quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.3% → 85.6%

Time: 12.6s

Alternatives: 8

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.3% 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: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-54}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(c \cdot \frac{a}{b\_2}\right) - b\_2 \cdot 2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e-54)
   (* -0.5 (/ c b_2))
   (if (<= b_2 5e+127)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (- (* 0.5 (* c (/ a b_2))) (* b_2 2.0)) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e-54) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 5e+127) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((0.5 * (c * (a / b_2))) - (b_2 * 2.0)) / 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 <= (-2.2d-54)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 5d+127) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((0.5d0 * (c * (a / b_2))) - (b_2 * 2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e-54) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 5e+127) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((0.5 * (c * (a / b_2))) - (b_2 * 2.0)) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.2e-54:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 5e+127:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = ((0.5 * (c * (a / b_2))) - (b_2 * 2.0)) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e-54)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 5e+127)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(c * Float64(a / b_2))) - Float64(b_2 * 2.0)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.2e-54)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 5e+127)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = ((0.5 * (c * (a / b_2))) - (b_2 * 2.0)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.2e-54], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 5e+127], 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[(N[(0.5 * N[(c * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 * 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.2e-54

   1. Initial program 14.4%

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

   3. Taylor expanded in b_2 around -inf 91.9%

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

   if -2.2e-54 < b_2 < 5.0000000000000004e127

   1. Initial program 83.5%

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

   if 5.0000000000000004e127 < b_2

   1. Initial program 54.3%

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

   3. Taylor expanded in a around 0 91.1%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{a \cdot c}{b\_2} - 2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. associate-/l* 97.7%

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

   5. Applied egg-rr 97.7%

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

      1. associate-*r/ 91.1%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\frac{a \cdot c}{b\_2}} - 2 \cdot b\_2}{a} \]

      2. *-commutative 91.1%

         \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{c \cdot a}}{b\_2} - 2 \cdot b\_2}{a} \]

      3. associate-*r/ 97.7%

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

   7. Simplified 97.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-54}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(c \cdot \frac{a}{b\_2}\right) - b\_2 \cdot 2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-54}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-54)
   (* -0.5 (/ c b_2))
   (if (<= b_2 4.8e-6)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-54) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 4.8e-6) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((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 <= (-2.8d-54)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 4.8d-6) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((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 <= -2.8e-54) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 4.8e-6) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-54:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 4.8e-6:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-54)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 4.8e-6)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + 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 <= -2.8e-54)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 4.8e-6)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-54], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.8e-6], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.8000000000000002e-54

   1. Initial program 14.4%

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

   3. Taylor expanded in b_2 around -inf 91.9%

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

   if -2.8000000000000002e-54 < b_2 < 4.7999999999999998e-6

   1. Initial program 81.0%

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

   3. Taylor expanded in b_2 around 0 71.5%

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

      1. associate-*r* 71.5%

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

      2. neg-mul-1 71.5%

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

   5. Simplified 71.5%

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

   if 4.7999999999999998e-6 < b_2

   1. Initial program 68.8%

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

   3. Taylor expanded in c around 0 87.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-54}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 7.0× speedup

Math

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

FPCore

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

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)) + ((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 <= (-5d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((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 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	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)) + ((c / b_2) * 0.5)
	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(Float64(-2.0 * Float64(b_2 / a)) + 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 <= -5e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	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[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 28.5%

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

   3. Taylor expanded in b_2 around -inf 73.1%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 76.7%

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

   3. Taylor expanded in c around 0 63.4%

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

4. Final simplification 68.4%

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

Alternative 4: 67.8% accurate, 11.2× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (* -0.5 (/ c b_2)) (/ (* b_2 -2.0) 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 * -2.0) / 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 * (-2.0d0)) / 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 * -2.0) / 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 * -2.0) / 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(Float64(b_2 * -2.0) / 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 * -2.0) / 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[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 28.5%

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

   3. Taylor expanded in b_2 around -inf 73.1%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 76.7%

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

   3. Taylor expanded in b_2 around inf 62.8%

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

      1. associate-*r/ 62.8%

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

      2. *-commutative 62.8%

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

   5. Simplified 62.8%

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

Alternative 5: 36.5% accurate, 11.2× speedup

Math

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

FPCore

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

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 * (c * 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 <= (-5d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = b_2 * (c * 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 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = b_2 * (c * 0.5);
	}
	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 * (c * 0.5)
	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(c * 0.5));
	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 * (c * 0.5);
	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 * N[(c * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 28.5%

