quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.5% → 85.9%

Time: 12.7s

Alternatives: 9

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: 51.5% 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: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+92}:\\ \;\;\;\;\frac{\left|b\_2\right| \cdot \sqrt{1 - a \cdot \frac{c}{{b\_2}^{2}}} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+92)
   (/ (- (* (fabs b_2) (sqrt (- 1.0 (* a (/ c (pow b_2 2.0)))))) b_2) a)
   (if (<= b_2 3e-88)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+92) {
		tmp = ((fabs(b_2) * sqrt((1.0 - (a * (c / pow(b_2, 2.0)))))) - b_2) / a;
	} else if (b_2 <= 3e-88) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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+92)) then
        tmp = ((abs(b_2) * sqrt((1.0d0 - (a * (c / (b_2 ** 2.0d0)))))) - b_2) / a
    else if (b_2 <= 3d-88) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+92) {
		tmp = ((Math.abs(b_2) * Math.sqrt((1.0 - (a * (c / Math.pow(b_2, 2.0)))))) - b_2) / a;
	} else if (b_2 <= 3e-88) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e+92:
		tmp = ((math.fabs(b_2) * math.sqrt((1.0 - (a * (c / math.pow(b_2, 2.0)))))) - b_2) / a
	elif b_2 <= 3e-88:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+92)
		tmp = Float64(Float64(Float64(abs(b_2) * sqrt(Float64(1.0 - Float64(a * Float64(c / (b_2 ^ 2.0)))))) - b_2) / a);
	elseif (b_2 <= 3e-88)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e+92)
		tmp = ((abs(b_2) * sqrt((1.0 - (a * (c / (b_2 ^ 2.0)))))) - b_2) / a;
	elseif (b_2 <= 3e-88)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+92], N[(N[(N[(N[Abs[b$95$2], $MachinePrecision] * N[Sqrt[N[(1.0 - N[(a * N[(c / N[Power[b$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3e-88], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.00000000000000022e92

   1. Initial program 49.7%

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

      1. +-commutative 49.7%

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

      2. unsub-neg 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in b_2 around inf 49.6%

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

      1. associate-*r/ 49.6%

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

      2. associate-*r* 49.6%

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

      3. neg-mul-1 49.6%

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

      4. *-commutative 49.6%

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

   7. Simplified 49.6%

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

      1. sqrt-prod 49.6%

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

      2. fmm-def 49.6%

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

      3. +-commutative 49.6%

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

      4. associate-/l* 54.1%

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

      5. fma-define 54.1%

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

   9. Applied egg-rr 54.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{{b\_2}^{2}}, \sqrt{\mathsf{fma}\left(c, \frac{-a}{{b\_2}^{2}}, 1\right)}, -b\_2\right)}}{a} \]
   10. Step-by-step derivation

       1. fmm-undef 54.1%

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

       2. unpow2 54.1%

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

       3. rem-sqrt-square 99.3%

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

       4. fma-undefine 99.3%

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

       5. associate-*r/ 85.2%

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

       6. *-commutative 85.2%

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

       7. neg-mul-1 85.2%

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

       8. associate-*r* 85.2%

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

       9. associate-*r/ 85.2%

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

       10. +-commutative 85.2%

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

       11. mul-1-neg 85.2%

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

       12. unsub-neg 85.2%

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

       13. associate-/l* 99.3%

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

   11. Simplified 99.3%

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

   if -5.00000000000000022e92 < b_2 < 2.9999999999999999e-88

   1. Initial program 82.9%

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

      1. +-commutative 82.9%

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

      2. unsub-neg 82.9%

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

   3. Simplified 82.9%

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

   if 2.9999999999999999e-88 < b_2

   1. Initial program 16.0%

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

      1. +-commutative 16.0%

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

      2. unsub-neg 16.0%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in b_2 around inf 84.6%

