quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.8% → 85.1%

Time: 11.8s

Alternatives: 11

Speedup: 11.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.92 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(a \cdot \frac{c}{b\_2}\right) \cdot \left(--0.5\right) - b\_2\right) - b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.8e-12)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3.92e+151)
     (/ (- (- b_2) (sqrt (fma b_2 b_2 (* c (- a))))) a)
     (/ (- (- (* (* a (/ c b_2)) (- -0.5)) b_2) b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e-12) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.92e+151) {
		tmp = (-b_2 - sqrt(fma(b_2, b_2, (c * -a)))) / a;
	} else {
		tmp = ((((a * (c / b_2)) * -(-0.5)) - b_2) - b_2) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.8e-12)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3.92e+151)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(c * Float64(-a))))) / a);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * Float64(c / b_2)) * Float64(-(-0.5))) - b_2) - b_2) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.8e-12], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.92e+151], N[(N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] * (--0.5)), $MachinePrecision] - b$95$2), $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.79999999999999996e-12

   1. Initial program 13.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 85.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   5. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   if -3.79999999999999996e-12 < b_2 < 3.92000000000000006e151

   1. Initial program 80.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. fma-neg 80.5%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}}}{a} \]

      2. distribute-rgt-neg-in 80.5%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{a \cdot \left(-c\right)}\right)}}{a} \]

   4. Applied egg-rr 80.5%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, a \cdot \left(-c\right)\right)}}}{a} \]

   if 3.92000000000000006e151 < b_2

   1. Initial program 39.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 94.6%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(b\_2 + -0.5 \cdot \frac{a \cdot c}{b\_2}\right)}}{a} \]
   4. Step-by-step derivation

      1. associate-/l* 97.3%

         \[\leadsto \frac{\left(-b\_2\right) - \left(b\_2 + -0.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b\_2}\right)}\right)}{a} \]

   5. Simplified 97.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(b\_2 + -0.5 \cdot \left(a \cdot \frac{c}{b\_2}\right)\right)}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.92 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(a \cdot \frac{c}{b\_2}\right) \cdot \left(--0.5\right) - b\_2\right) - b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.92 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(a \cdot \frac{c}{b\_2}\right) \cdot \left(--0.5\right) - b\_2\right) - b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.2e-15)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3.92e+151)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (- (- (* (* a (/ c b_2)) (- -0.5)) b_2) b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.2e-15) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.92e+151) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((((a * (c / b_2)) * -(-0.5)) - b_2) - b_2) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5.2d-15)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 3.92d+151) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((((a * (c / b_2)) * -(-0.5d0)) - b_2) - b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.2e-15) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.92e+151) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((((a * (c / b_2)) * -(-0.5)) - b_2) - b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.2e-15:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 3.92e+151:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = ((((a * (c / b_2)) * -(-0.5)) - b_2) - b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.2e-15)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3.92e+151)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * Float64(c / b_2)) * Float64(-(-0.5))) - b_2) - b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.2e-15)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 3.92e+151)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = ((((a * (c / b_2)) * -(-0.5)) - b_2) - b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.2e-15], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.92e+151], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] * (--0.5)), $MachinePrecision] - b$95$2), $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.20000000000000009e-15

   1. Initial program 13.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 85.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   5. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   if -5.20000000000000009e-15 < b_2 < 3.92000000000000006e151

   1. Initial program 80.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 3.92000000000000006e151 < b_2

   1. Initial program 39.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 94.6%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(b\_2 + -0.5 \cdot \frac{a \cdot c}{b\_2}\right)}}{a} \]
   4. Step-by-step derivation

      1. associate-/l* 97.3%

         \[\leadsto \frac{\left(-b\_2\right) - \left(b\_2 + -0.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b\_2}\right)}\right)}{a} \]

   5. Simplified 97.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(b\_2 + -0.5 \cdot \left(a \cdot \frac{c}{b\_2}\right)\right)}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.92 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(a \cdot \frac{c}{b\_2}\right) \cdot \left(--0.5\right) - b\_2\right) - b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.65 \cdot 10^{-15}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.65e-15)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5.2e-91)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.65e-15) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.2e-91) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.65d-15)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 5.2d-91) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.65e-15) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.2e-91) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.65e-15:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 5.2e-91:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.65e-15)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5.2e-91)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.65e-15)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 5.2e-91)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.65e-15], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5.2e-91], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.6500000000000001e-15

   1. Initial program 13.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 85.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   5. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   if -2.6500000000000001e-15 < b_2 < 5.20000000000000028e-91

   1. Initial program 75.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0 72.2%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. mul-1-neg 72.2%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-a \cdot c}}}{a} \]

      2. distribute-rgt-neg-out 72.2%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{a \cdot \left(-c\right)}}}{a} \]

