quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.6% → 85.2%

Time: 13.3s

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: 51.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.55 \cdot 10^{-120}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2} + -2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.55e-120)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.5e+41)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* 0.5 (/ c b_2)) (* -2.0 (/ b_2 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.55e-120) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.5e+41) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (0.5 * (c / b_2)) + (-2.0 * (b_2 / a));
	}
	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 <= (-1.55d-120)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.5d+41) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (0.5d0 * (c / b_2)) + ((-2.0d0) * (b_2 / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.55e-120) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.5e+41) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (0.5 * (c / b_2)) + (-2.0 * (b_2 / a));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.55e-120:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.5e+41:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (0.5 * (c / b_2)) + (-2.0 * (b_2 / a))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.55e-120)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.5e+41)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(0.5 * Float64(c / b_2)) + Float64(-2.0 * Float64(b_2 / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.55e-120)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.5e+41)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (0.5 * (c / b_2)) + (-2.0 * (b_2 / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.55e-120], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.5e+41], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.5500000000000001e-120

   1. Initial program 23.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 83.9%

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

      1. associate-*r/ 83.9%

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

   5. Applied egg-rr 83.9%

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

   if -1.5500000000000001e-120 < b_2 < 1.4999999999999999e41

   1. Initial program 74.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 1.4999999999999999e41 < b_2

   1. Initial program 53.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 96.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.55 \cdot 10^{-120}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2} + -2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.55 \cdot 10^{-120}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{-120}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.55e-120)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6.2e-120) (/ (sqrt (* c (- a))) (- a)) (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.55e-120) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.2e-120) {
		tmp = sqrt((c * -a)) / -a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	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 <= (-1.55d-120)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 6.2d-120) then
        tmp = sqrt((c * -a)) / -a
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.55e-120) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.2e-120) {
		tmp = Math.sqrt((c * -a)) / -a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.55e-120:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 6.2e-120:
		tmp = math.sqrt((c * -a)) / -a
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.55e-120)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6.2e-120)
		tmp = Float64(sqrt(Float64(c * Float64(-a))) / Float64(-a));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.55e-120)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 6.2e-120)
		tmp = sqrt((c * -a)) / -a;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.55e-120], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6.2e-120], N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] / (-a)), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.5500000000000001e-120

   1. Initial program 23.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 83.9%

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

      1. associate-*r/ 83.9%

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

   5. Applied egg-rr 83.9%

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

   if -1.5500000000000001e-120 < b_2 < 6.20000000000000038e-120

   1. Initial program 77.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 77.2%

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

      1. add-cube-cbrt 76.6%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\sqrt[3]{a \cdot \left(\frac{{b\_2}^{2}}{a} - c\right)} \cdot \sqrt[3]{a \cdot \left(\frac{{b\_2}^{2}}{a} - c\right)}\right) \cdot \sqrt[3]{a \cdot \left(\frac{{b\_2}^{2}}{a} - c\right)}}}}{a} \]

      2. pow3 76.7%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{{\left(\sqrt[3]{a \cdot \left(\frac{{b\_2}^{2}}{a} - c\right)}\right)}^{3}}}}{a} \]

   5. Applied egg-rr 76.7%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{{\left(\sqrt[3]{a \cdot \left(\frac{{b\_2}^{2}}{a} - c\right)}\right)}^{3}}}}{a} \]

   6. Taylor expanded in a around -inf 0.0%

      \[\leadsto \frac{\color{blue}{\sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)} \cdot {\left(\sqrt{-1}\right)}^{2}}}{a} \]
   7. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)}}}{a} \]

      2. unpow2 0.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)}}{a} \]

      3. rem-square-sqrt 77.2%

         \[\leadsto \frac{\color{blue}{-1} \cdot \sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)}}{a} \]

      4. mul-1-neg 77.2%

         \[\leadsto \frac{\color{blue}{-\sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)}}}{a} \]