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

Alternative 2: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{b\_2 \cdot -2 + c \cdot \frac{0.5}{\frac{b\_2}{a}}}{a}\\ \mathbf{elif}\;b\_2 \leq 2.45 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.1e+154)
   (/ (+ (* b_2 -2.0) (* c (/ 0.5 (/ b_2 a)))) a)
   (if (<= b_2 2.45e-88)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.1e+154) {
		tmp = ((b_2 * -2.0) + (c * (0.5 / (b_2 / a)))) / a;
	} else if (b_2 <= 2.45e-88) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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 <= (-1.1d+154)) then
        tmp = ((b_2 * (-2.0d0)) + (c * (0.5d0 / (b_2 / a)))) / a
    else if (b_2 <= 2.45d-88) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.1e+154) {
		tmp = ((b_2 * -2.0) + (c * (0.5 / (b_2 / a)))) / a;
	} else if (b_2 <= 2.45e-88) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.1e+154:
		tmp = ((b_2 * -2.0) + (c * (0.5 / (b_2 / a)))) / a
	elif b_2 <= 2.45e-88:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.1e+154)
		tmp = Float64(Float64(Float64(b_2 * -2.0) + Float64(c * Float64(0.5 / Float64(b_2 / a)))) / a);
	elseif (b_2 <= 2.45e-88)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.1e+154)
		tmp = ((b_2 * -2.0) + (c * (0.5 / (b_2 / a)))) / a;
	elseif (b_2 <= 2.45e-88)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.1e+154], N[(N[(N[(b$95$2 * -2.0), $MachinePrecision] + N[(c * N[(0.5 / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.45e-88], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.1000000000000001e154

   1. Initial program 38.0%

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

      1. +-commutative 38.0%

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

      2. unsub-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in b_2 around -inf 98.8%

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

   6. Taylor expanded in a around 0 85.9%

      \[\leadsto \color{blue}{\frac{-2 \cdot b\_2 + 0.5 \cdot \frac{a \cdot c}{b\_2}}{a}} \]
   7. Step-by-step derivation

      1. clear-num 85.9%

         \[\leadsto \frac{-2 \cdot b\_2 + 0.5 \cdot \color{blue}{\frac{1}{\frac{b\_2}{a \cdot c}}}}{a} \]

      2. un-div-inv 85.9%

         \[\leadsto \frac{-2 \cdot b\_2 + \color{blue}{\frac{0.5}{\frac{b\_2}{a \cdot c}}}}{a} \]

      3. associate-/r* 100.0%

         \[\leadsto \frac{-2 \cdot b\_2 + \frac{0.5}{\color{blue}{\frac{\frac{b\_2}{a}}{c}}}}{a} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \frac{-2 \cdot b\_2 + \color{blue}{\frac{0.5}{\frac{\frac{b\_2}{a}}{c}}}}{a} \]
   9. Step-by-step derivation

      1. associate-/r/ 100.0%

         \[\leadsto \frac{-2 \cdot b\_2 + \color{blue}{\frac{0.5}{\frac{b\_2}{a}} \cdot c}}{a} \]

   10. Simplified 100.0%

       \[\leadsto \frac{-2 \cdot b\_2 + \color{blue}{\frac{0.5}{\frac{b\_2}{a}} \cdot c}}{a} \]

   if -1.1000000000000001e154 < b_2 < 2.45000000000000014e-88

   1. Initial program 83.0%

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

      1. +-commutative 83.0%

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

      2. unsub-neg 83.0%

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

   3. Simplified 83.0%

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

   if 2.45000000000000014e-88 < b_2

   1. Initial program 16.0%

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

      1. +-commutative 16.0%

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

      2. unsub-neg 16.0%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in b_2 around inf 84.6%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{b\_2 \cdot -2 + c \cdot \frac{0.5}{\frac{b\_2}{a}}}{a}\\ \mathbf{elif}\;b\_2 \leq 2.45 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.5 \cdot 10^{-71}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \mathbf{elif}\;b\_2 \leq 3.3 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.5e-71)
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))
   (if (<= b_2 3.3e-88) (/ (- (sqrt (* a (- c))) b_2) a) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.5e-71) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 3.3e-88) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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 <= (-6.5d-71)) then
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    else if (b_2 <= 3.3d-88) then
        tmp = (sqrt((a * -c)) - b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.5e-71) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 3.3e-88) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.5e-71:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	elif b_2 <= 3.3e-88:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.5e-71)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	elseif (b_2 <= 3.3e-88)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.5e-71)
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	elseif (b_2 <= 3.3e-88)
		tmp = (sqrt((a * -c)) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.5e-71], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.3e-88], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.50000000000000005e-71

