quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.0% → 83.9%

Time: 12.0s

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: 52.0% 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: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5 \cdot c}{b\_2}\\ \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq -3.8 \cdot 10^{-53}:\\ \;\;\;\;\left(b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right)\right) \cdot \frac{-1}{a}\\ \mathbf{elif}\;b\_2 \leq -2.5 \cdot 10^{-144}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq 7.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (/ (* -0.5 c) b_2)))
   (if (<= b_2 -1.8e-10)
     t_0
     (if (<= b_2 -3.8e-53)
       (* (+ b_2 (hypot b_2 (sqrt (* c (- a))))) (/ -1.0 a))
       (if (<= b_2 -2.5e-144)
         t_0
         (if (<= b_2 7.4e+88)
           (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
           (/ (* b_2 -2.0) a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = (-0.5 * c) / b_2;
	double tmp;
	if (b_2 <= -1.8e-10) {
		tmp = t_0;
	} else if (b_2 <= -3.8e-53) {
		tmp = (b_2 + hypot(b_2, sqrt((c * -a)))) * (-1.0 / a);
	} else if (b_2 <= -2.5e-144) {
		tmp = t_0;
	} else if (b_2 <= 7.4e+88) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = (-0.5 * c) / b_2;
	double tmp;
	if (b_2 <= -1.8e-10) {
		tmp = t_0;
	} else if (b_2 <= -3.8e-53) {
		tmp = (b_2 + Math.hypot(b_2, Math.sqrt((c * -a)))) * (-1.0 / a);
	} else if (b_2 <= -2.5e-144) {
		tmp = t_0;
	} else if (b_2 <= 7.4e+88) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = (-0.5 * c) / b_2
	tmp = 0
	if b_2 <= -1.8e-10:
		tmp = t_0
	elif b_2 <= -3.8e-53:
		tmp = (b_2 + math.hypot(b_2, math.sqrt((c * -a)))) * (-1.0 / a)
	elif b_2 <= -2.5e-144:
		tmp = t_0
	elif b_2 <= 7.4e+88:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(Float64(-0.5 * c) / b_2)
	tmp = 0.0
	if (b_2 <= -1.8e-10)
		tmp = t_0;
	elseif (b_2 <= -3.8e-53)
		tmp = Float64(Float64(b_2 + hypot(b_2, sqrt(Float64(c * Float64(-a))))) * Float64(-1.0 / a));
	elseif (b_2 <= -2.5e-144)
		tmp = t_0;
	elseif (b_2 <= 7.4e+88)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = (-0.5 * c) / b_2;
	tmp = 0.0;
	if (b_2 <= -1.8e-10)
		tmp = t_0;
	elseif (b_2 <= -3.8e-53)
		tmp = (b_2 + hypot(b_2, sqrt((c * -a)))) * (-1.0 / a);
	elseif (b_2 <= -2.5e-144)
		tmp = t_0;
	elseif (b_2 <= 7.4e+88)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]}, If[LessEqual[b$95$2, -1.8e-10], t$95$0, If[LessEqual[b$95$2, -3.8e-53], N[(N[(b$95$2 + N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -2.5e-144], t$95$0, If[LessEqual[b$95$2, 7.4e+88], 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[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.8e-10 or -3.7999999999999998e-53 < b_2 < -2.4999999999999999e-144

