quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.7% → 86.0%

Time: 11.4s

Alternatives: 7

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.7% 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: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.3 \cdot 10^{+118}:\\ \;\;\;\;2 \cdot \left(-\frac{b\_2}{a}\right) - -0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9.7 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.3e+118)
   (- (* 2.0 (- (/ b_2 a))) (* -0.5 (/ c b_2)))
   (if (<= b_2 9.7e-110)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.3e+118) {
		tmp = (2.0 * -(b_2 / a)) - (-0.5 * (c / b_2));
	} else if (b_2 <= 9.7e-110) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - 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 <= (-2.3d+118)) then
        tmp = (2.0d0 * -(b_2 / a)) - ((-0.5d0) * (c / b_2))
    else if (b_2 <= 9.7d-110) then
        tmp = (sqrt(((b_2 * b_2) - (c * a))) - 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 <= -2.3e+118) {
		tmp = (2.0 * -(b_2 / a)) - (-0.5 * (c / b_2));
	} else if (b_2 <= 9.7e-110) {
		tmp = (Math.sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.3e+118:
		tmp = (2.0 * -(b_2 / a)) - (-0.5 * (c / b_2))
	elif b_2 <= 9.7e-110:
		tmp = (math.sqrt(((b_2 * b_2) - (c * a))) - 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 <= -2.3e+118)
		tmp = Float64(Float64(2.0 * Float64(-Float64(b_2 / a))) - Float64(-0.5 * Float64(c / b_2)));
	elseif (b_2 <= 9.7e-110)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.3e+118)
		tmp = (2.0 * -(b_2 / a)) - (-0.5 * (c / b_2));
	elseif (b_2 <= 9.7e-110)
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - 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, -2.3e+118], N[(N[(2.0 * (-N[(b$95$2 / a), $MachinePrecision])), $MachinePrecision] - N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 9.7e-110], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.30000000000000016e118

   1. Initial program 38.8%

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

      1. +-commutative 38.8%

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

      2. unsub-neg 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in b_2 around -inf 92.0%

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

   6. Taylor expanded in c around 0 93.0%

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

   if -2.30000000000000016e118 < b_2 < 9.70000000000000047e-110

   1. Initial program 86.8%

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

      1. +-commutative 86.8%

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

      2. unsub-neg 86.8%

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

   3. Simplified 86.8%

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

   if 9.70000000000000047e-110 < b_2

   1. Initial program 12.9%

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

      1. +-commutative 12.9%

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

      2. unsub-neg 12.9%

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

   3. Simplified 12.9%

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

   5. Taylor expanded in b_2 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. *-commutative 86.5%

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

   7. Simplified 86.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.3 \cdot 10^{+118}:\\ \;\;\;\;2 \cdot \left(-\frac{b\_2}{a}\right) - -0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9.7 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.5 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \left(-\frac{b\_2}{a}\right) - -0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.5e-93)
   (- (* 2.0 (- (/ b_2 a))) (* -0.5 (/ c b_2)))
   (if (<= b_2 9e-110) (/ (- (sqrt (* c (- a))) b_2) a) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.5e-93) {
		tmp = (2.0 * -(b_2 / a)) - (-0.5 * (c / b_2));
	} else if (b_2 <= 9e-110) {
		tmp = (sqrt((c * -a)) - 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 <= (-4.5d-93)) then
        tmp = (2.0d0 * -(b_2 / a)) - ((-0.5d0) * (c / b_2))
    else if (b_2 <= 9d-110) then
        tmp = (sqrt((c * -a)) - 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 <= -4.5e-93) {
		tmp = (2.0 * -(b_2 / a)) - (-0.5 * (c / b_2));
	} else if (b_2 <= 9e-110) {
		tmp = (Math.sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.5e-93:
		tmp = (2.0 * -(b_2 / a)) - (-0.5 * (c / b_2))
	elif b_2 <= 9e-110:
		tmp = (math.sqrt((c * -a)) - 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 <= -4.5e-93)
		tmp = Float64(Float64(2.0 * Float64(-Float64(b_2 / a))) - Float64(-0.5 * Float64(c / b_2)));
	elseif (b_2 <= 9e-110)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(-a))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.5e-93)
		tmp = (2.0 * -(b_2 / a)) - (-0.5 * (c / b_2));
	elseif (b_2 <= 9e-110)
		tmp = (sqrt((c * -a)) - 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, -4.5e-93], N[(N[(2.0 * (-N[(b$95$2 / a), $MachinePrecision])), $MachinePrecision] - N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 9e-110], N[(N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.5000000000000002e-93

   1. Initial program 67.5%

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

      1. +-commutative 67.5%

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

      2. unsub-neg 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in b_2 around -inf 85.0%

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

   6. Taylor expanded in c around 0 85.6%

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

   if -4.5000000000000002e-93 < b_2 < 9.0000000000000002e-110

   1. Initial program 77.7%

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

      1. +-commutative 77.7%

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

      2. unsub-neg 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in b_2 around 0 73.3%

