quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.7% → 84.5%

Time: 12.0s

Alternatives: 9

Speedup: 15.9×

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: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -8.2 \cdot 10^{+165}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.2e+165)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 4.8e-95)
     (/ (- (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 <= -8.2e+165) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4.8e-95) {
		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 <= (-8.2d+165)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 4.8d-95) 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 <= -8.2e+165) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4.8e-95) {
		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 <= -8.2e+165:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 4.8e-95:
		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 <= -8.2e+165)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 4.8e-95)
		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 <= -8.2e+165)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 4.8e-95)
		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, -8.2e+165], 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, 4.8e-95], 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 < -8.2000000000000005e165

   1. Initial program 45.3%

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

      1. +-commutative 45.3%

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

      2. unsub-neg 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in b_2 around -inf 94.7%

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

   if -8.2000000000000005e165 < b_2 < 4.8e-95

   1. Initial program 84.9%

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

      1. +-commutative 84.9%

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

      2. unsub-neg 84.9%

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

   3. Simplified 84.9%

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

   if 4.8e-95 < b_2

   1. Initial program 16.4%

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

      1. +-commutative 16.4%

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

      2. unsub-neg 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in b_2 around inf 64.9%

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

   5. Taylor expanded in c around 0 87.0%

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

      1. associate-*r/ 87.0%

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

   7. Simplified 87.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8.2 \cdot 10^{+165}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -6.5 \cdot 10^{-76}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.5e-76)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 2.6e-95) (/ (- (sqrt (* c (- a))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.5e-76) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.6e-95) {
		tmp = (sqrt((c * -a)) - 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 <= (-6.5d-76)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 2.6d-95) then
        tmp = (sqrt((c * -a)) - 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 <= -6.5e-76) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.6e-95) {
		tmp = (Math.sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.5e-76:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 2.6e-95:
		tmp = (math.sqrt((c * -a)) - 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 <= -6.5e-76)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 2.6e-95)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(-a))) - 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 <= -6.5e-76)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 2.6e-95)
		tmp = (sqrt((c * -a)) - 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, -6.5e-76], 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, 2.6e-95], N[(N[(N[Sqrt[N[(c * (-a)), $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 < -6.5e-76

   1. Initial program 70.9%

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

      1. +-commutative 70.9%

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

      2. unsub-neg 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in b_2 around -inf 88.0%

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

   if -6.5e-76 < b_2 < 2.60000000000000001e-95

   1. Initial program 79.4%

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

      1. +-commutative 79.4%

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

      2. unsub-neg 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in b_2 around 0 75.8%

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

      1. mul-1-neg 75.8%

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

      2. distribute-rgt-neg-out 75.8%

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

   6. Simplified 75.8%

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

   if 2.60000000000000001e-95 < b_2

   1. Initial program 16.4%

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

      1. +-commutative 16.4%

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

      2. unsub-neg 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in b_2 around inf 64.9%

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

   5. Taylor expanded in c around 0 87.0%

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

      1. associate-*r/ 87.0%

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

   7. Simplified 87.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.5 \cdot 10^{-76}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3: 68.2% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-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 <= -2e-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 <= (-2d-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 <= -2e-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 <= -2e-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 <= -2e-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 <= -2e-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, -2e-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 < -1.999999999999994e-310

   1. Initial program 74.6%

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

      1. +-commutative 74.6%

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

      2. unsub-neg 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in b_2 around -inf 67.6%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 33.3%

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

      1. +-commutative 33.3%

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

      2. unsub-neg 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in b_2 around inf 48.3%

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

   5. Taylor expanded in c around 0 66.5%

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

      1. associate-*r/ 66.5%

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

   7. Simplified 66.5%

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

4. Final simplification 67.0%

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

Alternative 4: 48.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 6.8 \cdot 10^{-297}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 6.8e-297) (/ (- b_2) a) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 6.8e-297) {
		tmp = -b_2 / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 6.8d-297) then
        tmp = -b_2 / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 6.8e-297) {
		tmp = -b_2 / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 6.8e-297:
		tmp = -b_2 / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 6.8e-297)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 6.8e-297)
		tmp = -b_2 / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 6.8e-297], N[((-b$95$2) / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 6.79999999999999966e-297