   1. Initial program 17.9%

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

   3. Taylor expanded in b_2 around -inf 88.4%

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

      1. associate-*r/ 88.4%

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

   5. Simplified 88.4%

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

   if -1.8e-10 < b_2 < -3.7999999999999998e-53

   1. Initial program 83.2%

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

      1. frac-2neg 83.2%

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

      2. div-inv 83.3%

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

   4. Applied egg-rr 83.3%

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

   if -2.4999999999999999e-144 < b_2 < 7.39999999999999988e88

   1. Initial program 84.3%

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

   if 7.39999999999999988e88 < b_2

   1. Initial program 66.2%

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

   3. Taylor expanded in b_2 around inf 96.8%

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

      1. *-commutative 96.8%

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

   5. Simplified 96.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -3.8 \cdot 10^{-53}:\\ \;\;\;\;\left(b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right)\right) \cdot \frac{-1}{a}\\ \mathbf{elif}\;b\_2 \leq -2.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5 \cdot c}{b\_2}\\ t_1 := \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{if}\;b\_2 \leq -1.95 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq -2.6 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b\_2 \leq -2.9 \cdot 10^{-142}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq 2.6 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (/ (* -0.5 c) b_2))
        (t_1 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)))
   (if (<= b_2 -1.95e-10)
     t_0
     (if (<= b_2 -2.6e-53)
       t_1
       (if (<= b_2 -2.9e-142)
         t_0
         (if (<= b_2 2.6e+88) t_1 (/ (* b_2 -2.0) a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = (-0.5 * c) / b_2;
	double t_1 = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	double tmp;
	if (b_2 <= -1.95e-10) {
		tmp = t_0;
	} else if (b_2 <= -2.6e-53) {
		tmp = t_1;
	} else if (b_2 <= -2.9e-142) {
		tmp = t_0;
	} else if (b_2 <= 2.6e+88) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-0.5d0) * c) / b_2
    t_1 = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    if (b_2 <= (-1.95d-10)) then
        tmp = t_0
    else if (b_2 <= (-2.6d-53)) then
        tmp = t_1
    else if (b_2 <= (-2.9d-142)) then
        tmp = t_0
    else if (b_2 <= 2.6d+88) then
        tmp = t_1
    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 t_0 = (-0.5 * c) / b_2;
	double t_1 = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	double tmp;
	if (b_2 <= -1.95e-10) {
		tmp = t_0;
	} else if (b_2 <= -2.6e-53) {
		tmp = t_1;
	} else if (b_2 <= -2.9e-142) {
		tmp = t_0;
	} else if (b_2 <= 2.6e+88) {
		tmp = t_1;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = (-0.5 * c) / b_2
	t_1 = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	tmp = 0
	if b_2 <= -1.95e-10:
		tmp = t_0
	elif b_2 <= -2.6e-53:
		tmp = t_1
	elif b_2 <= -2.9e-142:
		tmp = t_0
	elif b_2 <= 2.6e+88:
		tmp = t_1
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(Float64(-0.5 * c) / b_2)
	t_1 = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a)
	tmp = 0.0
	if (b_2 <= -1.95e-10)
		tmp = t_0;
	elseif (b_2 <= -2.6e-53)
		tmp = t_1;
	elseif (b_2 <= -2.9e-142)
		tmp = t_0;
	elseif (b_2 <= 2.6e+88)
		tmp = t_1;
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = (-0.5 * c) / b_2;
	t_1 = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	tmp = 0.0;
	if (b_2 <= -1.95e-10)
		tmp = t_0;
	elseif (b_2 <= -2.6e-53)
		tmp = t_1;
	elseif (b_2 <= -2.9e-142)
		tmp = t_0;
	elseif (b_2 <= 2.6e+88)
		tmp = t_1;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[b$95$2, -1.95e-10], t$95$0, If[LessEqual[b$95$2, -2.6e-53], t$95$1, If[LessEqual[b$95$2, -2.9e-142], t$95$0, If[LessEqual[b$95$2, 2.6e+88], t$95$1, N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.95e-10 or -2.59999999999999996e-53 < b_2 < -2.8999999999999999e-142

   1. Initial program 17.9%

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

   3. Taylor expanded in b_2 around -inf 88.4%

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

      1. associate-*r/ 88.4%

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

   5. Simplified 88.4%

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

   if -1.95e-10 < b_2 < -2.59999999999999996e-53 or -2.8999999999999999e-142 < b_2 < 2.6000000000000001e88