      5. rem-cube-cbrt 77.2%

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

      6. *-commutative 77.2%

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

      7. neg-mul-1 77.2%

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

   8. Simplified 77.2%

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

   if 6.20000000000000038e-120 < b_2

   1. Initial program 57.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 84.1%

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

      1. *-commutative 84.1%

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

   5. Simplified 84.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.55 \cdot 10^{-120}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{-120}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.1 \cdot 10^{-123}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.7 \cdot 10^{-89}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2} + -2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.1e-123)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.7e-89)
     (- (sqrt (/ c (- a))))
     (+ (* 0.5 (/ c b_2)) (* -2.0 (/ b_2 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.1e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.7e-89) {
		tmp = -sqrt((c / -a));
	} else {
		tmp = (0.5 * (c / b_2)) + (-2.0 * (b_2 / a));
	}
	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 <= (-5.1d-123)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.7d-89) then
        tmp = -sqrt((c / -a))
    else
        tmp = (0.5d0 * (c / b_2)) + ((-2.0d0) * (b_2 / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.1e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.7e-89) {
		tmp = -Math.sqrt((c / -a));
	} else {
		tmp = (0.5 * (c / b_2)) + (-2.0 * (b_2 / a));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.1e-123:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.7e-89:
		tmp = -math.sqrt((c / -a))
	else:
		tmp = (0.5 * (c / b_2)) + (-2.0 * (b_2 / a))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.1e-123)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.7e-89)
		tmp = Float64(-sqrt(Float64(c / Float64(-a))));
	else
		tmp = Float64(Float64(0.5 * Float64(c / b_2)) + Float64(-2.0 * Float64(b_2 / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.1e-123)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.7e-89)
		tmp = -sqrt((c / -a));
	else
		tmp = (0.5 * (c / b_2)) + (-2.0 * (b_2 / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.1e-123], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.7e-89], (-N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision]), N[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.1000000000000001e-123

   1. Initial program 24.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 83.1%

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

      1. associate-*r/ 83.2%

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

   5. Applied egg-rr 83.2%

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

   if -5.1000000000000001e-123 < b_2 < 1.7e-89

   1. Initial program 71.7%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 71.7%

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

      1. add-cube-cbrt 71.2%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\sqrt[3]{a \cdot \left(\frac{{b\_2}^{2}}{a} - c\right)} \cdot \sqrt[3]{a \cdot \left(\frac{{b\_2}^{2}}{a} - c\right)}\right) \cdot \sqrt[3]{a \cdot \left(\frac{{b\_2}^{2}}{a} - c\right)}}}}{a} \]

      2. pow3 71.2%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{{\left(\sqrt[3]{a \cdot \left(\frac{{b\_2}^{2}}{a} - c\right)}\right)}^{3}}}}{a} \]

   5. Applied egg-rr 71.2%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{{\left(\sqrt[3]{a \cdot \left(\frac{{b\_2}^{2}}{a} - c\right)}\right)}^{3}}}}{a} \]

   6. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{\sqrt{\frac{c \cdot {\left(\sqrt[3]{-1}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-1}\right)}^{3}}{a}}} \]

      2. unpow2 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-1}\right)}^{3}}{a}} \]

      3. rem-square-sqrt 34.7%

         \[\leadsto \color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-1}\right)}^{3}}{a}} \]

      4. neg-mul-1 34.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{c \cdot {\left(\sqrt[3]{-1}\right)}^{3}}{a}}} \]

      5. rem-cube-cbrt 34.7%

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

      6. *-commutative 34.7%

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

      7. neg-mul-1 34.7%

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

   8. Simplified 34.7%

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

   if 1.7e-89 < b_2

   1. Initial program 59.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 87.9%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.1 \cdot 10^{-123}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.7 \cdot 10^{-89}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2} + -2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.8% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (0.5d0 * (c / b_2)) + ((-2.0d0) * (b_2 / a))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (0.5 * (c / b_2)) + (-2.0 * (b_2 / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 35.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 68.6%

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

      1. associate-*r/ 68.7%

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

   5. Applied egg-rr 68.7%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 61.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 69.5%

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

4. Final simplification 69.0%

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

Alternative 5: 68.6% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 35.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 68.6%

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

      1. associate-*r/ 68.7%

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

   5. Applied egg-rr 68.7%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 61.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 69.4%

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

      1. *-commutative 69.4%

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

   5. Simplified 69.4%

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

4. Final simplification 69.0%

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

Alternative 6: 68.6% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 35.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 68.6%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 61.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 69.4%

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

      1. *-commutative 69.4%

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

   5. Simplified 69.4%

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

4. Final simplification 69.0%

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

Alternative 7: 49.0% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = b_2 / -a
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 35.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 68.6%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 61.0%