   1. Initial program 74.8%

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

      1. +-commutative 74.8%

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

      2. unsub-neg 74.8%

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

   3. Simplified 74.8%

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

      1. add-cube-cbrt 74.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. sqrt-prod 74.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      3. pow2 74.4%

         \[\leadsto \frac{\sqrt{\color{blue}{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{2}}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}} - b_2}{a} \]

      4. fma-neg 74.4%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}}\right)}^{2}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}} - b_2}{a} \]

      5. *-commutative 74.4%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}} - b_2}{a} \]

      6. distribute-rgt-neg-in 74.4%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}} - b_2}{a} \]

      7. add-sqr-sqrt 74.4%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}}} - b_2}{a} \]

      8. cbrt-prod 74.4%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\color{blue}{\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}}}} - b_2}{a} \]

      9. cbrt-prod 74.4%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\color{blue}{\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}}} - b_2}{a} \]

      10. add-sqr-sqrt 74.4%

          \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{\color{blue}{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      11. fma-neg 74.4%

          \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{\color{blue}{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}}} - b_2}{a} \]

   5. Applied egg-rr 74.4%

      \[\leadsto \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow2 74.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      2. rem-sqrt-square 74.4%

         \[\leadsto \frac{\color{blue}{\left|\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right|} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      3. distribute-rgt-neg-out 74.4%

         \[\leadsto \frac{\left|\sqrt[3]{\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      4. *-commutative 74.4%

         \[\leadsto \frac{\left|\sqrt[3]{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      5. fma-neg 74.4%

         \[\leadsto \frac{\left|\sqrt[3]{\color{blue}{b_2 \cdot b_2 - a \cdot c}}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      6. *-commutative 74.4%

         \[\leadsto \frac{\left|\sqrt[3]{b_2 \cdot b_2 - \color{blue}{c \cdot a}}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      7. distribute-rgt-neg-out 74.4%

         \[\leadsto \frac{\left|\sqrt[3]{b_2 \cdot b_2 - c \cdot a}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)}} - b_2}{a} \]

      8. *-commutative 74.4%

         \[\leadsto \frac{\left|\sqrt[3]{b_2 \cdot b_2 - c \cdot a}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)}} - b_2}{a} \]

      9. fma-neg 74.4%

         \[\leadsto \frac{\left|\sqrt[3]{b_2 \cdot b_2 - c \cdot a}\right| \cdot \sqrt{\sqrt[3]{\color{blue}{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      10. *-commutative 74.4%

          \[\leadsto \frac{\left|\sqrt[3]{b_2 \cdot b_2 - c \cdot a}\right| \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - \color{blue}{c \cdot a}}} - b_2}{a} \]

   7. Simplified 74.4%

      \[\leadsto \frac{\color{blue}{\left|\sqrt[3]{b_2 \cdot b_2 - c \cdot a}\right| \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - c \cdot a}}} - b_2}{a} \]

   8. Taylor expanded in b_2 around inf 30.0%

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

      1. neg-mul-1 30.0%

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

   10. Simplified 30.0%

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

   if 6.79999999999999966e-297 < b_2

   1. Initial program 32.8%

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

      1. +-commutative 32.8%

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

      2. unsub-neg 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in b_2 around inf 67.0%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 6.8 \cdot 10^{-297}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 5: 67.8% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 6.8 \cdot 10^{-297}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 6.8e-297) (* b_2 (/ -2.0 a)) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 6.8e-297) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 6.8d-297) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 6.8e-297) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 6.8e-297:
		tmp = b_2 * (-2.0 / a)
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 6.8e-297)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 6.8e-297)
		tmp = b_2 * (-2.0 / a);
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 6.8e-297], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 6.79999999999999966e-297

   1. Initial program 74.8%

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

      1. +-commutative 74.8%

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

      2. unsub-neg 74.8%

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

   3. Simplified 74.8%

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

      1. div-sub 74.8%

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

      2. add-sqr-sqrt 74.6%

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

      3. *-un-lft-identity 74.6%

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

      4. times-frac 74.5%

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

      5. fma-neg 74.6%

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

   5. Applied egg-rr 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}}{1}, \frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}}{a}, -\frac{b_2}{a}\right)} \]

   6. Taylor expanded in b_2 around -inf 67.1%

      \[\leadsto \color{blue}{2 \cdot \left(-1 \cdot \frac{b_2}{a}\right) + 0.5 \cdot \frac{c}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r* 67.1%

         \[\leadsto \color{blue}{\left(2 \cdot -1\right) \cdot \frac{b_2}{a}} + 0.5 \cdot \frac{c}{b_2} \]

