quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.9% → 85.8%

Time: 12.0s

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: 51.9% 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: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 10^{+139}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b\_2} - \frac{b\_2}{a}\right) - \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.8e-65)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1e+139)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (- (- (* 0.5 (/ c b_2)) (/ b_2 a)) (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.8e-65) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1e+139) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (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 <= (-5.8d-65)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1d+139) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((0.5d0 * (c / b_2)) - (b_2 / a)) - (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 <= -5.8e-65) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1e+139) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.8e-65:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1e+139:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.8e-65)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1e+139)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(c / b_2)) - Float64(b_2 / a)) - Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.8e-65)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1e+139)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.8e-65], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1e+139], 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[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.7999999999999996e-65

   1. Initial program 16.0%

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

   3. Taylor expanded in b_2 around -inf 90.1%

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

      1. associate-*r/ 90.1%

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

   5. Simplified 90.1%

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

   if -5.7999999999999996e-65 < b_2 < 1.00000000000000003e139

   1. Initial program 83.8%

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

   if 1.00000000000000003e139 < b_2

   1. Initial program 52.0%

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

      1. div-sub 52.0%

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

      2. neg-mul-1 52.0%

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

      3. associate-/l* 52.0%

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

      4. add-sqr-sqrt 51.9%

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

      5. sqrt-prod 52.0%

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

      6. sqr-neg 52.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 2.2%

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

      9. fma-neg 2.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{-b\_2}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      11. sqrt-unprod 52.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      12. sqr-neg 52.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      13. sqrt-prod 51.9%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      14. add-sqr-sqrt 52.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{b\_2}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

   4. Applied egg-rr 40.8%

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

      1. fma-undefine 40.8%

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

      2. unsub-neg 40.8%

         \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}{a}} \]

      3. mul-1-neg 40.8%

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

      4. distribute-frac-neg2 40.8%

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

   6. Simplified 40.8%

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

   7. Taylor expanded in c around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. neg-mul-1 0.0%

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

      3. sub-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. distribute-lft-neg-in 0.0%

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

      7. distribute-rgt-neg-in 0.0%

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

      8. *-commutative 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 95.8%

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

      11. mul-1-neg 95.8%

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

      12. remove-double-neg 95.8%

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

      13. associate-*r/ 95.8%

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

   9. Simplified 95.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 10^{+139}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b\_2} - \frac{b\_2}{a}\right) - \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-65}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b\_2} - \frac{b\_2}{a}\right) - \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3e-65)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 9.2e-57)
     (/ (- (- b_2) (sqrt (* a (- c)))) a)
     (- (- (* 0.5 (/ c b_2)) (/ b_2 a)) (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-65) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 9.2e-57) {
		tmp = (-b_2 - sqrt((a * -c))) / a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (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-65)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 9.2d-57) then
        tmp = (-b_2 - sqrt((a * -c))) / a
    else
        tmp = ((0.5d0 * (c / b_2)) - (b_2 / a)) - (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-65) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 9.2e-57) {
		tmp = (-b_2 - Math.sqrt((a * -c))) / a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3e-65:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 9.2e-57:
		tmp = (-b_2 - math.sqrt((a * -c))) / a
	else:
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3e-65)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 9.2e-57)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(c / b_2)) - Float64(b_2 / a)) - Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3e-65)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 9.2e-57)
		tmp = (-b_2 - sqrt((a * -c))) / a;
	else
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3e-65], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 9.2e-57], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.99999999999999998e-65

   1. Initial program 16.0%

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

   3. Taylor expanded in b_2 around -inf 90.1%

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

      1. associate-*r/ 90.1%

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

   5. Simplified 90.1%

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

   if -2.99999999999999998e-65 < b_2 < 9.2000000000000001e-57

   1. Initial program 79.4%

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

   3. Taylor expanded in b_2 around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. distribute-rgt-neg-out 72.3%

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

   5. Simplified 72.3%

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

   if 9.2000000000000001e-57 < b_2

   1. Initial program 69.6%

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

      1. div-sub 69.6%

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

      2. neg-mul-1 69.6%

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

      3. associate-/l* 69.6%

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

      4. add-sqr-sqrt 69.5%

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

      5. sqrt-prod 69.6%

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

      6. sqr-neg 69.6%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 5.4%

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

      9. fma-neg 5.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{-b\_2}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      11. sqrt-unprod 69.6%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      12. sqr-neg 69.6%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      13. sqrt-prod 69.5%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      14. add-sqr-sqrt 69.6%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{b\_2}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

