quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.5% → 85.9%

Time: 10.9s

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: 52.5% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b\_2} \cdot \left(--0.5\right) - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{{\left({b\_2}^{2} - c \cdot a\right)}^{0.5} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+153)
   (- (* (/ c b_2) (- -0.5)) (* 2.0 (/ b_2 a)))
   (if (<= b_2 3.5e-33)
     (/ (- (pow (- (pow b_2 2.0) (* c a)) 0.5) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+153) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 3.5e-33) {
		tmp = (pow((pow(b_2, 2.0) - (c * a)), 0.5) - 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 <= (-1d+153)) then
        tmp = ((c / b_2) * -(-0.5d0)) - (2.0d0 * (b_2 / a))
    else if (b_2 <= 3.5d-33) then
        tmp = ((((b_2 ** 2.0d0) - (c * a)) ** 0.5d0) - 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 <= -1e+153) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 3.5e-33) {
		tmp = (Math.pow((Math.pow(b_2, 2.0) - (c * a)), 0.5) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e+153:
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a))
	elif b_2 <= 3.5e-33:
		tmp = (math.pow((math.pow(b_2, 2.0) - (c * a)), 0.5) - 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 <= -1e+153)
		tmp = Float64(Float64(Float64(c / b_2) * Float64(-(-0.5))) - Float64(2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 3.5e-33)
		tmp = Float64(Float64((Float64((b_2 ^ 2.0) - Float64(c * a)) ^ 0.5) - 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 <= -1e+153)
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	elseif (b_2 <= 3.5e-33)
		tmp = ((((b_2 ^ 2.0) - (c * a)) ^ 0.5) - 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, -1e+153], N[(N[(N[(c / b$95$2), $MachinePrecision] * (--0.5)), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-33], N[(N[(N[Power[N[(N[Power[b$95$2, 2.0], $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision], 0.5], $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 < -1e153

   1. Initial program 37.9%

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

      1. +-commutative 37.9%

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

      2. unsub-neg 37.9%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in b_2 around -inf 97.9%

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

   6. Taylor expanded in c around 0 98.4%

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

   if -1e153 < b_2 < 3.4999999999999999e-33

   1. Initial program 78.0%

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

      1. +-commutative 78.0%

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

      2. unsub-neg 78.0%

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

   3. Simplified 78.0%

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

      1. pow1/2 78.1%

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

      2. pow2 78.1%

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

   6. Applied egg-rr 78.1%

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

   if 3.4999999999999999e-33 < b_2

   1. Initial program 13.1%

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

      1. +-commutative 13.1%

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

      2. unsub-neg 13.1%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in b_2 around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. *-commutative 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b\_2} \cdot \left(--0.5\right) - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{{\left({b\_2}^{2} - c \cdot a\right)}^{0.5} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{c}{b\_2} \cdot \left(--0.5\right) - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.6e+149)
   (- (* (/ c b_2) (- -0.5)) (* 2.0 (/ b_2 a)))
   (if (<= b_2 3.5e-33)
     (/ (- (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 <= -3.6e+149) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 3.5e-33) {
		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 <= (-3.6d+149)) then
        tmp = ((c / b_2) * -(-0.5d0)) - (2.0d0 * (b_2 / a))
    else if (b_2 <= 3.5d-33) 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 <= -3.6e+149) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 3.5e-33) {
		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 <= -3.6e+149:
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a))
	elif b_2 <= 3.5e-33:
		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 <= -3.6e+149)
		tmp = Float64(Float64(Float64(c / b_2) * Float64(-(-0.5))) - Float64(2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 3.5e-33)
		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 <= -3.6e+149)
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	elseif (b_2 <= 3.5e-33)
		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, -3.6e+149], N[(N[(N[(c / b$95$2), $MachinePrecision] * (--0.5)), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-33], 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 < -3.59999999999999995e149

   1. Initial program 37.9%

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

      1. +-commutative 37.9%

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

      2. unsub-neg 37.9%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in b_2 around -inf 97.9%

