quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.2% → 86.0%

Time: 11.9s

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: 51.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{+117}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.9e+117)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 3e-69)
     (/ (- (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 <= -1.9e+117) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3e-69) {
		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 <= (-1.9d+117)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 3d-69) 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 <= -1.9e+117) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3e-69) {
		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 <= -1.9e+117:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 3e-69:
		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 <= -1.9e+117)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 3e-69)
		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 <= -1.9e+117)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 3e-69)
		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, -1.9e+117], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3e-69], 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 < -1.9000000000000001e117

   1. Initial program 38.4%

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

      1. +-commutative 38.4%

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

      2. unsub-neg 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in b_2 around -inf 92.9%

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

   if -1.9000000000000001e117 < b_2 < 2.99999999999999989e-69

   1. Initial program 79.5%

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

      1. +-commutative 79.5%

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

      2. unsub-neg 79.5%

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

   3. Simplified 79.5%

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

   if 2.99999999999999989e-69 < b_2

   1. Initial program 11.8%

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

      1. +-commutative 11.8%

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

      2. unsub-neg 11.8%

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

   3. Simplified 11.8%

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

   5. Taylor expanded in b_2 around inf 92.6%

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

      1. associate-*r/ 92.6%

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

      2. *-commutative 92.6%

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

   7. Simplified 92.6%

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

Alternative 2: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.5 \cdot 10^{-73}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 9.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.5e-73)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 9.5e-70) (/ (- (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 <= -7.5e-73) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 9.5e-70) {
		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 <= (-7.5d-73)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 9.5d-70) 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 <= -7.5e-73) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 9.5e-70) {
		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 <= -7.5e-73:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 9.5e-70:
		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 <= -7.5e-73)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 9.5e-70)
		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 <= -7.5e-73)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 9.5e-70)
		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, -7.5e-73], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 9.5e-70], 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 < -7.5e-73

   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 77.6%

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

   if -7.5e-73 < b_2 < 9.4999999999999994e-70

   1. Initial program 74.9%

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

      1. +-commutative 74.9%

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

      2. unsub-neg 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in b_2 around 0 68.5%

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

      1. associate-*r* 68.5%

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

      2. neg-mul-1 68.5%

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

      3. *-commutative 68.5%

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

   7. Simplified 68.5%

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

   if 9.4999999999999994e-70 < b_2

   1. Initial program 11.8%

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

      1. +-commutative 11.8%

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

      2. unsub-neg 11.8%

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

   3. Simplified 11.8%

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

   5. Taylor expanded in b_2 around inf 92.6%

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

      1. associate-*r/ 92.6%

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

      2. *-commutative 92.6%

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

   7. Simplified 92.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.5 \cdot 10^{-73}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 9.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 4 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-139)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 4e-69) (/ (sqrt (* a (- c))) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-139) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 4e-69) {
		tmp = sqrt((a * -c)) / 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 <= (-2d-139)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 4d-69) then
        tmp = sqrt((a * -c)) / 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 <= -2e-139) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 4e-69) {
		tmp = Math.sqrt((a * -c)) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-139:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 4e-69:
		tmp = math.sqrt((a * -c)) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-139)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 4e-69)
		tmp = Float64(sqrt(Float64(a * Float64(-c))) / 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 <= -2e-139)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 4e-69)
		tmp = sqrt((a * -c)) / 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, -2e-139], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4e-69], N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.00000000000000006e-139

   1. Initial program 67.6%

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

      1. +-commutative 67.6%

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

      2. unsub-neg 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in b_2 around -inf 74.6%

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

   if -2.00000000000000006e-139 < b_2 < 3.9999999999999999e-69

   1. Initial program 72.0%

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

      1. +-commutative 72.0%

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

      2. unsub-neg 72.0%

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

   3. Simplified 72.0%

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

      1. prod-diff 71.6%

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

      2. *-commutative 71.6%

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

      3. fma-neg 71.6%

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

      4. prod-diff 71.6%

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

      5. *-commutative 71.6%

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

      6. fma-neg 71.6%

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

      7. associate-+l+ 71.6%

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

      8. pow2 71.6%

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

      9. *-commutative 71.6%

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

      10. fma-undefine 71.6%

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

      11. distribute-lft-neg-in 71.6%

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

      12. *-commutative 71.6%

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

      13. distribute-rgt-neg-in 71.6%

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

      14. fma-define 71.6%

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

      15. *-commutative 71.6%

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

      16. fma-undefine 71.6%

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

      17. distribute-lft-neg-in 71.6%

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

      18. *-commutative 71.6%

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

      19. distribute-rgt-neg-in 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. associate-+l- 71.6%

