quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.4% → 86.7%

Time: 9.4s

Alternatives: 8

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: 52.4% 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: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{+153}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+153)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 2.7e-88)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+153) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 2.7e-88) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= (-5d+153)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 2.7d-88) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= -5e+153) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 2.7e-88) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e+153:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 2.7e-88:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - 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 <= -5e+153)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 2.7e-88)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - 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 <= -5e+153)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 2.7e-88)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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, -5e+153], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.7e-88], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.00000000000000018e153

   1. Initial program 38.7%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 38.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 38.7%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 38.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 38.7%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 38.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 100.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

   if -5.00000000000000018e153 < b_2 < 2.69999999999999995e-88

   1. Initial program 88.8%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 88.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 88.8%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 88.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 88.8%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 88.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   if 2.69999999999999995e-88 < b_2

   1. Initial program 12.8%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 12.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 12.8%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 12.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 12.8%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 12.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 90.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   5. Step-by-step derivation

      1. associate-*r/ 90.7%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

      2. *-commutative 90.7%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b_2} \]

   6. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{+153}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.4 \cdot 10^{-116}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.4e-116)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 2.7e-88) (/ (- (sqrt (* a (- c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.4e-116) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.7e-88) {
		tmp = (sqrt((a * -c)) - 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 <= (-2.4d-116)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 2.7d-88) then
        tmp = (sqrt((a * -c)) - 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 <= -2.4e-116) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.7e-88) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.4e-116:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 2.7e-88:
		tmp = (math.sqrt((a * -c)) - 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 <= -2.4e-116)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 2.7e-88)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - 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 <= -2.4e-116)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 2.7e-88)
		tmp = (sqrt((a * -c)) - 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, -2.4e-116], 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, 2.7e-88], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.39999999999999993e-116

   1. Initial program 75.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 75.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 75.4%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 75.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 75.4%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 87.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -2.39999999999999993e-116 < b_2 < 2.69999999999999995e-88

   1. Initial program 84.3%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 84.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 84.3%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 84.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 84.3%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around 0 83.2%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}} - b_2}{a} \]
   5. Step-by-step derivation

      1. associate-*r* 83.2%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}} - b_2}{a} \]

      2. neg-mul-1 83.2%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right)} \cdot c} - b_2}{a} \]

   6. Simplified 83.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right) \cdot c}} - b_2}{a} \]

   if 2.69999999999999995e-88 < b_2

   1. Initial program 12.8%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 12.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 12.8%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 12.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 12.8%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 12.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 90.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   5. Step-by-step derivation

      1. associate-*r/ 90.7%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

      2. *-commutative 90.7%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b_2} \]

   6. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.4 \cdot 10^{-116}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3: 67.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Python

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

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-310)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	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 <= -1e-310)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.999999999999969e-311

   1. Initial program 79.9%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 79.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 79.9%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 79.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 79.9%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 61.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -9.999999999999969e-311 < b_2

   1. Initial program 31.7%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 31.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 31.7%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 31.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 31.7%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 31.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 69.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   5. Step-by-step derivation

      1. associate-*r/ 69.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

      2. *-commutative 69.2%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b_2} \]

   6. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4: 43.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 1.25 \cdot 10^{+46}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.25e+46) (* -2.0 (/ b_2 a)) (* c (/ 0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.25e+46) {
		tmp = -2.0 * (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 <= 1.25d+46) then
        tmp = (-2.0d0) * (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 <= 1.25e+46) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = c * (0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.25e+46:
		tmp = -2.0 * (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 <= 1.25e+46)
		tmp = Float64(-2.0 * Float64(b_2 / 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 <= 1.25e+46)
		tmp = -2.0 * (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, 1.25e+46], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.2500000000000001e46

   1. Initial program 73.0%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 73.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 73.0%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 73.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 73.0%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 73.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 41.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

   if 1.2500000000000001e46 < b_2

   1. Initial program 7.9%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 7.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 7.9%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 7.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 7.9%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 7.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 74.5%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b_2}}}{a} \]
   5. Step-by-step derivation

      1. associate-/l* 83.7%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\frac{a}{\frac{b_2}{c}}}}{a} \]

      2. associate-/r/ 73.5%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{a}{b_2} \cdot c\right)}}{a} \]

   6. Simplified 73.5%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \left(\frac{a}{b_2} \cdot c\right)}}{a} \]

   8. Applied egg-rr 27.9%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(-a \cdot \frac{c}{b_2}\right)} \]

   9. Taylor expanded in a around 0 27.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2}} \]

   11. Simplified 27.7%

       \[\leadsto \color{blue}{c \cdot \frac{0.5}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 1.25 \cdot 10^{+46}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b_2}\\ \end{array} \]