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

   6. Taylor expanded in c around 0 98.4%

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

   if -3.59999999999999995e149 < b_2 < 3.4999999999999999e-33

   1. Initial program 78.0%

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

      1. +-commutative 78.0%

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

      2. unsub-neg 78.0%

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

   3. Simplified 78.0%

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

   if 3.4999999999999999e-33 < b_2

   1. Initial program 13.1%

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

      1. +-commutative 13.1%

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

      2. unsub-neg 13.1%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in b_2 around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. *-commutative 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{c}{b\_2} \cdot \left(--0.5\right) - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{c}{b\_2} \cdot \left(--0.5\right) - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{-c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-70)
   (- (* (/ c b_2) (- -0.5)) (* 2.0 (/ b_2 a)))
   (if (<= b_2 3e-33) (/ (- (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 <= -5e-70) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 3e-33) {
		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 <= (-5d-70)) then
        tmp = ((c / b_2) * -(-0.5d0)) - (2.0d0 * (b_2 / a))
    else if (b_2 <= 3d-33) 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 <= -5e-70) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 3e-33) {
		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 <= -5e-70:
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a))
	elif b_2 <= 3e-33:
		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 <= -5e-70)
		tmp = Float64(Float64(Float64(c / b_2) * Float64(-(-0.5))) - Float64(2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 3e-33)
		tmp = Float64(Float64(sqrt(Float64(-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 <= -5e-70)
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	elseif (b_2 <= 3e-33)
		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, -5e-70], N[(N[(N[(c / b$95$2), $MachinePrecision] * (--0.5)), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3e-33], 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 < -4.9999999999999998e-70

   1. Initial program 62.7%

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

      1. +-commutative 62.7%

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

      2. unsub-neg 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in b_2 around -inf 85.9%

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

   6. Taylor expanded in c around 0 86.4%

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

   if -4.9999999999999998e-70 < b_2 < 3.0000000000000002e-33

   1. Initial program 68.6%

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

      1. +-commutative 68.6%

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

      2. unsub-neg 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in b_2 around 0 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

      3. *-commutative 63.5%

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

   7. Simplified 63.5%

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

   if 3.0000000000000002e-33 < b_2

   1. Initial program 13.1%

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

      1. +-commutative 13.1%

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

      2. unsub-neg 13.1%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in b_2 around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. *-commutative 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-70}:\\ \;\;\;\;\frac{c}{b\_2} \cdot \left(--0.5\right) - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{-c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310)
   (- (* (/ c b_2) (- -0.5)) (* 2.0 (/ b_2 a)))
   (/ (* -0.5 c) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (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 <= (-2d-310)) then
        tmp = ((c / b_2) * -(-0.5d0)) - (2.0d0 * (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 <= -2e-310) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (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 <= -2e-310)
		tmp = Float64(Float64(Float64(c / b_2) * Float64(-(-0.5))) - Float64(2.0 * Float64(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 <= -2e-310)
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (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, -2e-310], N[(N[(N[(c / b$95$2), $MachinePrecision] * (--0.5)), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 64.9%

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

      1. +-commutative 64.9%

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

      2. unsub-neg 64.9%

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

   3. Simplified 64.9%

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

   5. Taylor expanded in b_2 around -inf 69.1%

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

   6. Taylor expanded in c around 0 69.6%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 28.9%

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

      1. +-commutative 28.9%

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

      2. unsub-neg 28.9%

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

   3. Simplified 28.9%

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

   5. Taylor expanded in b_2 around inf 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 69.5%

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

Alternative 5: 67.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.9e-303) (/ (* 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 <= 2.9e-303) {
		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 <= 2.9d-303) 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 <= 2.9e-303) {
		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 <= 2.9e-303:
		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 <= 2.9e-303)
		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 <= 2.9e-303)
		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, 2.9e-303], 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 < 2.90000000000000014e-303

   1. Initial program 65.4%

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

      1. +-commutative 65.4%

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

      2. unsub-neg 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in b_2 around -inf 68.5%

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

      1. *-commutative 68.5%

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

   7. Simplified 68.5%

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

   if 2.90000000000000014e-303 < b_2

   1. Initial program 27.8%

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

      1. +-commutative 27.8%

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

      2. unsub-neg 27.8%

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

   3. Simplified 27.8%

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

   5. Taylor expanded in b_2 around inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. *-commutative 70.5%

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

   7. Simplified 70.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.9e-303) (/ (* 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 <= 2.9e-303) {
		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 <= 2.9d-303) 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 <= 2.9e-303) {
		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 <= 2.9e-303:
		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 <= 2.9e-303)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.9e-303)
		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, 2.9e-303], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.90000000000000014e-303

   1. Initial program 65.4%

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

      1. +-commutative 65.4%

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

      2. unsub-neg 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in b_2 around -inf 68.5%