   1. Initial program 84.2%

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

   if 2.6000000000000001e88 < b_2

   1. Initial program 66.2%

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

   3. Taylor expanded in b_2 around inf 96.8%

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

      1. *-commutative 96.8%

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

   5. Simplified 96.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.95 \cdot 10^{-10}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -2.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{elif}\;b\_2 \leq -2.9 \cdot 10^{-142}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.6 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.6 \cdot 10^{-142}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.65 \cdot 10^{-20}:\\ \;\;\;\;\frac{b\_2 + \sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.6e-142)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.65e-20)
     (/ (+ b_2 (sqrt (* c (- a)))) (- a))
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.6e-142) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.65e-20) {
		tmp = (b_2 + sqrt((c * -a))) / -a;
	} 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 <= (-1.6d-142)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.65d-20) then
        tmp = (b_2 + sqrt((c * -a))) / -a
    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 <= -1.6e-142) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.65e-20) {
		tmp = (b_2 + Math.sqrt((c * -a))) / -a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.6e-142:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.65e-20:
		tmp = (b_2 + math.sqrt((c * -a))) / -a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.6e-142)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.65e-20)
		tmp = Float64(Float64(b_2 + sqrt(Float64(c * Float64(-a)))) / Float64(-a));
	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 <= -1.6e-142)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.65e-20)
		tmp = (b_2 + sqrt((c * -a))) / -a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.6e-142], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.65e-20], N[(N[(b$95$2 + N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.5999999999999999e-142

   1. Initial program 21.3%

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

   3. Taylor expanded in b_2 around -inf 84.3%

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

      1. associate-*r/ 84.3%

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

   5. Simplified 84.3%

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

   if -1.5999999999999999e-142 < b_2 < 1.65e-20

   1. Initial program 84.4%

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

   3. Taylor expanded in b_2 around 0 74.8%

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

      1. mul-1-neg 74.8%

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

      2. distribute-rgt-neg-out 74.8%

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

   5. Simplified 74.8%

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

   if 1.65e-20 < b_2

   1. Initial program 70.3%

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

   3. Taylor expanded in b_2 around inf 90.3%

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

      1. *-commutative 90.3%

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

   5. Simplified 90.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.6 \cdot 10^{-142}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.65 \cdot 10^{-20}:\\ \;\;\;\;\frac{b\_2 + \sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-296}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.1e-296) (* c (/ -0.5 b_2)) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.1e-296) {
		tmp = c * (-0.5 / 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 <= (-3.1d-296)) then
        tmp = c * ((-0.5d0) / 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 <= -3.1e-296) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.1e-296:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.1e-296)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.1e-296)
		tmp = c * (-0.5 / 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, -3.1e-296], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.1000000000000002e-296

   1. Initial program 27.5%

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

      1. frac-2neg 27.5%

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

      2. div-inv 27.5%

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

   4. Applied egg-rr 33.9%

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

   5. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b\_2}} \]
   6. 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 76.5%

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

      5. associate-*r* 76.5%

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

      6. metadata-eval 76.5%

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

      7. *-commutative 76.5%

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

      8. associate-/l* 76.3%

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

   7. Simplified 76.3%

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

   if -3.1000000000000002e-296 < b_2

   1. Initial program 76.4%

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

      1. frac-2neg 76.4%

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

      2. div-inv 76.4%

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

   4. Applied egg-rr 59.6%

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

   5. Taylor expanded in b_2 around inf 63.0%

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

      1. *-commutative 63.0%

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

   7. Simplified 63.0%

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

   8. Taylor expanded in b_2 around 0 63.1%

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

      1. metadata-eval 63.1%

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

      2. distribute-lft-neg-in 63.1%

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

      3. associate-*r/ 63.1%

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

      4. *-commutative 63.1%

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

      5. associate-*r/ 63.0%

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

      6. distribute-rgt-neg-in 63.0%

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

      7. distribute-neg-frac 63.0%

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

      8. metadata-eval 63.0%

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

   10. Simplified 63.0%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-296}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-296}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.1e-296) (/ (* -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 <= -3.1e-296) {
		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 <= (-3.1d-296)) 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 <= -3.1e-296) {
		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 <= -3.1e-296:
		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 <= -3.1e-296)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.1e-296)
		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, -3.1e-296], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.1000000000000002e-296

