jeff quadratic root 2

Percentage Accurate: 71.8% → 90.9%

Time: 19.4s

Alternatives: 7

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: 71.8% 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: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{+133}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(b, 2, a \cdot \left(-2 \cdot \frac{c}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -4.5e+152)
     (if (>= b 0.0) (* -2.0 (/ c (+ b b))) (- (/ c b) (/ b a)))
     (if (<= b 2.25e+133)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0)
         (* -2.0 (/ c (fma b 2.0 (* a (* -2.0 (/ c b))))))
         (/ (- c) b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -4.5e+152) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + b));
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.25e+133) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / fma(b, 2.0, (a * (-2.0 * (c / b)))));
	} else {
		tmp_1 = -c / b;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -4.5e+152)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(c / Float64(b + b)));
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.25e+133)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(c / fma(b, 2.0, Float64(a * Float64(-2.0 * Float64(c / b))))));
	else
		tmp_1 = Float64(Float64(-c) / b);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -4.5e+152], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.25e+133], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b * 2.0 + N[(a * N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5000000000000001e152

   1. Initial program 42.2%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in b around inf 42.2%

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

   5. Taylor expanded in b around -inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. unsub-neg 97.3%

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

   7. Simplified 97.3%

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

   if -4.5000000000000001e152 < b < 2.24999999999999992e133

   1. Initial program 88.9%

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

   if 2.24999999999999992e133 < b

   1. Initial program 35.3%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in b around inf 86.5%

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

      1. +-commutative 86.5%

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

      2. fma-def 86.5%

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

      3. associate-/l* 93.0%

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

      4. associate-/r/ 93.0%

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

   6. Simplified 93.0%

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

   7. Taylor expanded in b around inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. distribute-neg-frac 93.0%

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

   9. Simplified 93.0%

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

   10. Taylor expanded in b around 0 86.5%

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

       1. *-commutative 86.5%

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

       2. fma-def 86.5%

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

       3. *-commutative 86.5%

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

       4. *-commutative 86.5%

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

       5. associate-*r/ 93.0%

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

       6. associate-*l* 93.0%

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

   12. Simplified 93.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{+133}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(b, 2, a \cdot \left(-2 \cdot \frac{c}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \left(c \cdot \frac{1}{b + \mathsf{fma}\left(-2, \frac{c}{b} \cdot a, b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+152)
   (if (>= b 0.0) (* -2.0 (/ c (+ b b))) (- (/ c b) (/ b a)))
   (if (>= b 0.0)
     (* -2.0 (* c (/ 1.0 (+ b (fma -2.0 (* (/ c b) a) b)))))
     (* (- b (sqrt (- (* b b) (* c (* a 4.0))))) (/ -0.5 a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -4.5e+152) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + b));
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c * (1.0 / (b + fma(-2.0, ((c / b) * a), b))));
	} else {
		tmp_1 = (b - sqrt(((b * b) - (c * (a * 4.0))))) * (-0.5 / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -4.5e+152)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(c / Float64(b + b)));
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(c * Float64(1.0 / Float64(b + fma(-2.0, Float64(Float64(c / b) * a), b)))));
	else
		tmp_1 = Float64(Float64(b - sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))) * Float64(-0.5 / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+152], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c * N[(1.0 / N[(b + N[(-2.0 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5000000000000001e152

   1. Initial program 42.2%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in b around inf 42.2%

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

   5. Taylor expanded in b around -inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. unsub-neg 97.3%

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

   7. Simplified 97.3%

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

   if -4.5000000000000001e152 < b

   1. Initial program 76.9%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in b around inf 73.1%

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

      1. +-commutative 73.1%

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

      2. fma-def 73.1%

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

      3. associate-/l* 74.7%

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

      4. associate-/r/ 74.7%

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

   6. Simplified 74.7%

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

      1. associate-*r* 74.7%

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

      2. metadata-eval 74.7%

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

      3. distribute-rgt-neg-in 74.7%

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

      4. associate-*r* 74.7%

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

      5. fma-neg 74.7%

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

   8. Applied egg-rr 74.7%

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

      1. div-inv 74.6%

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

      2. *-commutative 74.6%

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

   10. Applied egg-rr 74.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \left(c \cdot \frac{1}{b + \mathsf{fma}\left(-2, \frac{c}{b} \cdot a, b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

Alternative 3: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{fma}\left(-2, \frac{c}{b} \cdot a, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+152)
   (if (>= b 0.0) (* -2.0 (/ c (+ b b))) (- (/ c b) (/ b a)))
   (if (>= b 0.0)
     (* -2.0 (/ c (+ b (fma -2.0 (* (/ c b) a) b))))
     (* (- b (sqrt (- (* b b) (* c (* a 4.0))))) (/ -0.5 a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -4.5e+152) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + b));
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + fma(-2.0, ((c / b) * a), b)));
	} else {
		tmp_1 = (b - sqrt(((b * b) - (c * (a * 4.0))))) * (-0.5 / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -4.5e+152)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(c / Float64(b + b)));
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(c / Float64(b + fma(-2.0, Float64(Float64(c / b) * a), b))));
	else
		tmp_1 = Float64(Float64(b - sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))) * Float64(-0.5 / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+152], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[(-2.0 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5000000000000001e152