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

      1. *-commutative 68.5%

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

   7. Simplified 68.5%

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

   if 2.90000000000000014e-303 < b_2

   1. Initial program 27.8%

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

      1. +-commutative 27.8%

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

      2. unsub-neg 27.8%

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

   3. Simplified 27.8%

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

   5. Taylor expanded in c around 0 62.9%

      \[\leadsto \color{blue}{c \cdot \left(-0.125 \cdot \frac{a \cdot c}{{b\_2}^{3}} - 0.5 \cdot \frac{1}{b\_2}\right)} \]
   6. Step-by-step derivation

      1. fma-neg 62.9%

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{a \cdot c}{{b\_2}^{3}}, -0.5 \cdot \frac{1}{b\_2}\right)} \]

      2. associate-/l* 65.6%

         \[\leadsto c \cdot \mathsf{fma}\left(-0.125, \color{blue}{a \cdot \frac{c}{{b\_2}^{3}}}, -0.5 \cdot \frac{1}{b\_2}\right) \]

      3. associate-*r/ 65.6%

         \[\leadsto c \cdot \mathsf{fma}\left(-0.125, a \cdot \frac{c}{{b\_2}^{3}}, -\color{blue}{\frac{0.5 \cdot 1}{b\_2}}\right) \]

      4. metadata-eval 65.6%

         \[\leadsto c \cdot \mathsf{fma}\left(-0.125, a \cdot \frac{c}{{b\_2}^{3}}, -\frac{\color{blue}{0.5}}{b\_2}\right) \]

   7. Simplified 65.6%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-0.125, a \cdot \frac{c}{{b\_2}^{3}}, -\frac{0.5}{b\_2}\right)} \]

   8. Taylor expanded in a around 0 70.3%

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

Alternative 7: 66.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.9e-303) (* 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 <= 2.9e-303) {
		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 <= 2.9d-303) 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 <= 2.9e-303) {
		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 <= 2.9e-303:
		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 <= 2.9e-303)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.9e-303)
		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, 2.9e-303], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.90000000000000014e-303

   1. Initial program 65.4%

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

      1. +-commutative 65.4%

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

      2. unsub-neg 65.4%

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

   3. Simplified 65.4%

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

      1. clear-num 65.3%

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

      2. associate-/r/ 65.2%

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

      3. sub-neg 65.2%

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

      4. add-sqr-sqrt 47.7%

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

      5. hypot-define 62.4%

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

      6. *-commutative 62.4%

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

      7. distribute-rgt-neg-in 62.4%

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

   6. Applied egg-rr 62.4%

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

   7. Taylor expanded in b_2 around -inf 68.5%

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

      1. metadata-eval 68.5%

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

      2. distribute-lft-neg-in 68.5%

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

      3. associate-*r/ 68.5%

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

      4. *-commutative 68.5%

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

      5. associate-*r/ 68.2%

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

      6. distribute-rgt-neg-in 68.2%

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

      7. distribute-neg-frac 68.2%

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

      8. metadata-eval 68.2%

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

   9. Simplified 68.2%

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

   if 2.90000000000000014e-303 < b_2

   1. Initial program 27.8%

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

      1. +-commutative 27.8%

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

      2. unsub-neg 27.8%

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

   3. Simplified 27.8%

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

   5. Taylor expanded in c around 0 62.9%

      \[\leadsto \color{blue}{c \cdot \left(-0.125 \cdot \frac{a \cdot c}{{b\_2}^{3}} - 0.5 \cdot \frac{1}{b\_2}\right)} \]
   6. Step-by-step derivation

      1. fma-neg 62.9%

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{a \cdot c}{{b\_2}^{3}}, -0.5 \cdot \frac{1}{b\_2}\right)} \]

      2. associate-/l* 65.6%

         \[\leadsto c \cdot \mathsf{fma}\left(-0.125, \color{blue}{a \cdot \frac{c}{{b\_2}^{3}}}, -0.5 \cdot \frac{1}{b\_2}\right) \]

      3. associate-*r/ 65.6%

         \[\leadsto c \cdot \mathsf{fma}\left(-0.125, a \cdot \frac{c}{{b\_2}^{3}}, -\color{blue}{\frac{0.5 \cdot 1}{b\_2}}\right) \]

      4. metadata-eval 65.6%

         \[\leadsto c \cdot \mathsf{fma}\left(-0.125, a \cdot \frac{c}{{b\_2}^{3}}, -\frac{\color{blue}{0.5}}{b\_2}\right) \]

   7. Simplified 65.6%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-0.125, a \cdot \frac{c}{{b\_2}^{3}}, -\frac{0.5}{b\_2}\right)} \]