   1. Initial program 27.5%

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

   3. Taylor expanded in b_2 around -inf 76.5%

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

      1. associate-*r/ 76.5%

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

   5. Simplified 76.5%

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

   if -3.1000000000000002e-296 < b_2

   1. Initial program 76.4%

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

      1. frac-2neg 76.4%

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

      2. div-inv 76.4%

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

   4. Applied egg-rr 59.6%

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

   5. Taylor expanded in b_2 around inf 63.0%

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

      1. *-commutative 63.0%

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

   7. Simplified 63.0%

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

   8. Taylor expanded in b_2 around 0 63.1%

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

      1. metadata-eval 63.1%

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

      2. distribute-lft-neg-in 63.1%

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

      3. associate-*r/ 63.1%

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

      4. *-commutative 63.1%

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

      5. associate-*r/ 63.0%

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

      6. distribute-rgt-neg-in 63.0%

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

      7. distribute-neg-frac 63.0%

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

      8. metadata-eval 63.0%

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

   10. Simplified 63.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-296}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-296}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.1e-296) (/ (* -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 <= -3.1e-296) {
		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 <= (-3.1d-296)) 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 <= -3.1e-296) {
		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 <= -3.1e-296:
		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 <= -3.1e-296)
		tmp = Float64(Float64(-0.5 * 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 <= -3.1e-296)
		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, -3.1e-296], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.1000000000000002e-296

   1. Initial program 27.5%

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

   3. Taylor expanded in b_2 around -inf 76.5%

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

      1. associate-*r/ 76.5%

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

   5. Simplified 76.5%

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

   if -3.1000000000000002e-296 < b_2

   1. Initial program 76.4%

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

   3. Taylor expanded in b_2 around inf 63.1%

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

      1. *-commutative 63.1%

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

   5. Simplified 63.1%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-296}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.0% accurate, 22.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

   1. frac-2neg 51.6%

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

   2. div-inv 51.5%

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

4. Applied egg-rr 46.6%

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

5. Taylor expanded in b_2 around inf 32.3%

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

   1. *-commutative 32.3%

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

7. Simplified 32.3%

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

8. Taylor expanded in b_2 around 0 32.3%

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

   1. metadata-eval 32.3%

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

   2. distribute-lft-neg-in 32.3%

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

   3. associate-*r/ 32.3%

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

   4. *-commutative 32.3%

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

   5. associate-*r/ 32.3%

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

   6. distribute-rgt-neg-in 32.3%

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

   7. distribute-neg-frac 32.3%

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

   8. metadata-eval 32.3%

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

10. Simplified 32.3%

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

11. Final simplification 32.3%

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

Alternative 8: 15.0% 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 51.6%

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

   1. add-sqr-sqrt 49.9%

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

   2. pow2 49.9%

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

   3. pow1/2 49.9%

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

   4. sqrt-pow1 49.9%

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

   5. pow2 49.9%

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

   6. metadata-eval 49.9%

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

4. Applied egg-rr 49.9%

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

5. Taylor expanded in a around -inf 16.0%

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

6. Taylor expanded in b_2 around inf 16.5%

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

   1. mul-1-neg 16.5%

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

   2. distribute-frac-neg 16.5%

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

8. Simplified 16.5%

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

9. Final simplification 16.5%

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

Developer target: 99.6% accurate, 0.1× speedup

Math

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

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 2024054 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b_2 0.0) (/ c (- (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))) b_2)) (/ (+ b_2 (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c)))))) (- a)))

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