   1. Initial program 69.8%

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

      1. +-commutative 69.8%

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

      2. unsub-neg 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in b_2 around -inf 81.0%

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

   6. Taylor expanded in c around 0 81.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]

   if -6.50000000000000005e-71 < b_2 < 3.29999999999999994e-88

   1. Initial program 77.3%

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

      1. +-commutative 77.3%

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

      2. unsub-neg 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in b_2 around 0 75.1%

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

      1. associate-*r* 75.1%

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

      2. neg-mul-1 75.1%

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

      3. *-commutative 75.1%

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

   7. Simplified 75.1%

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

   if 3.29999999999999994e-88 < b_2

   1. Initial program 16.0%

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

      1. +-commutative 16.0%

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

      2. unsub-neg 16.0%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in b_2 around inf 84.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.5 \cdot 10^{-71}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \mathbf{elif}\;b\_2 \leq 3.3 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6 \cdot 10^{-71}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \mathbf{elif}\;b\_2 \leq 3.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6e-71)
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))
   (if (<= b_2 3.2e-88) (/ (sqrt (* a (- c))) a) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e-71) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 3.2e-88) {
		tmp = sqrt((a * -c)) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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 <= (-6d-71)) then
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    else if (b_2 <= 3.2d-88) then
        tmp = sqrt((a * -c)) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e-71) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 3.2e-88) {
		tmp = Math.sqrt((a * -c)) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6e-71:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	elif b_2 <= 3.2e-88:
		tmp = math.sqrt((a * -c)) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6e-71)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	elseif (b_2 <= 3.2e-88)
		tmp = Float64(sqrt(Float64(a * Float64(-c))) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6e-71)
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	elseif (b_2 <= 3.2e-88)
		tmp = sqrt((a * -c)) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6e-71], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.2e-88], N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.0000000000000003e-71

   1. Initial program 69.8%

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

      1. +-commutative 69.8%

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

      2. unsub-neg 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in b_2 around -inf 81.0%