   8. Taylor expanded in a around 0 70.3%

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

Alternative 8: 43.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 0.0068:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 0.0068) (* b_2 (/ -2.0 a)) (* (/ c b_2) 0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 0.0068) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = (c / b_2) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 0.0068d0) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = (c / b_2) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 0.0068) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = (c / b_2) * 0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 0.0068:
		tmp = b_2 * (-2.0 / a)
	else:
		tmp = (c / b_2) * 0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 0.0068)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = Float64(Float64(c / b_2) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 0.0068)
		tmp = b_2 * (-2.0 / a);
	else
		tmp = (c / b_2) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 0.0068], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 0.00679999999999999962

   1. Initial program 63.9%

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

      1. +-commutative 63.9%

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

      2. unsub-neg 63.9%

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

   3. Simplified 63.9%

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

      1. clear-num 63.7%

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

      2. associate-/r/ 63.7%

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

      3. sub-neg 63.7%

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

      4. add-sqr-sqrt 49.6%

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

      5. hypot-define 60.9%

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

      6. *-commutative 60.9%

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

      7. distribute-rgt-neg-in 60.9%

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

   6. Applied egg-rr 60.9%

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

   7. Taylor expanded in b_2 around -inf 53.5%

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

      1. metadata-eval 53.5%

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

      2. distribute-lft-neg-in 53.5%

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

      3. associate-*r/ 53.5%

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

      4. *-commutative 53.5%

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

      5. associate-*r/ 53.3%

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

      6. distribute-rgt-neg-in 53.3%

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

      7. distribute-neg-frac 53.3%

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

      8. metadata-eval 53.3%

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

   9. Simplified 53.3%

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

   if 0.00679999999999999962 < b_2

   1. Initial program 12.5%

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

      1. +-commutative 12.5%

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

      2. unsub-neg 12.5%

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

   3. Simplified 12.5%

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

      1. clear-num 12.5%

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

      2. associate-/r/ 12.5%

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

      3. sub-neg 12.5%

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

      4. add-sqr-sqrt 10.2%

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

      5. hypot-define 21.7%

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

      6. *-commutative 21.7%

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

      7. distribute-rgt-neg-in 21.7%

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

   6. Applied egg-rr 21.7%

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

   7. Taylor expanded in b_2 around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 71.6%

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

      5. neg-mul-1 71.6%

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

      6. distribute-rgt-neg-out 71.6%

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

      7. *-commutative 71.6%

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

      8. distribute-rgt-neg-in 71.6%

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

   9. Simplified 71.6%

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

       1. frac-times 75.7%

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

       2. associate-*r* 75.7%

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

       3. *-un-lft-identity 75.7%

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

       4. add-sqr-sqrt 36.5%

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

       5. sqrt-unprod 40.5%

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

       6. sqr-neg 40.5%

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

       7. sqrt-unprod 14.2%

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

       8. add-sqr-sqrt 26.6%

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

   11. Applied egg-rr 26.6%

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

   12. Taylor expanded in c around 0 26.5%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 0.0068:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 10.9% accurate, 22.4× speedup

Math

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

FPCore

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

C

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

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 = (c / b_2) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.6%

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

   1. +-commutative 47.6%

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

   2. unsub-neg 47.6%

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

3. Simplified 47.6%

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

   1. clear-num 47.5%

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

   2. associate-/r/ 47.5%

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

   3. sub-neg 47.5%

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

   4. add-sqr-sqrt 37.1%

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

   5. hypot-define 48.5%

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

   6. *-commutative 48.5%

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

   7. distribute-rgt-neg-in 48.5%

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

6. Applied egg-rr 48.5%

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

7. Taylor expanded in b_2 around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 26.0%

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

   5. neg-mul-1 26.0%

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

   6. distribute-rgt-neg-out 26.0%

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

   7. *-commutative 26.0%

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

   8. distribute-rgt-neg-in 26.0%

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

9. Simplified 26.0%

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

    1. frac-times 26.9%

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

    2. associate-*r* 26.9%

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

    3. *-un-lft-identity 26.9%

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

    4. add-sqr-sqrt 12.8%

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

    5. sqrt-unprod 14.2%

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

    6. sqr-neg 14.2%

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

    7. sqrt-unprod 5.4%

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

    8. add-sqr-sqrt 10.2%

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

11. Applied egg-rr 10.2%

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

12. Taylor expanded in c around 0 10.5%

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

13. Final simplification 10.5%

    \[\leadsto \frac{c}{b\_2} \cdot 0.5 \]
14. 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 2024108 
(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))