      2. metadata-eval 67.1%

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

      3. *-commutative 67.1%

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

   8. Simplified 67.1%

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

   9. Taylor expanded in b_2 around inf 66.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 66.8%

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

       2. associate-*l/ 66.6%

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

       3. *-commutative 66.6%

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

   11. Simplified 66.6%

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

   if 6.79999999999999966e-297 < b_2

   1. Initial program 32.8%

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

      1. +-commutative 32.8%

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

      2. unsub-neg 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in b_2 around inf 67.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 6.8 \cdot 10^{-297}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 6: 67.8% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 7.2 \cdot 10^{-297}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 7.2e-297) (* b_2 (/ -2.0 a)) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.2e-297) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 7.2d-297) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.2e-297) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 7.2e-297:
		tmp = b_2 * (-2.0 / a)
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 7.2e-297)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 7.2e-297)
		tmp = b_2 * (-2.0 / a);
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 7.2e-297], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 7.19999999999999988e-297

   1. Initial program 74.8%

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

      1. +-commutative 74.8%

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

      2. unsub-neg 74.8%

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

   3. Simplified 74.8%

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

      1. div-sub 74.8%

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

      2. add-sqr-sqrt 74.6%

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

      3. *-un-lft-identity 74.6%

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

      4. times-frac 74.5%

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

      5. fma-neg 74.6%

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

   5. Applied egg-rr 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}}{1}, \frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}}{a}, -\frac{b_2}{a}\right)} \]

   6. Taylor expanded in b_2 around -inf 67.1%

      \[\leadsto \color{blue}{2 \cdot \left(-1 \cdot \frac{b_2}{a}\right) + 0.5 \cdot \frac{c}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r* 67.1%

         \[\leadsto \color{blue}{\left(2 \cdot -1\right) \cdot \frac{b_2}{a}} + 0.5 \cdot \frac{c}{b_2} \]

      2. metadata-eval 67.1%

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

      3. *-commutative 67.1%

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

   8. Simplified 67.1%

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

   9. Taylor expanded in b_2 around inf 66.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 66.8%

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

       2. associate-*l/ 66.6%

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

       3. *-commutative 66.6%

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

   11. Simplified 66.6%

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

   if 7.19999999999999988e-297 < b_2

   1. Initial program 32.8%

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

      1. +-commutative 32.8%

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

      2. unsub-neg 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in b_2 around inf 48.7%

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

   5. Taylor expanded in c around 0 67.0%

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

      1. associate-*r/ 67.0%

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

   7. Simplified 67.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 7.2 \cdot 10^{-297}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 7: 68.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 8.4 \cdot 10^{-297}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 8.4e-297) (/ (* b_2 -2.0) a) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 8.4e-297) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 8.4d-297) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 8.4e-297) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 8.4e-297:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 8.4e-297)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 8.4e-297)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 8.4e-297], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 8.40000000000000054e-297

   1. Initial program 74.8%

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

      1. +-commutative 74.8%

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

      2. unsub-neg 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in b_2 around -inf 66.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 66.8%

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

   6. Simplified 66.8%

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

   if 8.40000000000000054e-297 < b_2

   1. Initial program 32.8%

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

      1. +-commutative 32.8%

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

      2. unsub-neg 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in b_2 around inf 48.7%

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

   5. Taylor expanded in c around 0 67.0%

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

      1. associate-*r/ 67.0%

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

   7. Simplified 67.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 8.4 \cdot 10^{-297}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 8: 23.8% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (if (<= b_2 -2e-310) (/ (- b_2) a) 0.0))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		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-310)) 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-310) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = -b_2 / a
	else:
		tmp = 0.0
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 74.6%

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

      1. +-commutative 74.6%

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

      2. unsub-neg 74.6%

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

   3. Simplified 74.6%

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

      1. add-cube-cbrt 74.1%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. sqrt-prod 74.2%

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt[3]{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      3. pow2 74.2%

         \[\leadsto \frac{\sqrt{\color{blue}{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{2}}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}} - b_2}{a} \]

      4. fma-neg 74.2%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}}\right)}^{2}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}} - b_2}{a} \]

      5. *-commutative 74.2%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}} - b_2}{a} \]