   4. Applied egg-rr 45.6%

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

      1. fma-undefine 45.6%

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

      2. unsub-neg 45.6%

         \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}{a}} \]

      3. mul-1-neg 45.6%

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

      4. distribute-frac-neg2 45.6%

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

   6. Simplified 45.6%

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

   7. Taylor expanded in c around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. neg-mul-1 0.0%

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

      3. sub-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. distribute-lft-neg-in 0.0%

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

      7. distribute-rgt-neg-in 0.0%

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

      8. *-commutative 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 89.8%

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

      11. mul-1-neg 89.8%

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

      12. remove-double-neg 89.8%

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

      13. associate-*r/ 89.8%

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

   9. Simplified 89.8%

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

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.15 \cdot 10^{-64}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.05 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b\_2} - \frac{b\_2}{a}\right) - \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.15e-64)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.05e-57)
     (/ (sqrt (* a (- c))) (- a))
     (- (- (* 0.5 (/ c b_2)) (/ b_2 a)) (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.15e-64) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.05e-57) {
		tmp = sqrt((a * -c)) / -a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (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 <= (-2.15d-64)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.05d-57) then
        tmp = sqrt((a * -c)) / -a
    else
        tmp = ((0.5d0 * (c / b_2)) - (b_2 / a)) - (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 <= -2.15e-64) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.05e-57) {
		tmp = Math.sqrt((a * -c)) / -a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.15e-64:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.05e-57:
		tmp = math.sqrt((a * -c)) / -a
	else:
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.15e-64)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.05e-57)
		tmp = Float64(sqrt(Float64(a * Float64(-c))) / Float64(-a));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(c / b_2)) - Float64(b_2 / a)) - Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.15e-64)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.05e-57)
		tmp = sqrt((a * -c)) / -a;
	else
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.15e-64], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.05e-57], N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.14999999999999987e-64

   1. Initial program 16.0%

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

   3. Taylor expanded in b_2 around -inf 90.1%

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

      1. associate-*r/ 90.1%

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

   5. Simplified 90.1%

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

   if -2.14999999999999987e-64 < b_2 < 2.0500000000000001e-57

   1. Initial program 79.4%

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

      1. prod-diff 79.0%

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

      2. *-commutative 79.0%

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

      3. fma-neg 79.0%

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

      4. prod-diff 79.0%

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

      5. *-commutative 79.0%

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

      6. fma-neg 79.0%

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

      7. associate-+l+ 79.1%

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

      8. pow2 79.1%

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

      9. *-commutative 79.1%

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

      10. fma-undefine 79.0%

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

      11. distribute-lft-neg-in 79.0%

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

      12. *-commutative 79.0%

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

      13. distribute-rgt-neg-in 79.0%

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

      14. fma-define 79.1%

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

      15. *-commutative 79.1%

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

      16. fma-undefine 79.0%

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

      17. distribute-lft-neg-in 79.0%

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

      18. *-commutative 79.0%

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

      19. distribute-rgt-neg-in 79.0%

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

   4. Applied egg-rr 79.1%

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

      1. count-2 79.1%

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

   6. Simplified 79.1%

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

   7. Taylor expanded in b_2 around 0 70.6%

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

      1. mul-1-neg 70.6%

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

      2. *-commutative 70.6%

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

      3. distribute-lft1-in 70.6%

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

      4. metadata-eval 70.6%

         \[\leadsto -\sqrt{2 \cdot \left(\color{blue}{0} \cdot \left(a \cdot c\right)\right) - a \cdot c} \cdot \frac{1}{a} \]

      5. mul0-lft 71.0%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{0} - a \cdot c} \cdot \frac{1}{a} \]

      6. metadata-eval 71.0%

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

      7. neg-sub0 71.0%

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

      8. *-commutative 71.0%

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

      9. distribute-rgt-neg-in 71.0%

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

      10. sub0-neg 71.0%

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

      11. metadata-eval 71.0%

          \[\leadsto -\sqrt{c \cdot \left(\color{blue}{2 \cdot 0} - a\right)} \cdot \frac{1}{a} \]