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

      2. count-2 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in b_2 around 0 70.4%

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

       1. distribute-lft1-in 70.4%

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

       2. metadata-eval 70.4%

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

       3. mul0-lft 70.8%

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

       4. metadata-eval 70.8%

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

       5. cancel-sign-sub-inv 70.8%

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

       6. mul0-lft 70.8%

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

       7. distribute-rgt-in 70.8%

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

       8. metadata-eval 70.8%

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

       9. mul0-lft 70.8%

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

       10. metadata-eval 70.8%

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

       11. distribute-rgt1-in 70.8%

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

       12. sub-neg 70.8%

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

       13. associate-*l/ 71.0%

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

   11. Simplified 71.0%

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

   if 3.9999999999999999e-69 < b_2

   1. Initial program 11.8%

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

      1. +-commutative 11.8%

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

      2. unsub-neg 11.8%

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

   3. Simplified 11.8%

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

   5. Taylor expanded in b_2 around inf 92.6%

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

      1. associate-*r/ 92.6%

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

      2. *-commutative 92.6%

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

   7. Simplified 92.6%

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

Alternative 4: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{-141}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.9 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.2e-141)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 3.9e-148) (sqrt (/ c (- a))) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.2e-141) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3.9e-148) {
		tmp = sqrt((c / -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 <= (-5.2d-141)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 3.9d-148) then
        tmp = sqrt((c / -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 <= -5.2e-141) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3.9e-148) {
		tmp = Math.sqrt((c / -a));
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.2e-141:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 3.9e-148:
		tmp = math.sqrt((c / -a))
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.2e-141)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 3.9e-148)
		tmp = sqrt(Float64(c / Float64(-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 <= -5.2e-141)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 3.9e-148)
		tmp = sqrt((c / -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, -5.2e-141], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.9e-148], N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.20000000000000022e-141

   1. Initial program 68.2%

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

      1. +-commutative 68.2%

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

      2. unsub-neg 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in b_2 around -inf 73.3%

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

   if -5.20000000000000022e-141 < b_2 < 3.89999999999999994e-148

   1. Initial program 72.8%

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

      1. +-commutative 72.8%

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

      2. unsub-neg 72.8%

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

   3. Simplified 72.8%

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

      1. prod-diff 72.3%

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

      2. *-commutative 72.3%

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

      3. fma-neg 72.3%

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

      4. prod-diff 72.3%

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

      5. *-commutative 72.3%

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

      6. fma-neg 72.3%

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

      7. associate-+l+ 72.3%

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

      8. pow2 72.3%

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

      9. *-commutative 72.3%

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

      10. fma-undefine 72.3%

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

      11. distribute-lft-neg-in 72.3%

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

      12. *-commutative 72.3%

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

      13. distribute-rgt-neg-in 72.3%

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

      14. fma-define 72.3%

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

      15. *-commutative 72.3%

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

      16. fma-undefine 72.3%

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

      17. distribute-lft-neg-in 72.3%

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

      18. *-commutative 72.3%

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

      19. distribute-rgt-neg-in 72.3%

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

   6. Applied egg-rr 72.3%

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

      1. associate-+l- 72.3%

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

      2. count-2 72.3%

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

   8. Simplified 72.3%

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

   9. Taylor expanded in a around inf 46.0%

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

       1. *-commutative 46.0%

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

       2. distribute-rgt1-in 46.0%

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

       3. metadata-eval 46.0%

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

       4. mul0-lft 46.0%

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

       5. metadata-eval 46.0%

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

       6. neg-sub0 46.0%

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

   11. Simplified 46.0%

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

   if 3.89999999999999994e-148 < b_2

   1. Initial program 18.5%

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

      1. +-commutative 18.5%

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

      2. unsub-neg 18.5%

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

   3. Simplified 18.5%

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

   5. Taylor expanded in b_2 around inf 85.1%

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

      1. associate-*r/ 85.1%

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

      2. *-commutative 85.1%

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

   7. Simplified 85.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{-141}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.9 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311) (* -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 <= -2e-311) {
		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 <= (-2d-311)) 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 <= -2e-311) {
		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 <= -2e-311:
		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 <= -2e-311)
		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 <= -2e-311)
		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, -2e-311], 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 < -1.9999999999999e-311