   5. Simplified 72.2%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{a \cdot \left(-c\right)}}}{a} \]

   if 5.20000000000000028e-91 < b_2

   1. Initial program 68.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 85.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.65 \cdot 10^{-15}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-15)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.5e-119)
     (/ (sqrt (* c (- a))) (- a))
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-15) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.5e-119) {
		tmp = sqrt((c * -a)) / -a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.8d-15)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.5d-119) then
        tmp = sqrt((c * -a)) / -a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-15) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.5e-119) {
		tmp = Math.sqrt((c * -a)) / -a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-15:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.5e-119:
		tmp = math.sqrt((c * -a)) / -a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-15)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.5e-119)
		tmp = Float64(sqrt(Float64(c * Float64(-a))) / Float64(-a));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-15)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.5e-119)
		tmp = sqrt((c * -a)) / -a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-15], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.5e-119], N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.80000000000000014e-15

   1. Initial program 13.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 85.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   5. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   if -2.80000000000000014e-15 < b_2 < 1.5000000000000001e-119

   1. Initial program 76.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. prod-diff 75.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}}{a} \]

      2. *-commutative 75.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      3. fma-neg 75.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      4. prod-diff 75.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\mathsf{fma}\left(b\_2, b\_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      5. *-commutative 75.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{fma}\left(b\_2, b\_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      6. fma-neg 75.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      7. associate-+l+ 75.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}}{a} \]

      8. pow2 75.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      9. *-commutative 75.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      10. fma-undefine 75.7%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      11. distribute-lft-neg-in 75.7%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      12. *-commutative 75.7%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      13. distribute-rgt-neg-in 75.7%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      14. fma-define 75.7%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      15. *-commutative 75.7%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)}}{a} \]

      16. fma-undefine 75.7%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)}\right)}}{a} \]

      17. distribute-lft-neg-in 75.7%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)\right)}}{a} \]

      18. *-commutative 75.7%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)\right)}}{a} \]

      19. distribute-rgt-neg-in 75.7%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)\right)}}{a} \]

   4. Applied egg-rr 75.7%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   5. Step-by-step derivation

      1. count-2 75.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

   6. Simplified 75.7%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} - a \cdot c\right) + 2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

   7. Taylor expanded in b_2 around 0 72.1%

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

      1. mul-1-neg 72.1%

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

      2. distribute-lft1-in 72.1%

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

      3. metadata-eval 72.1%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\color{blue}{0} \cdot \left(a \cdot c\right)\right) - a \cdot c}}{a} \]

      4. mul0-lft 72.5%

         \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{0} - a \cdot c}}{a} \]

      5. metadata-eval 72.5%

         \[\leadsto \frac{-\sqrt{\color{blue}{0} - a \cdot c}}{a} \]

      6. neg-sub0 72.5%

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

      7. distribute-rgt-neg-in 72.5%

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

   9. Simplified 72.5%

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

   if 1.5000000000000001e-119 < b_2

   1. Initial program 68.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 83.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.25 \cdot 10^{-157}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{-138}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.25e-157)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 7e-138)
     (- (sqrt (/ c (- a))))
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.25e-157) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7e-138) {
		tmp = -sqrt((c / -a));
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.25d-157)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 7d-138) then
        tmp = -sqrt((c / -a))
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.25e-157) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7e-138) {
		tmp = -Math.sqrt((c / -a));
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.25e-157:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 7e-138:
		tmp = -math.sqrt((c / -a))
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.25e-157)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 7e-138)
		tmp = Float64(-sqrt(Float64(c / Float64(-a))));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.25e-157)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 7e-138)
		tmp = -sqrt((c / -a));
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.25e-157], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7e-138], (-N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision]), N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.25000000000000005e-157

   1. Initial program 26.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 72.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 72.3%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   5. Simplified 72.3%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   if -1.25000000000000005e-157 < b_2 < 6.9999999999999997e-138

   1. Initial program 78.7%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. prod-diff 78.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}}{a} \]

      2. *-commutative 78.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      3. fma-neg 78.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      4. prod-diff 78.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\mathsf{fma}\left(b\_2, b\_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      5. *-commutative 78.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{fma}\left(b\_2, b\_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      6. fma-neg 78.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      7. associate-+l+ 78.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}}{a} \]

      8. pow2 78.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      9. *-commutative 78.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      10. fma-undefine 78.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      11. distribute-lft-neg-in 78.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      12. *-commutative 78.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      13. distribute-rgt-neg-in 78.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      14. fma-define 78.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      15. *-commutative 78.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)}}{a} \]

      16. fma-undefine 78.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)}\right)}}{a} \]

      17. distribute-lft-neg-in 78.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)\right)}}{a} \]

      18. *-commutative 78.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)\right)}}{a} \]

      19. distribute-rgt-neg-in 78.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)\right)}}{a} \]