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

   6. Taylor expanded in c around 0 81.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]

   if -6.0000000000000003e-71 < b_2 < 3.20000000000000012e-88

   1. Initial program 77.3%

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

      1. +-commutative 77.3%

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

      2. unsub-neg 77.3%

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

   3. Simplified 77.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. prod-diff 76.7%

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

      2. *-commutative 76.7%

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

      3. fmm-def 76.7%

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

      4. prod-diff 76.7%

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

      5. *-commutative 76.7%

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

      6. fmm-def 76.7%

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

      7. associate-+l+ 76.6%

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

      8. pow2 76.6%

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

      9. *-commutative 76.6%

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

      10. fma-undefine 76.7%

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

      11. distribute-lft-neg-in 76.7%

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

      12. *-commutative 76.7%

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

      13. distribute-rgt-neg-in 76.7%

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

      14. fma-define 76.6%

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

      15. *-commutative 76.6%

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

      16. fma-undefine 76.7%

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

      17. distribute-lft-neg-in 76.7%

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

      18. *-commutative 76.7%

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

      19. distribute-rgt-neg-in 76.7%

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

   6. Applied egg-rr 76.6%

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

      1. *-commutative 76.6%

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

      2. count-2 76.6%

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

      3. *-commutative 76.6%

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

   8. Simplified 76.6%

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

   9. Taylor expanded in b_2 around 0 73.8%

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

       1. associate-*l/ 73.8%

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

       2. *-lft-identity 73.8%

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

       3. distribute-lft1-in 73.8%

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

       4. metadata-eval 73.8%

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

       5. mul0-lft 74.5%

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

       6. metadata-eval 74.5%

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

       7. neg-sub0 74.5%

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

       8. distribute-rgt-neg-out 74.5%

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

   11. Simplified 74.5%

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

   if 3.20000000000000012e-88 < b_2

   1. Initial program 16.0%

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

      1. +-commutative 16.0%

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

      2. unsub-neg 16.0%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in b_2 around inf 84.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -6 \cdot 10^{-71}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \mathbf{elif}\;b\_2 \leq 3.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.5 \cdot 10^{-116}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \mathbf{elif}\;b\_2 \leq 2.45 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{c \cdot \frac{-1}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.5e-116)
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))
   (if (<= b_2 2.45e-88) (sqrt (* c (/ -1.0 a))) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.5e-116) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 2.45e-88) {
		tmp = sqrt((c * (-1.0 / a)));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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 <= (-5.5d-116)) then
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    else if (b_2 <= 2.45d-88) then
        tmp = sqrt((c * ((-1.0d0) / a)))
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.5e-116) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 2.45e-88) {
		tmp = Math.sqrt((c * (-1.0 / a)));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.5e-116:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	elif b_2 <= 2.45e-88:
		tmp = math.sqrt((c * (-1.0 / a)))
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.5e-116)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	elseif (b_2 <= 2.45e-88)
		tmp = sqrt(Float64(c * Float64(-1.0 / a)));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.5e-116)
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	elseif (b_2 <= 2.45e-88)
		tmp = sqrt((c * (-1.0 / a)));
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.5e-116], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.45e-88], N[Sqrt[N[(c * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.4999999999999998e-116

   1. Initial program 72.6%

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

      1. +-commutative 72.6%

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

      2. unsub-neg 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in b_2 around -inf 75.9%

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

   6. Taylor expanded in c around 0 76.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]

   if -5.4999999999999998e-116 < b_2 < 2.45000000000000014e-88

   1. Initial program 74.2%

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

      1. +-commutative 74.2%

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

      2. unsub-neg 74.2%

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

   3. Simplified 74.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. prod-diff 73.4%

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

      2. *-commutative 73.4%

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

      3. fmm-def 73.4%

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

      4. prod-diff 73.4%

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

      5. *-commutative 73.4%

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

      6. fmm-def 73.4%

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

      7. associate-+l+ 73.4%

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

      8. pow2 73.4%

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

      9. *-commutative 73.4%

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

      10. fma-undefine 73.4%

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

      11. distribute-lft-neg-in 73.4%

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

      12. *-commutative 73.4%

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

      13. distribute-rgt-neg-in 73.4%

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

      14. fma-define 73.4%

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

      15. *-commutative 73.4%

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

      16. fma-undefine 73.4%

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

      17. distribute-lft-neg-in 73.4%

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

      18. *-commutative 73.4%

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

      19. distribute-rgt-neg-in 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. count-2 73.4%

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

      3. *-commutative 73.4%

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

   8. Simplified 73.4%

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

   9. Taylor expanded in a around inf 38.9%

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

       1. distribute-rgt1-in 38.9%

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

       2. metadata-eval 38.9%

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

       3. mul0-lft 38.9%

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

       4. metadata-eval 38.9%

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

       5. neg-sub0 38.9%

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

   11. Simplified 38.9%

       \[\leadsto \color{blue}{\sqrt{\frac{-c}{a}}} \]
   12. Step-by-step derivation

       1. frac-2neg 38.9%

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

       2. div-inv 39.0%

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

       3. add-sqr-sqrt 37.8%

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

       4. sqrt-unprod 26.5%

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

       5. sqr-neg 26.5%

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

       6. sqrt-unprod 0.7%

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

       7. add-sqr-sqrt 0.9%

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

       8. add-sqr-sqrt 0.2%

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

       9. sqrt-unprod 1.3%

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

       10. sqr-neg 1.3%

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

       11. sqrt-unprod 1.1%

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

       12. add-sqr-sqrt 39.0%

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

   13. Applied egg-rr 39.0%

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

   if 2.45000000000000014e-88 < b_2

   1. Initial program 16.0%

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

      1. +-commutative 16.0%

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

      2. unsub-neg 16.0%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in b_2 around inf 84.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.5 \cdot 10^{-116}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \mathbf{elif}\;b\_2 \leq 2.45 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{c \cdot \frac{-1}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.7 \cdot 10^{-116}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \mathbf{elif}\;b\_2 \leq 2.35 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.7e-116)
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))
   (if (<= b_2 2.35e-88) (sqrt (/ c (- a))) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.7e-116) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 2.35e-88) {
		tmp = sqrt((c / -a));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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 <= (-4.7d-116)) then
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    else if (b_2 <= 2.35d-88) then
        tmp = sqrt((c / -a))
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.7e-116) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 2.35e-88) {
		tmp = Math.sqrt((c / -a));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.7e-116:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	elif b_2 <= 2.35e-88:
		tmp = math.sqrt((c / -a))
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.7e-116)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	elseif (b_2 <= 2.35e-88)
		tmp = sqrt(Float64(c / Float64(-a)));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.7e-116)
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	elseif (b_2 <= 2.35e-88)
		tmp = sqrt((c / -a));
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.7e-116], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.35e-88], N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.69999999999999994e-116