      12. mul0-lft 71.0%

          \[\leadsto -\sqrt{c \cdot \left(2 \cdot \color{blue}{\left(0 \cdot a\right)} - a\right)} \cdot \frac{1}{a} \]

      13. metadata-eval 71.0%

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

      14. distribute-rgt1-in 71.0%

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

   9. Simplified 71.0%

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

   if 2.0500000000000001e-57 < b_2

   1. Initial program 69.6%

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

      1. div-sub 69.6%

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

      2. neg-mul-1 69.6%

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

      3. associate-/l* 69.6%

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

      4. add-sqr-sqrt 69.5%

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

      5. sqrt-prod 69.6%

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

      6. sqr-neg 69.6%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 5.4%

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

      9. fma-neg 5.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{-b\_2}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      11. sqrt-unprod 69.6%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      12. sqr-neg 69.6%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      13. sqrt-prod 69.5%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      14. add-sqr-sqrt 69.6%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{b\_2}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

   4. Applied egg-rr 45.6%

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

      1. fma-undefine 45.6%

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

      2. unsub-neg 45.6%

         \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}{a}} \]

      3. mul-1-neg 45.6%

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

      4. distribute-frac-neg2 45.6%

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

   6. Simplified 45.6%

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

   7. Taylor expanded in c around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. neg-mul-1 0.0%

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

      3. sub-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. distribute-lft-neg-in 0.0%

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

      7. distribute-rgt-neg-in 0.0%

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

      8. *-commutative 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 89.8%

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

      11. mul-1-neg 89.8%

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

      12. remove-double-neg 89.8%

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

      13. associate-*r/ 89.8%

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

   9. Simplified 89.8%

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

Alternative 4: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{-185}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b\_2} - \frac{b\_2}{a}\right) - \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.2e-110)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5e-185)
     (- (sqrt (/ (- c) a)))
     (- (- (* 0.5 (/ c b_2)) (/ b_2 a)) (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-110) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e-185) {
		tmp = -sqrt((-c / a));
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (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 <= (-4.2d-110)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 5d-185) then
        tmp = -sqrt((-c / a))
    else
        tmp = ((0.5d0 * (c / b_2)) - (b_2 / a)) - (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 <= -4.2e-110) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e-185) {
		tmp = -Math.sqrt((-c / a));
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.2e-110:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 5e-185:
		tmp = -math.sqrt((-c / a))
	else:
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.2e-110)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5e-185)
		tmp = Float64(-sqrt(Float64(Float64(-c) / a)));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(c / b_2)) - Float64(b_2 / a)) - Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.2e-110)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 5e-185)
		tmp = -sqrt((-c / a));
	else
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.2e-110], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5e-185], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), N[(N[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.20000000000000004e-110

   1. Initial program 20.0%

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

   3. Taylor expanded in b_2 around -inf 86.2%

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

      1. associate-*r/ 86.2%

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

   5. Simplified 86.2%

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

   if -4.20000000000000004e-110 < b_2 < 5.0000000000000003e-185

   1. Initial program 77.3%

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

      1. prod-diff 76.8%

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

      2. *-commutative 76.8%

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

      3. fma-neg 76.8%

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

      4. prod-diff 76.8%

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

      5. *-commutative 76.8%

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

      6. fma-neg 76.8%

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

      7. associate-+l+ 77.0%

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

      8. pow2 77.0%

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

      9. *-commutative 77.0%

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

      10. fma-undefine 76.8%

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

      11. distribute-lft-neg-in 76.8%

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

      12. *-commutative 76.8%

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

      13. distribute-rgt-neg-in 76.8%

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

      14. fma-define 77.0%

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

      15. *-commutative 77.0%

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

      16. fma-undefine 76.8%

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

      17. distribute-lft-neg-in 76.8%

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

      18. *-commutative 76.8%

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

      19. distribute-rgt-neg-in 76.8%

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

   4. Applied egg-rr 77.0%

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

      1. count-2 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in a around inf 32.8%