   1. Initial program 69.4%

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

      1. +-commutative 69.4%

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

      2. unsub-neg 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in b_2 around -inf 62.0%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 27.0%

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

      1. +-commutative 27.0%

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

      2. unsub-neg 27.0%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in b_2 around inf 71.8%

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

      1. associate-*r/ 71.8%

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

      2. *-commutative 71.8%

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

   7. Simplified 71.8%

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

Alternative 6: 68.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311) (* -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 <= -2e-311) {
		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 <= (-2d-311)) 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 <= -2e-311) {
		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 <= -2e-311:
		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 <= -2e-311)
		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 <= -2e-311)
		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, -2e-311], 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 < -1.9999999999999e-311

   1. Initial program 69.4%

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

      1. +-commutative 69.4%

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

      2. unsub-neg 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in b_2 around -inf 62.0%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 27.0%

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

      1. +-commutative 27.0%

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

      2. unsub-neg 27.0%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in c around 0 65.6%

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

      1. fma-neg 65.6%

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

      2. associate-/l* 67.3%

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

      3. associate-*r/ 67.3%

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

      4. metadata-eval 67.3%

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

   7. Simplified 67.3%

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

   8. Taylor expanded in a around 0 71.6%

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

Alternative 7: 43.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311) (* -2.0 (/ b_2 a)) 0.0))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-311)) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-311:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-311)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-311)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-311], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999e-311

   1. Initial program 69.4%

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

      1. +-commutative 69.4%

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

      2. unsub-neg 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in b_2 around -inf 62.0%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 27.0%

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

      1. +-commutative 27.0%

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

      2. unsub-neg 27.0%

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

   3. Simplified 27.0%

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

      1. div-sub 26.7%

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

      2. add-sqr-sqrt 25.6%

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

      3. associate-/l* 25.5%

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

      4. fma-neg 24.0%

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

      5. pow1/2 24.0%

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

      6. sqrt-pow1 24.0%

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

      7. pow2 24.0%

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

      8. metadata-eval 24.0%

         \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}, \frac{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a}, -\frac{b\_2}{a}\right) \]

      9. pow1/2 24.0%

         \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{\sqrt{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{0.5}}}}{a}, -\frac{b\_2}{a}\right) \]

      10. sqrt-pow1 24.0%

          \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}}}{a}, -\frac{b\_2}{a}\right) \]

      11. pow2 24.0%

          \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}}{a}, -\frac{b\_2}{a}\right) \]

      12. metadata-eval 24.0%

          \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}}{a}, -\frac{b\_2}{a}\right) \]

   6. Applied egg-rr 24.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, -\frac{b\_2}{a}\right)} \]
   7. Step-by-step derivation

      1. distribute-neg-frac2 24.0%

         \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, \color{blue}{\frac{b\_2}{-a}}\right) \]

   8. Simplified 24.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, \frac{b\_2}{-a}\right)} \]

   9. Taylor expanded in a around 0 23.0%

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

       1. distribute-rgt1-in 23.0%

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

       2. metadata-eval 23.0%

          \[\leadsto \frac{\color{blue}{0} \cdot b\_2}{a} \]

       3. mul0-lft 23.0%

          \[\leadsto \frac{\color{blue}{0}}{a} \]

       4. div0 23.0%

          \[\leadsto \color{blue}{0} \]

   11. Simplified 23.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 23.6% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (if (<= b_2 -2e-311) (/ (- b_2) a) 0.0))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-311)) then
        tmp = -b_2 / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-311:
		tmp = -b_2 / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-311)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-311)
		tmp = -b_2 / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-311], N[((-b$95$2) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999e-311

   1. Initial program 69.4%

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

      1. +-commutative 69.4%

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

      2. unsub-neg 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in b_2 around 0 38.1%