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

      1. add-sqr-sqrt 60.8%

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

      2. pow2 60.8%

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

      3. pow1/2 60.8%

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

      4. sqrt-pow1 60.8%

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

      5. sub-neg 60.8%

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

      6. +-commutative 60.8%

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

      7. distribute-rgt-neg-in 60.8%

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

      8. fma-define 60.9%

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

      9. pow2 60.9%

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

      10. metadata-eval 60.9%

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

   4. Applied egg-rr 60.9%

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

   5. Taylor expanded in b_2 around 0 32.7%

      \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(-1 \cdot \left(a \cdot c\right)\right)}^{0.25}\right)}}^{2}}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 32.7%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\color{blue}{\left(\left(-1 \cdot a\right) \cdot c\right)}}^{0.25}\right)}^{2}}{a} \]

      2. mul-1-neg 32.7%

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

   7. Simplified 32.7%

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

   8. Taylor expanded in b_2 around inf 23.9%

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

      1. associate-*r/ 23.9%

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

      2. mul-1-neg 23.9%

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

   10. Simplified 23.9%

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

4. Final simplification 46.8%

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

Alternative 8: 15.8% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.2%

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

   1. add-sqr-sqrt 46.1%

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

   2. pow2 46.1%

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

   3. pow1/2 46.1%

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

   4. sqrt-pow1 46.1%

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

   5. sub-neg 46.1%

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

   6. +-commutative 46.1%

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

   7. distribute-rgt-neg-in 46.1%

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

   8. fma-define 46.1%

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

   9. pow2 46.1%

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

   10. metadata-eval 46.1%

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

4. Applied egg-rr 46.1%

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

5. Taylor expanded in b_2 around 0 30.3%

   \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(-1 \cdot \left(a \cdot c\right)\right)}^{0.25}\right)}}^{2}}{a} \]
6. Step-by-step derivation

   1. associate-*r* 30.3%

      \[\leadsto \frac{\left(-b\_2\right) - {\left({\color{blue}{\left(\left(-1 \cdot a\right) \cdot c\right)}}^{0.25}\right)}^{2}}{a} \]

   2. mul-1-neg 30.3%

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

7. Simplified 30.3%

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

8. Taylor expanded in b_2 around inf 13.0%

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

   1. associate-*r/ 13.0%

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

   2. mul-1-neg 13.0%

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

10. Simplified 13.0%

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

11. Final simplification 13.0%

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

Alternative 9: 2.5% accurate, 37.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.2%

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

   1. add-sqr-sqrt 46.1%

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

   2. pow2 46.1%

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

   3. pow1/2 46.1%

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

   4. sqrt-pow1 46.1%

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

   5. sub-neg 46.1%

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

   6. +-commutative 46.1%

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

   7. distribute-rgt-neg-in 46.1%

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

   8. fma-define 46.1%

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

   9. pow2 46.1%

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

   10. metadata-eval 46.1%

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

4. Applied egg-rr 46.1%

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

5. Taylor expanded in b_2 around 0 30.3%

   \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(-1 \cdot \left(a \cdot c\right)\right)}^{0.25}\right)}}^{2}}{a} \]
6. Step-by-step derivation

   1. associate-*r* 30.3%

      \[\leadsto \frac{\left(-b\_2\right) - {\left({\color{blue}{\left(\left(-1 \cdot a\right) \cdot c\right)}}^{0.25}\right)}^{2}}{a} \]

   2. mul-1-neg 30.3%

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

7. Simplified 30.3%

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

8. Taylor expanded in b_2 around inf 13.0%

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

   1. associate-*r/ 13.0%

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

   2. mul-1-neg 13.0%

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

10. Simplified 13.0%

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

    1. div-inv 13.0%

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

    2. add-sqr-sqrt 1.4%

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

    3. sqrt-unprod 1.9%

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

    4. sqr-neg 1.9%

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

    5. sqrt-prod 0.7%

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

    6. add-sqr-sqrt 2.4%

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

12. Applied egg-rr 2.4%

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

    1. associate-*r/ 2.4%

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

    2. *-rgt-identity 2.4%

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

14. Simplified 2.4%

    \[\leadsto \color{blue}{\frac{b\_2}{a}} \]
15. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

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

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) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

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 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	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 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	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 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	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(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	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 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	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[(c / N[(t$95$1 - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + t$95$1), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024165 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :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) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

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