      6. distribute-rgt-neg-in 74.2%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}} - b_2}{a} \]

      7. add-sqr-sqrt 74.2%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}}} - b_2}{a} \]

      8. cbrt-prod 74.2%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\color{blue}{\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}}}} - b_2}{a} \]

      9. cbrt-prod 74.2%

         \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\color{blue}{\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}}} - b_2}{a} \]

      10. add-sqr-sqrt 74.2%

          \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{\color{blue}{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      11. fma-neg 74.2%

          \[\leadsto \frac{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{\color{blue}{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}}} - b_2}{a} \]

   5. Applied egg-rr 74.2%

      \[\leadsto \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow2 74.2%

         \[\leadsto \frac{\sqrt{\color{blue}{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      2. rem-sqrt-square 74.2%

         \[\leadsto \frac{\color{blue}{\left|\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}\right|} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      3. distribute-rgt-neg-out 74.2%

         \[\leadsto \frac{\left|\sqrt[3]{\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      4. *-commutative 74.2%

         \[\leadsto \frac{\left|\sqrt[3]{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      5. fma-neg 74.2%

         \[\leadsto \frac{\left|\sqrt[3]{\color{blue}{b_2 \cdot b_2 - a \cdot c}}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      6. *-commutative 74.2%

         \[\leadsto \frac{\left|\sqrt[3]{b_2 \cdot b_2 - \color{blue}{c \cdot a}}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} - b_2}{a} \]

      7. distribute-rgt-neg-out 74.2%

         \[\leadsto \frac{\left|\sqrt[3]{b_2 \cdot b_2 - c \cdot a}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)}} - b_2}{a} \]

      8. *-commutative 74.2%

         \[\leadsto \frac{\left|\sqrt[3]{b_2 \cdot b_2 - c \cdot a}\right| \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)}} - b_2}{a} \]

      9. fma-neg 74.2%

         \[\leadsto \frac{\left|\sqrt[3]{b_2 \cdot b_2 - c \cdot a}\right| \cdot \sqrt{\sqrt[3]{\color{blue}{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      10. *-commutative 74.2%

          \[\leadsto \frac{\left|\sqrt[3]{b_2 \cdot b_2 - c \cdot a}\right| \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - \color{blue}{c \cdot a}}} - b_2}{a} \]

   7. Simplified 74.2%

      \[\leadsto \frac{\color{blue}{\left|\sqrt[3]{b_2 \cdot b_2 - c \cdot a}\right| \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - c \cdot a}}} - b_2}{a} \]

   8. Taylor expanded in b_2 around inf 30.2%

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

      1. neg-mul-1 30.2%

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

   10. Simplified 30.2%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 33.3%

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

      1. +-commutative 33.3%

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

      2. unsub-neg 33.3%

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

   3. Simplified 33.3%

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

      1. div-sub 33.0%

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

      2. add-sqr-sqrt 31.8%

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

      3. *-un-lft-identity 31.8%

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

      4. times-frac 32.6%

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

      5. fma-neg 29.6%

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

   5. Applied egg-rr 29.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}}{1}, \frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}}{a}, -\frac{b_2}{a}\right)} \]

   6. Taylor expanded in b_2 around inf 10.4%

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

      1. distribute-rgt1-in 10.4%

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

      2. metadata-eval 10.4%

         \[\leadsto \color{blue}{0} \cdot \frac{b_2}{a} \]

      3. mul0-lft 13.1%

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

   8. Simplified 13.1%

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

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 11.3% 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 53.8%

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

   1. +-commutative 53.8%

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

   2. unsub-neg 53.8%

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

3. Simplified 53.8%

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

   1. div-sub 53.7%

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

   2. add-sqr-sqrt 52.9%

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

   3. *-un-lft-identity 52.9%

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

   4. times-frac 53.3%

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

   5. fma-neg 51.8%

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

5. Applied egg-rr 51.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}}{1}, \frac{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}}{a}, -\frac{b_2}{a}\right)} \]

6. Taylor expanded in b_2 around inf 6.5%

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

   1. distribute-rgt1-in 6.5%

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

   2. metadata-eval 6.5%

      \[\leadsto \color{blue}{0} \cdot \frac{b_2}{a} \]

   3. mul0-lft 8.0%

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

8. Simplified 8.0%

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

9. Final simplification 8.0%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))