   1. Initial program 72.6%

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

      1. +-commutative 72.6%

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

      2. unsub-neg 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in b_2 around -inf 75.9%

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

   6. Taylor expanded in c around 0 76.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]

   if -4.69999999999999994e-116 < b_2 < 2.35e-88

   1. Initial program 74.2%

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

      1. +-commutative 74.2%

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

      2. unsub-neg 74.2%

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

   3. Simplified 74.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. prod-diff 73.4%

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

      2. *-commutative 73.4%

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

      3. fmm-def 73.4%

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

      4. prod-diff 73.4%

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

      5. *-commutative 73.4%

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

      6. fmm-def 73.4%

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

      7. associate-+l+ 73.4%

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

      8. pow2 73.4%

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

      9. *-commutative 73.4%

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

      10. fma-undefine 73.4%

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

      11. distribute-lft-neg-in 73.4%

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

      12. *-commutative 73.4%

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

      13. distribute-rgt-neg-in 73.4%

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

      14. fma-define 73.4%

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

      15. *-commutative 73.4%

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

      16. fma-undefine 73.4%

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

      17. distribute-lft-neg-in 73.4%

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

      18. *-commutative 73.4%

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

      19. distribute-rgt-neg-in 73.4%

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

   6. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. count-2 73.4%

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

      3. *-commutative 73.4%

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

   8. Simplified 73.4%

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

   9. Taylor expanded in a around inf 38.9%

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

       1. distribute-rgt1-in 38.9%

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

       2. metadata-eval 38.9%

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

       3. mul0-lft 38.9%

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

       4. metadata-eval 38.9%

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

       5. neg-sub0 38.9%

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

   11. Simplified 38.9%

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

   if 2.35e-88 < b_2

   1. Initial program 16.0%

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

      1. +-commutative 16.0%

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

      2. unsub-neg 16.0%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in b_2 around inf 84.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.7 \cdot 10^{-116}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \mathbf{elif}\;b\_2 \leq 2.35 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-310)
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))
   (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-310) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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-310)) then
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-310) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-310:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-310)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-310)
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-310], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.999999999999969e-311

   1. Initial program 74.9%

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

      1. +-commutative 74.9%

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

      2. unsub-neg 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in b_2 around -inf 58.0%

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

   6. Taylor expanded in c around 0 59.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]

   if -9.999999999999969e-311 < b_2

   1. Initial program 28.7%

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

      1. +-commutative 28.7%

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

      2. unsub-neg 28.7%

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

   3. Simplified 28.7%

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

   5. Taylor expanded in b_2 around inf 65.1%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 3.8 \cdot 10^{-304}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 3.8e-304) (/ (* b_2 -2.0) a) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 3.8e-304) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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 <= 3.8d-304) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 3.8e-304) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 3.8e-304:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 3.8e-304)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 3.8e-304)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 3.8e-304], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 3.79999999999999961e-304

   1. Initial program 75.3%

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

      1. +-commutative 75.3%

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

      2. unsub-neg 75.3%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in b_2 around -inf 57.6%

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

      1. *-commutative 57.6%

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

   7. Simplified 57.6%

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

   if 3.79999999999999961e-304 < b_2

   1. Initial program 27.6%

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

      1. +-commutative 27.6%

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

      2. unsub-neg 27.6%

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

   3. Simplified 27.6%

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

   5. Taylor expanded in b_2 around inf 66.1%

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

Alternative 9: 35.4% 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 52.0%

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

   1. +-commutative 52.0%

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

   2. unsub-neg 52.0%

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

3. Simplified 52.0%

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

5. Taylor expanded in b_2 around inf 33.4%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

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) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	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(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	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 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	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[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024170 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :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) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

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