   1. *-commutative 64.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 2\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

7. Simplified 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 2\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

8. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b \cdot 2\right)\\ \end{array} \]

Alternative 7: 68.1% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* -2.0 (/ c (+ b b))) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * (c / (b + b));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-2.0d0) * (c / (b + b))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * (c / (b + b));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -2.0 * (c / (b + b))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-2.0 * Float64(c / Float64(b + b)));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -2.0 * (c / (b + b));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Initial program 74.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
2. Step-by-step derivation

3. Simplified 74.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

4. Taylor expanded in b around inf 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

5. Taylor expanded in b around -inf 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
6. Step-by-step derivation

   1. associate-*r/ 64.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot b}{a}\\ \end{array} \]

   2. neg-mul-1 64.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

7. Simplified 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

8. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8: 68.2% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{0.5 \cdot c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* -2.0 (/ (* 0.5 c) b)) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * ((0.5 * c) / b);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-2.0d0) * ((0.5d0 * c) / b)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * ((0.5 * c) / b);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -2.0 * ((0.5 * c) / b)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-2.0 * Float64(Float64(0.5 * c) / b));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -2.0 * ((0.5 * c) / b);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-2.0 * N[(N[(0.5 * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
2. Step-by-step derivation

3. Simplified 74.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

4. Taylor expanded in b around -inf 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
5. Step-by-step derivation

   1. mul-1-neg 70.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]

   2. unsub-neg 70.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

6. Simplified 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

7. Taylor expanded in c around 0 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\left(0.5 \cdot \frac{c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
8. Step-by-step derivation

   1. associate-*r/ 64.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

9. Simplified 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

10. Final simplification 64.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{0.5 \cdot c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 9: 68.2% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (/ (* -2.0 (* 0.5 c)) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-2.0 * (0.5 * c)) / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = ((-2.0d0) * (0.5d0 * c)) / b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-2.0 * (0.5 * c)) / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (-2.0 * (0.5 * c)) / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-2.0 * Float64(0.5 * c)) / b);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-2.0 * (0.5 * c)) / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(-2.0 * N[(0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
2. Step-by-step derivation

3. Simplified 74.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

4. Taylor expanded in b around -inf 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
5. Step-by-step derivation

   1. mul-1-neg 70.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]

   2. unsub-neg 70.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

6. Simplified 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

7. Taylor expanded in c around 0 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\left(0.5 \cdot \frac{c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
8. Step-by-step derivation

   1. associate-*r/ 64.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

9. Simplified 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{0.5 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
10. Step-by-step derivation

    1. associate-*r/ 64.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

    2. *-commutative 64.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{\left(c \cdot 0.5\right)}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

11. Applied egg-rr 64.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-2 \cdot \left(c \cdot 0.5\right)}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

12. Final simplification 64.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \left(0.5 \cdot c\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2023181 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))