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

      1. associate-*r* 73.3%

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

      2. neg-mul-1 73.3%

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

      3. *-commutative 73.3%

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

   7. Simplified 73.3%

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

   if 9.0000000000000002e-110 < b_2

   1. Initial program 12.9%

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

      1. +-commutative 12.9%

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

      2. unsub-neg 12.9%

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

   3. Simplified 12.9%

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

   5. Taylor expanded in b_2 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. *-commutative 86.5%

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

   7. Simplified 86.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.5 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \left(-\frac{b\_2}{a}\right) - -0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 6.6× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (2.0 * -(b_2 / a)) - (-0.5 * (c / b_2));
	} 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-310)) then
        tmp = (2.0d0 * -(b_2 / a)) - ((-0.5d0) * (c / b_2))
    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-310) {
		tmp = (2.0 * -(b_2 / a)) - (-0.5 * (c / b_2));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (2.0 * -(b_2 / a)) - (-0.5 * (c / b_2));
	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-310], N[(N[(2.0 * (-N[(b$95$2 / a), $MachinePrecision])), $MachinePrecision] - N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 70.8%

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

      1. +-commutative 70.8%

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

      2. unsub-neg 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in b_2 around -inf 70.7%

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

   6. Taylor expanded in c around 0 73.5%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 26.3%

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

      1. +-commutative 26.3%

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

      2. unsub-neg 26.3%

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

   3. Simplified 26.3%

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

   5. Taylor expanded in b_2 around inf 69.4%

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

      1. associate-*r/ 69.4%

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

      2. *-commutative 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 71.5%

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

Alternative 4: 23.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{+52}:\\ \;\;\;\;-\frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 9e+52) (- (/ b_2 a)) (/ (* c 0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e+52) {
		tmp = -(b_2 / a);
	} else {
		tmp = (c * 0.5) / 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 <= 9d+52) then
        tmp = -(b_2 / a)
    else
        tmp = (c * 0.5d0) / 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 <= 9e+52) {
		tmp = -(b_2 / a);
	} else {
		tmp = (c * 0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 9e+52:
		tmp = -(b_2 / a)
	else:
		tmp = (c * 0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 9e+52)
		tmp = Float64(-Float64(b_2 / a));
	else
		tmp = Float64(Float64(c * 0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 9e+52)
		tmp = -(b_2 / a);
	else
		tmp = (c * 0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 9e+52], (-N[(b$95$2 / a), $MachinePrecision]), N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 8.9999999999999999e52

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      2. unsub-neg 61.6%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in b_2 around 0 39.0%

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

      1. associate-*r* 39.0%

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

      2. neg-mul-1 39.0%

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

      3. *-commutative 39.0%

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

   7. Simplified 39.0%

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

   8. Taylor expanded in b_2 around inf 19.4%

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

      1. neg-mul-1 19.4%

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

      2. distribute-frac-neg2 19.4%

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

   10. Simplified 19.4%

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

   if 8.9999999999999999e52 < b_2

   1. Initial program 11.4%

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

      1. +-commutative 11.4%

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

      2. unsub-neg 11.4%

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

   3. Simplified 11.4%

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

   5. Taylor expanded in b_2 around inf 71.8%

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

      1. associate-*r/ 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r* 71.8%

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

      4. *-commutative 71.8%

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

   7. Simplified 71.8%

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

      1. associate-/l* 62.9%

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

      2. *-un-lft-identity 62.9%

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

      3. times-frac 70.4%

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

   9. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{1} \cdot \frac{\frac{a}{b\_2}}{a}} \]
   10. Step-by-step derivation

       1. /-rgt-identity 70.4%

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

       2. *-commutative 70.4%

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

       3. frac-2neg 70.4%

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

       4. /-rgt-identity 70.4%

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

       5. frac-2neg 70.4%

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

       6. metadata-eval 70.4%

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

       7. frac-times 62.9%

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

       8. distribute-neg-frac 62.9%

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

       9. add-sqr-sqrt 35.7%

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

       10. sqrt-unprod 37.1%

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

       11. sqr-neg 37.1%

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

       12. sqrt-unprod 17.8%

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

       13. add-sqr-sqrt 31.0%

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

       14. distribute-rgt-neg-in 31.0%

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

       15. metadata-eval 31.0%

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

       16. add-sqr-sqrt 13.2%

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

       17. sqrt-unprod 33.6%

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

       18. sqr-neg 33.6%

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

       19. sqrt-unprod 27.1%

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

       20. add-sqr-sqrt 62.9%

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

       21. *-commutative 62.9%

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

       22. neg-mul-1 62.9%

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

       23. add-sqr-sqrt 35.6%

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

       24. sqrt-unprod 36.6%

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

       25. sqr-neg 36.6%

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

   11. Applied egg-rr 31.0%

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

       1. associate-/l* 30.6%

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

       2. times-frac 30.5%

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

       3. *-commutative 30.5%

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

       4. *-commutative 30.5%

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

       5. associate-*r/ 30.8%

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

       6. associate-*l* 30.8%

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

       7. associate-/r* 30.5%

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

       8. *-inverses 30.5%

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

       9. associate-*r/ 30.5%

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

       10. metadata-eval 30.5%

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

       11. associate-*r/ 30.5%

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

       12. *-commutative 30.5%

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

   13. Simplified 30.5%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{+52}:\\ \;\;\;\;-\frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{+52}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 9e+52) (/ (* b_2 -2.0) a) (/ (* c 0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e+52) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * 0.5) / 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 <= 9d+52) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (c * 0.5d0) / 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 <= 9e+52) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * 0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 9e+52:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (c * 0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 9e+52)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(c * 0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 9e+52)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (c * 0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 9e+52], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 8.9999999999999999e52