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

   3. Taylor expanded in b_2 around -inf 73.1%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 76.7%

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

      1. div-sub 76.7%

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

      2. sub-neg 76.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 32.4%

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

      5. sqr-neg 32.4%

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

      6. sqrt-prod 33.6%

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

      7. add-sqr-sqrt 33.1%

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

      8. sub-neg 33.1%

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

      9. add-sqr-sqrt 32.6%

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

      10. hypot-define 32.4%

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

      11. distribute-rgt-neg-in 32.4%

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

   4. Applied egg-rr 32.4%

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

      1. sub-neg 32.4%

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

      2. div-sub 32.7%

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

      3. distribute-rgt-neg-out 32.7%

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

      4. distribute-lft-neg-in 32.7%

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

   6. Simplified 32.7%

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

   7. Taylor expanded in b_2 around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 3.7%

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

      5. neg-mul-1 3.7%

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

      6. distribute-rgt-neg-in 3.7%

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

      7. distribute-lft-neg-in 3.7%

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

      8. metadata-eval 3.7%

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

      9. associate-*r/ 3.7%

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

   9. Simplified 3.7%

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

   11. Applied egg-rr 7.0%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{0.5 \cdot \left(c \cdot b\_2\right)}\right)}^{3}} \]
   12. Step-by-step derivation

       1. rem-cube-cbrt 7.0%

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

       2. associate-*r* 7.0%

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

       3. *-commutative 7.0%

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

   13. Simplified 7.0%

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

4. Final simplification 41.6%

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

Alternative 6: 34.3% accurate, 22.4× speedup

Math

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

FPCore

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

C

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

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 = (-0.5d0) * (c / b_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in b_2 around -inf 39.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
4. Add Preprocessing

Alternative 7: 6.1% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \frac{a}{b\_2} \cdot -2 \end{array} \]

FPCore

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

C

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

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 = (a / b_2) * (-2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

   1. add-cbrt-cube 34.1%

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

   2. pow3 34.1%

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

   3. add-sqr-sqrt 8.4%

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

   4. sqrt-unprod 19.3%

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

   5. sqr-neg 19.3%

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

   6. sqrt-prod 17.8%

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

   7. add-sqr-sqrt 25.1%

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

   8. sub-neg 25.1%

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

   9. add-sqr-sqrt 24.3%

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

   10. hypot-define 17.3%

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

   11. distribute-rgt-neg-in 17.3%

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

4. Applied egg-rr 17.3%

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

5. Taylor expanded in b_2 around -inf 2.5%

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2 \cdot \sqrt[3]{-1}}{a}} \]
6. Step-by-step derivation

   1. associate-/l* 2.5%

      \[\leadsto -2 \cdot \color{blue}{\left(b\_2 \cdot \frac{\sqrt[3]{-1}}{a}\right)} \]

7. Simplified 2.5%

   \[\leadsto \color{blue}{-2 \cdot \left(b\_2 \cdot \frac{\sqrt[3]{-1}}{a}\right)} \]

9. Applied egg-rr 7.4%

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

10. Final simplification 7.4%

    \[\leadsto \frac{a}{b\_2} \cdot -2 \]
11. Add Preprocessing

Alternative 8: 3.4% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(b\_2 \cdot a\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-2.0d0) * (b_2 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

   1. add-cbrt-cube 34.1%

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

   2. pow3 34.1%

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

   3. add-sqr-sqrt 8.4%

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

   4. sqrt-unprod 19.3%

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

   5. sqr-neg 19.3%

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

   6. sqrt-prod 17.8%

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

   7. add-sqr-sqrt 25.1%

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

   8. sub-neg 25.1%

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

   9. add-sqr-sqrt 24.3%

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

   10. hypot-define 17.3%

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

   11. distribute-rgt-neg-in 17.3%

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

4. Applied egg-rr 17.3%

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

5. Taylor expanded in b_2 around -inf 2.5%

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2 \cdot \sqrt[3]{-1}}{a}} \]
6. Step-by-step derivation

   1. associate-/l* 2.5%

      \[\leadsto -2 \cdot \color{blue}{\left(b\_2 \cdot \frac{\sqrt[3]{-1}}{a}\right)} \]

7. Simplified 2.5%

   \[\leadsto \color{blue}{-2 \cdot \left(b\_2 \cdot \frac{\sqrt[3]{-1}}{a}\right)} \]

9. Applied egg-rr 3.5%

   \[\leadsto -2 \cdot \left(b\_2 \cdot \color{blue}{\left({\left(\sqrt[3]{a}\right)}^{2} \cdot \sqrt[3]{a}\right)}\right) \]
10. Step-by-step derivation

    1. unpow2 3.5%

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

    2. rem-3cbrt-lft 3.5%

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

11. Simplified 3.5%

    \[\leadsto -2 \cdot \left(b\_2 \cdot \color{blue}{a}\right) \]
12. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

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 2024177 
(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))