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

      1. associate-*r* 38.1%

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

      2. neg-mul-1 38.1%

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

      3. *-commutative 38.1%

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

   7. Simplified 38.1%

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

   8. Taylor expanded in b_2 around inf 22.7%

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

      1. associate-*r/ 22.7%

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

      2. neg-mul-1 22.7%

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

   10. Simplified 22.7%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 27.0%

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

      1. +-commutative 27.0%

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

      2. unsub-neg 27.0%

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

   3. Simplified 27.0%

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

      1. div-sub 26.7%

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

      2. add-sqr-sqrt 25.6%

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

      3. associate-/l* 25.5%

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

      4. fma-neg 24.0%

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

      5. pow1/2 24.0%

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

      6. sqrt-pow1 24.0%

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

      7. pow2 24.0%

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

      8. metadata-eval 24.0%

         \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}, \frac{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a}, -\frac{b\_2}{a}\right) \]

      9. pow1/2 24.0%

         \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{\sqrt{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{0.5}}}}{a}, -\frac{b\_2}{a}\right) \]

      10. sqrt-pow1 24.0%

          \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}}}{a}, -\frac{b\_2}{a}\right) \]

      11. pow2 24.0%

          \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}}{a}, -\frac{b\_2}{a}\right) \]

      12. metadata-eval 24.0%

          \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}}{a}, -\frac{b\_2}{a}\right) \]

   6. Applied egg-rr 24.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, -\frac{b\_2}{a}\right)} \]
   7. Step-by-step derivation

      1. distribute-neg-frac2 24.0%

         \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, \color{blue}{\frac{b\_2}{-a}}\right) \]

   8. Simplified 24.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, \frac{b\_2}{-a}\right)} \]

   9. Taylor expanded in a around 0 23.0%

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

       1. distribute-rgt1-in 23.0%

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

       2. metadata-eval 23.0%

          \[\leadsto \frac{\color{blue}{0} \cdot b\_2}{a} \]

       3. mul0-lft 23.0%

          \[\leadsto \frac{\color{blue}{0}}{a} \]

       4. div0 23.0%

          \[\leadsto \color{blue}{0} \]

   11. Simplified 23.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 11.2% accurate, 112.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 0.0)

C

double code(double a, double b_2, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0;
}

Python

def code(a, b_2, c):
	return 0.0

Julia

function code(a, b_2, c)
	return 0.0
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0;
end

Wolfram

code[a_, b$95$2_, c_] := 0.0

Derivation

1. Initial program 49.2%

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

   1. +-commutative 49.2%

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

   2. unsub-neg 49.2%

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

3. Simplified 49.2%

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

   1. div-sub 49.1%

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

   2. add-sqr-sqrt 48.4%

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

   3. associate-/l* 48.4%

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

   4. fma-neg 47.7%

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

   5. pow1/2 47.7%

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

   6. sqrt-pow1 47.7%

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

   7. pow2 47.7%

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

   8. metadata-eval 47.7%

      \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}, \frac{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a}, -\frac{b\_2}{a}\right) \]

   9. pow1/2 47.7%

      \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{\sqrt{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{0.5}}}}{a}, -\frac{b\_2}{a}\right) \]

   10. sqrt-pow1 47.7%

       \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}}}{a}, -\frac{b\_2}{a}\right) \]

   11. pow2 47.7%

       \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}}{a}, -\frac{b\_2}{a}\right) \]

   12. metadata-eval 47.7%

       \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}}{a}, -\frac{b\_2}{a}\right) \]

6. Applied egg-rr 47.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, -\frac{b\_2}{a}\right)} \]
7. Step-by-step derivation

   1. distribute-neg-frac2 47.7%

      \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, \color{blue}{\frac{b\_2}{-a}}\right) \]

8. Simplified 47.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, \frac{b\_2}{-a}\right)} \]

9. Taylor expanded in a around 0 12.5%

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

    1. distribute-rgt1-in 12.5%

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

    2. metadata-eval 12.5%

       \[\leadsto \frac{\color{blue}{0} \cdot b\_2}{a} \]

    3. mul0-lft 12.5%

       \[\leadsto \frac{\color{blue}{0}}{a} \]

    4. div0 12.5%

       \[\leadsto \color{blue}{0} \]

11. Simplified 12.5%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024135 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

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