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      2. unsub-neg 61.6%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in b_2 around -inf 51.0%

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

      1. *-commutative 51.0%

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

   7. Simplified 51.0%

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

   if 8.9999999999999999e52 < b_2

   1. Initial program 11.4%

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

      1. +-commutative 11.4%

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

      2. unsub-neg 11.4%

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

   3. Simplified 11.4%

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

   5. Taylor expanded in b_2 around inf 71.8%

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

      1. associate-*r/ 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r* 71.8%

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

      4. *-commutative 71.8%

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

   7. Simplified 71.8%

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

      1. associate-/l* 62.9%

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

      2. *-un-lft-identity 62.9%

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

      3. times-frac 70.4%

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

   9. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{1} \cdot \frac{\frac{a}{b\_2}}{a}} \]
   10. Step-by-step derivation

       1. /-rgt-identity 70.4%

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

       2. *-commutative 70.4%

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

       3. frac-2neg 70.4%

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

       4. /-rgt-identity 70.4%

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

       5. frac-2neg 70.4%

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

       6. metadata-eval 70.4%

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

       7. frac-times 62.9%

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

       8. distribute-neg-frac 62.9%

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

       9. add-sqr-sqrt 35.7%

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

       10. sqrt-unprod 37.1%

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

       11. sqr-neg 37.1%

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

       12. sqrt-unprod 17.8%

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

       13. add-sqr-sqrt 31.0%

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

       14. distribute-rgt-neg-in 31.0%

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

       15. metadata-eval 31.0%

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

       16. add-sqr-sqrt 13.2%

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

       17. sqrt-unprod 33.6%

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

       18. sqr-neg 33.6%

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

       19. sqrt-unprod 27.1%

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

       20. add-sqr-sqrt 62.9%

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

       21. *-commutative 62.9%

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

       22. neg-mul-1 62.9%

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

       23. add-sqr-sqrt 35.6%

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

       24. sqrt-unprod 36.6%

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

       25. sqr-neg 36.6%

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

   11. Applied egg-rr 31.0%

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

       1. associate-/l* 30.6%

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

       2. times-frac 30.5%

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

       3. *-commutative 30.5%

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

       4. *-commutative 30.5%

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

       5. associate-*r/ 30.8%

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

       6. associate-*l* 30.8%

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

       7. associate-/r* 30.5%

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

       8. *-inverses 30.5%

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

       9. associate-*r/ 30.5%

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

       10. metadata-eval 30.5%

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

       11. associate-*r/ 30.5%

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

       12. *-commutative 30.5%

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

   13. Simplified 30.5%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{+52}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 4e-310) (/ (* 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 <= 4e-310) {
		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 <= 4d-310) 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 <= 4e-310) {
		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 <= 4e-310:
		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 <= 4e-310)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 4e-310)
		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, 4e-310], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 3.999999999999988e-310

   1. Initial program 70.8%

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

      1. +-commutative 70.8%

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

      2. unsub-neg 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in b_2 around -inf 73.0%

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

      1. *-commutative 73.0%

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

   7. Simplified 73.0%

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

   if 3.999999999999988e-310 < b_2

   1. Initial program 26.3%

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

      1. +-commutative 26.3%

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

      2. unsub-neg 26.3%

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

   3. Simplified 26.3%

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

   5. Taylor expanded in b_2 around inf 69.4%

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

      1. associate-*r/ 69.4%

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

      2. *-commutative 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 15.4% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ -\frac{b\_2}{a} \end{array} \]

FPCore

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

C

double code(double a, double b_2, double c) {
	return -(b_2 / 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 / a)
end function

Java

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

Python

def code(a, b_2, c):
	return -(b_2 / a)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

   1. +-commutative 49.4%

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

   2. unsub-neg 49.4%

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

3. Simplified 49.4%

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

5. Taylor expanded in b_2 around 0 30.6%

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

   1. associate-*r* 30.6%

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

   2. neg-mul-1 30.6%

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

   3. *-commutative 30.6%

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

7. Simplified 30.6%

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

8. Taylor expanded in b_2 around inf 15.2%

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

   1. neg-mul-1 15.2%

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

   2. distribute-frac-neg2 15.2%

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

10. Simplified 15.2%

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

11. Final simplification 15.2%

    \[\leadsto -\frac{b\_2}{a} \]
12. Add Preprocessing

Developer target: 99.6% accurate, 0.1× speedup

Math

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

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 2024074 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b_2 0.0) (/ (- (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) a) (/ (- c) (+ 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))))))))

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