Alternative 5: 67.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 4.8 \cdot 10^{-307}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 4.8e-307) (* -2.0 (/ b_2 a)) (/ -0.5 (/ b_2 c))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 4.8e-307) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = -0.5 / (b_2 / c);
	}
	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 <= 4.8d-307) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = (-0.5d0) / (b_2 / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 4.8e-307) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = -0.5 / (b_2 / c);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 4.8e-307:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = -0.5 / (b_2 / c)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 4.8e-307)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(-0.5 / Float64(b_2 / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 4.8e-307)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = -0.5 / (b_2 / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 4.8e-307], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 4.80000000000000036e-307

   1. Initial program 80.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 80.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 80.1%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 80.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 80.1%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 60.7%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

   if 4.80000000000000036e-307 < b_2

   1. Initial program 31.2%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 31.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 31.2%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 31.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 31.2%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 31.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 53.3%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b_2}}}{a} \]
   5. Step-by-step derivation

      1. associate-/l* 60.8%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\frac{a}{\frac{b_2}{c}}}}{a} \]

      2. associate-/r/ 55.2%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{a}{b_2} \cdot c\right)}}{a} \]

   6. Simplified 55.2%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \left(\frac{a}{b_2} \cdot c\right)}}{a} \]

   7. Taylor expanded in a around 0 69.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   8. Step-by-step derivation

      1. associate-*r/ 69.7%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

      2. associate-/l* 69.4%

         \[\leadsto \color{blue}{\frac{-0.5}{\frac{b_2}{c}}} \]

   9. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{b_2}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 4.8 \cdot 10^{-307}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternative 6: 67.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 4.8 \cdot 10^{-307}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 4.8e-307) (* -2.0 (/ b_2 a)) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 4.8e-307) {
		tmp = -2.0 * (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 <= 4.8d-307) then
        tmp = (-2.0d0) * (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 <= 4.8e-307) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 4.8e-307:
		tmp = -2.0 * (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 <= 4.8e-307)
		tmp = Float64(-2.0 * 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 <= 4.8e-307)
		tmp = -2.0 * (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, 4.8e-307], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 4.80000000000000036e-307

   1. Initial program 80.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 80.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 80.1%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 80.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 80.1%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 60.7%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

   if 4.80000000000000036e-307 < b_2

   1. Initial program 31.2%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 31.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. *-commutative 31.2%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

      3. unsub-neg 31.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

      4. *-commutative 31.2%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

   3. Simplified 31.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 69.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   5. Step-by-step derivation

      1. associate-*r/ 69.7%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

      2. *-commutative 69.7%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b_2} \]

   6. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 4.8 \cdot 10^{-307}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 7: 34.7% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{b_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -2.0 (/ b_2 a)))

C

double code(double a, double b_2, double c) {
	return -2.0 * (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 = (-2.0d0) * (b_2 / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / a);
}

Python

def code(a, b_2, c):
	return -2.0 * (b_2 / a)

Julia

function code(a, b_2, c)
	return Float64(-2.0 * Float64(b_2 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -2.0 * (b_2 / a);
end

Wolfram

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

Derivation

1. Initial program 53.9%

   \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 53.9%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

   2. *-commutative 53.9%

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

   3. unsub-neg 53.9%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

   4. *-commutative 53.9%

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

3. Simplified 53.9%

   \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

4. Taylor expanded in b_2 around -inf 29.8%

   \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

5. Final simplification 29.8%

   \[\leadsto -2 \cdot \frac{b_2}{a} \]

Alternative 8: 14.9% 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 53.9%

   \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 53.9%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

   2. *-commutative 53.9%

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} + \left(-b_2\right)}{a} \]

   3. unsub-neg 53.9%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a} \]

   4. *-commutative 53.9%

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{a \cdot c}} - b_2}{a} \]

3. Simplified 53.9%

   \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

4. Taylor expanded in b_2 around 0 37.8%

   \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}} - b_2}{a} \]
5. Step-by-step derivation

   1. associate-*r* 37.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}} - b_2}{a} \]

   2. neg-mul-1 37.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right)} \cdot c} - b_2}{a} \]

6. Simplified 37.8%

   \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right) \cdot c}} - b_2}{a} \]

7. Taylor expanded in a around 0 13.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 13.8%

      \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

   2. mul-1-neg 13.8%

      \[\leadsto \frac{\color{blue}{-b_2}}{a} \]

9. Simplified 13.8%

   \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

10. Final simplification 13.8%

    \[\leadsto \frac{-b_2}{a} \]

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 2023336 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (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))