   4. Applied egg-rr 78.3%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   5. Step-by-step derivation

      1. count-2 78.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - a \cdot c\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

   6. Simplified 78.3%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} - a \cdot c\right) + 2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

   7. Taylor expanded in a around inf 42.1%

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

      1. mul-1-neg 42.1%

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

      2. distribute-rgt1-in 42.1%

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

      3. metadata-eval 42.1%

         \[\leadsto -\sqrt{\frac{2 \cdot \left(\color{blue}{0} \cdot c\right) - c}{a}} \]

      4. mul0-lft 42.1%

         \[\leadsto -\sqrt{\frac{2 \cdot \color{blue}{0} - c}{a}} \]

      5. metadata-eval 42.1%

         \[\leadsto -\sqrt{\frac{\color{blue}{0} - c}{a}} \]

      6. neg-sub0 42.1%

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

   9. Simplified 42.1%

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

   if 6.9999999999999997e-138 < b_2

   1. Initial program 68.7%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 82.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.25 \cdot 10^{-157}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{-138}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (/ (* -0.5 c) b_2)
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 36.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 60.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 60.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   5. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 71.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 66.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (* -0.5 c) b_2) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 36.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 60.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 60.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   5. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 71.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 65.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-309}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3e-309) (* c (/ -0.5 b_2)) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-309) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3d-309)) then
        tmp = c * ((-0.5d0) / b_2)
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-309) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3e-309:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3e-309)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3e-309)
		tmp = c * (-0.5 / b_2);
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3e-309], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.000000000000001e-309

   1. Initial program 36.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. fma-neg 36.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}}}{a} \]

      2. distribute-rgt-neg-in 36.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{a \cdot \left(-c\right)}\right)}}{a} \]

   4. Applied egg-rr 36.3%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, a \cdot \left(-c\right)\right)}}}{a} \]

   5. Taylor expanded in b_2 around -inf 60.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. metadata-eval 60.2%

         \[\leadsto \color{blue}{\left(-0.5\right)} \cdot \frac{c}{b\_2} \]

      2. distribute-lft-neg-in 60.2%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

      3. associate-*r/ 60.2%

         \[\leadsto -\color{blue}{\frac{0.5 \cdot c}{b\_2}} \]

      4. *-commutative 60.2%

         \[\leadsto -\frac{\color{blue}{c \cdot 0.5}}{b\_2} \]

      5. associate-/l* 60.1%

         \[\leadsto -\color{blue}{c \cdot \frac{0.5}{b\_2}} \]

      6. metadata-eval 60.1%

         \[\leadsto -c \cdot \frac{\color{blue}{0.5 \cdot 1}}{b\_2} \]

      7. associate-*r/ 60.1%

         \[\leadsto -c \cdot \color{blue}{\left(0.5 \cdot \frac{1}{b\_2}\right)} \]

      8. distribute-rgt-neg-in 60.1%

         \[\leadsto \color{blue}{c \cdot \left(-0.5 \cdot \frac{1}{b\_2}\right)} \]

      9. associate-*r/ 60.1%

         \[\leadsto c \cdot \left(-\color{blue}{\frac{0.5 \cdot 1}{b\_2}}\right) \]

      10. metadata-eval 60.1%

          \[\leadsto c \cdot \left(-\frac{\color{blue}{0.5}}{b\_2}\right) \]

      11. distribute-neg-frac 60.1%

          \[\leadsto c \cdot \color{blue}{\frac{-0.5}{b\_2}} \]

      12. metadata-eval 60.1%

          \[\leadsto c \cdot \frac{\color{blue}{-0.5}}{b\_2} \]

   7. Simplified 60.1%

      \[\leadsto \color{blue}{c \cdot \frac{-0.5}{b\_2}} \]

   if -3.000000000000001e-309 < b_2

   1. Initial program 71.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 65.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 43.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) 0.0 (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = 0.0d0
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = 0.0
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = 0.0;
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = 0.0;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], 0.0, N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 36.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 14.2%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{-1 \cdot \left(b\_2 \cdot \left(1 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* 14.2%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(1 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}}{a} \]

      2. mul-1-neg 14.2%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(-b\_2\right)} \cdot \left(1 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}{a} \]

      3. associate-/l* 14.5%

         \[\leadsto \frac{\left(-b\_2\right) - \left(-b\_2\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(a \cdot \frac{c}{{b\_2}^{2}}\right)}\right)}{a} \]

   5. Simplified 14.5%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(-b\_2\right) \cdot \left(1 + -0.5 \cdot \left(a \cdot \frac{c}{{b\_2}^{2}}\right)\right)}}{a} \]

   6. Taylor expanded in a around 0 13.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 + -1 \cdot b\_2\right)}}{a} \]
   7. Step-by-step derivation

      1. distribute-rgt1-in 13.2%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\_2\right)}}{a} \]