   1. Initial program 42.2%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in b around inf 42.2%

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

   5. Taylor expanded in b around -inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. unsub-neg 97.3%

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

   7. Simplified 97.3%

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

   if -4.5000000000000001e152 < b

   1. Initial program 76.9%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in b around inf 73.1%

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

      1. +-commutative 73.1%

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

      2. fma-def 73.1%

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

      3. associate-/l* 74.7%

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

      4. associate-/r/ 74.7%

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

   6. Simplified 74.7%

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

      1. associate-*r* 74.7%

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

      2. metadata-eval 74.7%

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

      3. distribute-rgt-neg-in 74.7%

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

      4. associate-*r* 74.7%

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

      5. fma-neg 74.7%

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

   8. Applied egg-rr 74.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{fma}\left(-2, \frac{c}{b} \cdot a, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

Alternative 4: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \frac{c}{b + b}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{-76}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -2.0 (/ c (+ b b)))))
   (if (<= b -3.5e-76)
     (if (>= b 0.0) t_0 (- (/ c b) (/ b a)))
     (if (>= b 0.0) t_0 (* (/ -0.5 a) (- b (sqrt (* (* c a) -4.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = -2.0 * (c / (b + b));
	double tmp_1;
	if (b <= -3.5e-76) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (-0.5 / a) * (b - sqrt(((c * a) * -4.0)));
	}
	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
    t_0 = (-2.0d0) * (c / (b + b))
    if (b <= (-3.5d-76)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = (c / b) - (b / a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = ((-0.5d0) / a) * (b - sqrt(((c * a) * (-4.0d0))))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -2.0 * (c / (b + b));
	double tmp_1;
	if (b <= -3.5e-76) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (-0.5 / a) * (b - Math.sqrt(((c * a) * -4.0)));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -2.0 * (c / (b + b))
	tmp_1 = 0
	if b <= -3.5e-76:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = (c / b) - (b / a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = (-0.5 / a) * (b - math.sqrt(((c * a) * -4.0)))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(-2.0 * Float64(c / Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -3.5e-76)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(Float64(c * a) * -4.0))));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = -2.0 * (c / (b + b));
	tmp_2 = 0.0;
	if (b <= -3.5e-76)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = (c / b) - (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = (-0.5 / a) * (b - sqrt(((c * a) * -4.0)));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-2.0 * N[(c / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.5e-76], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.49999999999999997e-76

   1. Initial program 71.3%

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

   3. Simplified 71.1%

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

   4. Taylor expanded in b around inf 71.1%

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

   5. Taylor expanded in b around -inf 86.3%

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

      1. mul-1-neg 86.3%

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

      2. unsub-neg 86.3%

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

   7. Simplified 86.3%

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

   if -3.49999999999999997e-76 < b

   1. Initial program 73.0%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in b around inf 69.8%

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

   5. Taylor expanded in b around 0 66.8%

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

      1. *-commutative 66.8%

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

   7. Simplified 66.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-76}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right)\\ \end{array} \]

Alternative 5: 67.3% 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 72.4%

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

3. Simplified 72.3%

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

4. Taylor expanded in b around inf 70.2%

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

5. Taylor expanded in b around -inf 65.2%

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

   1. *-commutative 65.2%

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

7. Simplified 65.2%

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

8. Final simplification 65.2%

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

Alternative 6: 67.6% accurate, 13.0× speedup

Math

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

FPCore

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

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -2.0 * (c / (b + 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(c / Float64(b + 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 * (c / (b + 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[(c / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.4%

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

3. Simplified 72.3%

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

4. Taylor expanded in b around inf 70.2%

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

5. Taylor expanded in b around -inf 65.3%

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

   1. mul-1-neg 65.3%

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

   2. unsub-neg 65.3%

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

7. Simplified 65.3%

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

8. Final simplification 65.3%

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

Alternative 7: 3.0% accurate, 19.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		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 >= 0.0d0) 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 >= 0.0) {
		tmp = b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.4%

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

3. Simplified 72.3%

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

4. Taylor expanded in b around inf 69.1%

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

   1. +-commutative 69.1%

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

   2. fma-def 69.1%

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

   3. associate-/l* 70.5%

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

   4. associate-/r/ 70.5%

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

6. Simplified 70.5%

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

7. Taylor expanded in b around inf 35.9%

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

   1. mul-1-neg 35.9%

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

   2. distribute-neg-frac 35.9%

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

9. Simplified 35.9%

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

10. Taylor expanded in c around inf 3.1%

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

11. Taylor expanded in b around 0 3.1%

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

12. Final simplification 3.1%

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

Reproduce

herbie shell --seed 2023252 
(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))))