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

      1. mul-1-neg 32.8%

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

      2. distribute-rgt1-in 32.8%

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

      3. metadata-eval 32.8%

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

      4. mul0-lft 32.8%

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

      5. metadata-eval 32.8%

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

      6. neg-sub0 32.8%

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

   9. Simplified 32.8%

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

   if 5.0000000000000003e-185 < b_2

   1. Initial program 72.8%

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

      1. div-sub 72.8%

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

      2. neg-mul-1 72.8%

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

      3. associate-/l* 72.8%

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

      4. add-sqr-sqrt 72.7%

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

      5. sqrt-prod 72.7%

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

      6. sqr-neg 72.7%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 17.7%

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

      9. fma-neg 17.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{-b\_2}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      11. sqrt-unprod 72.7%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      12. sqr-neg 72.7%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      13. sqrt-prod 72.7%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      14. add-sqr-sqrt 72.8%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{b\_2}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

   4. Applied egg-rr 54.1%

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

      1. fma-undefine 54.1%

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

      2. unsub-neg 54.1%

         \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}{a}} \]

      3. mul-1-neg 54.1%

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

      4. distribute-frac-neg2 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in c around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. neg-mul-1 0.0%

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

      3. sub-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. distribute-lft-neg-in 0.0%

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

      7. distribute-rgt-neg-in 0.0%

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

      8. *-commutative 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 76.9%

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

      11. mul-1-neg 76.9%

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

      12. remove-double-neg 76.9%

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

      13. associate-*r/ 76.9%

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

   9. Simplified 76.9%

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

Alternative 5: 67.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b\_2} - \frac{b\_2}{a}\right) - \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309)
   (/ (* -0.5 c) b_2)
   (- (- (* 0.5 (/ c b_2)) (/ b_2 a)) (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (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 <= (-1d-309)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((0.5d0 * (c / b_2)) - (b_2 / a)) - (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 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(c / b_2)) - Float64(b_2 / a)) - Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 31.8%

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

   3. Taylor expanded in b_2 around -inf 71.0%

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

      1. associate-*r/ 71.0%

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

   5. Simplified 71.0%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 72.9%

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

      1. div-sub 72.9%

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

      2. neg-mul-1 72.9%

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

      3. associate-/l* 72.9%

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

      4. add-sqr-sqrt 72.8%

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

      5. sqrt-prod 72.8%

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

      6. sqr-neg 72.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 23.9%

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

      9. fma-neg 23.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{-b\_2}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      11. sqrt-unprod 72.8%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      12. sqr-neg 72.8%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      13. sqrt-prod 72.8%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      14. add-sqr-sqrt 72.9%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{b\_2}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

   4. Applied egg-rr 56.3%

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

      1. fma-undefine 56.3%

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

      2. unsub-neg 56.3%

         \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}{a}} \]

      3. mul-1-neg 56.3%

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

      4. distribute-frac-neg2 56.3%

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

   6. Simplified 56.3%

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

   7. Taylor expanded in c around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. neg-mul-1 0.0%

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

      3. sub-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. metadata-eval 0.0%

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

      6. distribute-lft-neg-in 0.0%

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

      7. distribute-rgt-neg-in 0.0%

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

      8. *-commutative 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 68.9%

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

      11. mul-1-neg 68.9%

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

      12. remove-double-neg 68.9%

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

      13. associate-*r/ 68.9%

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

   9. Simplified 68.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{c}{b\_2} - \frac{b\_2}{a}\right) - \frac{b\_2}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2} + \frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309)
   (/ (* -0.5 c) b_2)
   (+ (* 0.5 (/ c b_2)) (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (0.5 * (c / b_2)) + ((b_2 / a) * -2.0);
	}
	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 <= (-1d-309)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (0.5d0 * (c / b_2)) + ((b_2 / a) * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (0.5 * (c / b_2)) + ((b_2 / a) * -2.0);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (0.5 * (c / b_2)) + ((b_2 / a) * -2.0)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(0.5 * Float64(c / b_2)) + Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (0.5 * (c / b_2)) + ((b_2 / a) * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 31.8%

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

   3. Taylor expanded in b_2 around -inf 71.0%

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

      1. associate-*r/ 71.0%

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

   5. Simplified 71.0%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 72.9%

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

   3. Taylor expanded in c around 0 68.9%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2} + \frac{b\_2}{a} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309) (/ (* -0.5 c) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / 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 <= (-1d-309)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 31.8%

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

   3. Taylor expanded in b_2 around -inf 71.0%

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

      1. associate-*r/ 71.0%

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

   5. Simplified 71.0%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 72.9%

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

   3. Taylor expanded in b_2 around inf 68.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 68.4%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   5. Simplified 68.4%