      2. metadata-eval 13.2%

         \[\leadsto \frac{-1 \cdot \left(\color{blue}{0} \cdot b\_2\right)}{a} \]

      3. mul0-lft 13.2%

         \[\leadsto \frac{-1 \cdot \color{blue}{0}}{a} \]

      4. metadata-eval 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in a around 0 13.2%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 71.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 65.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 23.4% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (if (<= b_2 -5e-310) 0.0 (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = 0.0d0
    else
        tmp = b_2 / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = 0.0
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = 0.0;
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = 0.0;
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], 0.0, N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 36.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 14.2%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{-1 \cdot \left(b\_2 \cdot \left(1 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* 14.2%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(1 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}}{a} \]

      2. mul-1-neg 14.2%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(-b\_2\right)} \cdot \left(1 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}{a} \]

      3. associate-/l* 14.5%

         \[\leadsto \frac{\left(-b\_2\right) - \left(-b\_2\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(a \cdot \frac{c}{{b\_2}^{2}}\right)}\right)}{a} \]

   5. Simplified 14.5%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(-b\_2\right) \cdot \left(1 + -0.5 \cdot \left(a \cdot \frac{c}{{b\_2}^{2}}\right)\right)}}{a} \]

   6. Taylor expanded in a around 0 13.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 + -1 \cdot b\_2\right)}}{a} \]
   7. Step-by-step derivation

      1. distribute-rgt1-in 13.2%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\_2\right)}}{a} \]

      2. metadata-eval 13.2%

         \[\leadsto \frac{-1 \cdot \left(\color{blue}{0} \cdot b\_2\right)}{a} \]

      3. mul0-lft 13.2%

         \[\leadsto \frac{-1 \cdot \color{blue}{0}}{a} \]

      4. metadata-eval 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in a around 0 13.2%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 71.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. fma-neg 71.5%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}}}{a} \]

      2. distribute-rgt-neg-in 71.5%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{a \cdot \left(-c\right)}\right)}}{a} \]

   4. Applied egg-rr 71.5%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, a \cdot \left(-c\right)\right)}}}{a} \]

   5. Taylor expanded in b_2 around 0 36.7%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   6. Step-by-step derivation

      1. neg-mul-1 36.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-a \cdot c}}}{a} \]

      2. *-commutative 36.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{-\color{blue}{c \cdot a}}}{a} \]

      3. distribute-rgt-neg-in 36.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   7. Simplified 36.7%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   8. Taylor expanded in b_2 around inf 23.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} \]
   9. Step-by-step derivation

      1. mul-1-neg 23.8%

         \[\leadsto \color{blue}{-\frac{b\_2}{a}} \]

      2. distribute-frac-neg 23.8%

         \[\leadsto \color{blue}{\frac{-b\_2}{a}} \]

   10. Simplified 23.8%

       \[\leadsto \color{blue}{\frac{-b\_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 11.0% accurate, 112.0× speedup

Math

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

FPCore

(FPCore (a b_2 c) :precision binary64 0.0)

C

double code(double a, double b_2, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0;
}

Python

def code(a, b_2, c):
	return 0.0

Julia

function code(a, b_2, c)
	return 0.0
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0;
end

Wolfram

code[a_, b$95$2_, c_] := 0.0

Derivation

1. Initial program 53.3%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in b_2 around -inf 8.3%

   \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{-1 \cdot \left(b\_2 \cdot \left(1 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)}}{a} \]
4. Step-by-step derivation

   1. associate-*r* 8.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(1 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}}{a} \]

   2. mul-1-neg 8.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(-b\_2\right)} \cdot \left(1 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}{a} \]

   3. associate-/l* 8.5%

      \[\leadsto \frac{\left(-b\_2\right) - \left(-b\_2\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(a \cdot \frac{c}{{b\_2}^{2}}\right)}\right)}{a} \]

5. Simplified 8.5%

   \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(-b\_2\right) \cdot \left(1 + -0.5 \cdot \left(a \cdot \frac{c}{{b\_2}^{2}}\right)\right)}}{a} \]

6. Taylor expanded in a around 0 8.2%

   \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 + -1 \cdot b\_2\right)}}{a} \]
7. Step-by-step derivation

   1. distribute-rgt1-in 8.2%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\_2\right)}}{a} \]

   2. metadata-eval 8.2%

      \[\leadsto \frac{-1 \cdot \left(\color{blue}{0} \cdot b\_2\right)}{a} \]

   3. mul0-lft 8.2%

      \[\leadsto \frac{-1 \cdot \color{blue}{0}}{a} \]

   4. metadata-eval 8.2%

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

8. Simplified 8.2%

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

9. Taylor expanded in a around 0 8.2%

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

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

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

Reproduce

herbie shell --seed 2024118 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))