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

Alternative 8: 67.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309) (/ (* -0.5 c) b_2) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = b_2 * (-2.0 / 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 <= (-1d-309)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 31.8%

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

   3. Taylor expanded in b_2 around -inf 71.0%

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

      1. associate-*r/ 71.0%

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

   5. Simplified 71.0%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 72.9%

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

   3. Taylor expanded in b_2 around inf 67.6%

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

      1. associate-*r/ 67.6%

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

      2. metadata-eval 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in c around 0 68.1%

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

Alternative 9: 67.4% accurate, 11.2× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3e-309) (* c (/ -0.5 b_2)) (* b_2 (/ -2.0 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 = b_2 * (-2.0 / 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 = b_2 * ((-2.0d0) / 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 = b_2 * (-2.0 / 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 = b_2 * (-2.0 / 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(b_2 * Float64(-2.0 / 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 = b_2 * (-2.0 / 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[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.000000000000001e-309

   1. Initial program 31.8%

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

   3. Taylor expanded in b_2 around -inf 71.0%

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

      1. associate-*r/ 71.0%

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

   5. Simplified 71.0%

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

   6. Taylor expanded in c around 0 71.0%

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

      1. associate-*r/ 71.0%

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

      2. *-commutative 71.0%

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

      3. *-rgt-identity 71.0%

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

      4. associate-*r/ 70.7%

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

      5. associate-*l* 70.7%

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

      6. metadata-eval 70.7%

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

      7. distribute-lft-neg-in 70.7%

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

      8. associate-*r/ 70.7%

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

      9. metadata-eval 70.7%

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

      10. distribute-neg-frac 70.7%

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

      11. metadata-eval 70.7%

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

   8. Simplified 70.7%

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

   if -3.000000000000001e-309 < b_2

   1. Initial program 72.9%

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

   3. Taylor expanded in b_2 around inf 67.6%

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

      1. associate-*r/ 67.6%

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

      2. metadata-eval 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in c around 0 68.1%

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

Alternative 10: 43.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -68000000000:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -68000000000.0) (* 0.5 (/ c b_2)) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -68000000000.0) {
		tmp = 0.5 * (c / b_2);
	} else {
		tmp = b_2 * (-2.0 / 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 <= (-68000000000.0d0)) then
        tmp = 0.5d0 * (c / b_2)
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -68000000000.0) {
		tmp = 0.5 * (c / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -68000000000.0:
		tmp = 0.5 * (c / b_2)
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -68000000000.0)
		tmp = Float64(0.5 * Float64(c / b_2));
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -68000000000.0)
		tmp = 0.5 * (c / b_2);
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -68000000000.0], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -6.8e10

   1. Initial program 13.8%

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

   3. Taylor expanded in b_2 around inf 2.4%

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

      1. associate-*r/ 2.4%

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

      2. metadata-eval 2.4%

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

   5. Simplified 2.4%

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

   6. Taylor expanded in b_2 around 0 29.5%

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

   if -6.8e10 < b_2

   1. Initial program 71.9%

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

   3. Taylor expanded in b_2 around inf 52.5%

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

      1. associate-*r/ 52.5%

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

      2. metadata-eval 52.5%

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

   5. Simplified 52.5%

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

   6. Taylor expanded in c around 0 53.2%

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

Alternative 11: 10.8% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* 0.5 (/ c b_2)))

C

double code(double a, double b_2, double c) {
	return 0.5 * (c / b_2);
}

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 = 0.5d0 * (c / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.5 * (c / b_2);
}

Python

def code(a, b_2, c):
	return 0.5 * (c / b_2)

Julia

function code(a, b_2, c)
	return Float64(0.5 * Float64(c / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.5 * (c / b_2);
end

Wolfram

code[a_, b$95$2_, c_] := N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in b_2 around inf 36.7%

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

   1. associate-*r/ 36.7%

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

   2. metadata-eval 36.7%

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

5. Simplified 36.7%

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

6. Taylor expanded in b_2 around 0 11.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b\_2}} \]
7. Add Preprocessing

Developer target: 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 2024110 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b_2 0.0) (/ c (- (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))) b_2)) (/ (+ b_2 (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c)))))) (- a)))

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