quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.8% → 85.3%

Time: 11.3s

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: 51.8% 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: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.8 \cdot 10^{-25}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.22 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.8e-25)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.22e+135)
     (/ (- (- 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 tmp;
	if (b_2 <= -5.8e-25) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.22e+135) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (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 <= (-5.8d-25)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 1.22d+135) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (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 <= -5.8e-25) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.22e+135) {
		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):
	tmp = 0
	if b_2 <= -5.8e-25:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 1.22e+135:
		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)
	tmp = 0.0
	if (b_2 <= -5.8e-25)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.22e+135)
		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)
	tmp = 0.0;
	if (b_2 <= -5.8e-25)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 1.22e+135)
		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_] := If[LessEqual[b$95$2, -5.8e-25], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.22e+135], 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 3 regimes

2. if b_2 < -5.8000000000000001e-25

   1. Initial program 11.2%

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

   3. Taylor expanded in b_2 around -inf 88.8%

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

   if -5.8000000000000001e-25 < b_2 < 1.21999999999999996e135

   1. Initial program 75.5%

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

   if 1.21999999999999996e135 < b_2

   1. Initial program 41.7%

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

   3. Taylor expanded in b_2 around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.8 \cdot 10^{-25}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.22 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.7 \cdot 10^{-25}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.75 \cdot 10^{-68}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.7e-25)
   (* -0.5 (/ c b_2))
   (if (<= b_2 2.75e-68)
     (/ (- (- 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 <= -3.7e-25) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.75e-68) {
		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 <= (-3.7d-25)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 2.75d-68) 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 <= -3.7e-25) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.75e-68) {
		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 <= -3.7e-25:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 2.75e-68:
		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 <= -3.7e-25)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 2.75e-68)
		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 <= -3.7e-25)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 2.75e-68)
		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, -3.7e-25], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.75e-68], 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 < -3.70000000000000009e-25

   1. Initial program 11.2%

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

   3. Taylor expanded in b_2 around -inf 88.8%

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

   if -3.70000000000000009e-25 < b_2 < 2.7500000000000001e-68

   1. Initial program 68.6%

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

   3. Taylor expanded in b_2 around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. neg-mul-1 66.7%

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

   5. Simplified 66.7%

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

   if 2.7500000000000001e-68 < b_2

   1. Initial program 66.0%

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

   3. Taylor expanded in c around 0 88.3%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.7 \cdot 10^{-25}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.75 \cdot 10^{-68}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.08 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.4e-25)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.08e-67)
     (/ (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 <= -3.4e-25) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.08e-67) {
		tmp = 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 <= (-3.4d-25)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 1.08d-67) then
        tmp = 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 <= -3.4e-25) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.08e-67) {
		tmp = 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 <= -3.4e-25:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 1.08e-67:
		tmp = 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 <= -3.4e-25)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.08e-67)
		tmp = Float64(sqrt(Float64(c * Float64(-a))) / Float64(-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 <= -3.4e-25)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 1.08e-67)
		tmp = 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, -3.4e-25], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.08e-67], N[(N[Sqrt[N[(c * (-a)), $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 < -3.40000000000000002e-25

   1. Initial program 11.2%

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

   3. Taylor expanded in b_2 around -inf 88.8%

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

   if -3.40000000000000002e-25 < b_2 < 1.0800000000000001e-67

   1. Initial program 68.6%

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

      1. prod-diff 68.1%

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

      2. *-commutative 68.1%

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

      3. fma-define 68.1%

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

      4. associate-+l+ 68.1%

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

      5. pow2 68.1%

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

      6. distribute-rgt-neg-in 68.1%

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

      7. fma-define 68.2%

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

      8. *-commutative 68.2%

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

      9. fma-undefine 68.1%

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

      10. distribute-lft-neg-in 68.1%

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

      11. *-commutative 68.1%

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

      12. distribute-rgt-neg-in 68.1%

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

      13. fma-define 68.2%

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

   4. Applied egg-rr 68.2%

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

      1. unpow2 68.2%

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

      2. fma-undefine 68.2%

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

      3. *-commutative 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in b_2 around 0 65.3%

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

      1. mul-1-neg 65.3%

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

      2. distribute-lft1-in 65.7%

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

      3. metadata-eval 65.7%

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

      4. neg-mul-1 65.7%

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

      5. distribute-rgt-neg-in 65.7%

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

   9. Simplified 65.7%

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

   if 1.0800000000000001e-67 < b_2

   1. Initial program 66.0%

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

   3. Taylor expanded in c around 0 88.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.08 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.3 \cdot 10^{-187}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.8 \cdot 10^{-87}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.3e-187)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.8e-87)
     (- (sqrt (/ c (- 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 <= -1.3e-187) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.8e-87) {
		tmp = -sqrt((c / -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 <= (-1.3d-187)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 1.8d-87) then
        tmp = -sqrt((c / -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 <= -1.3e-187) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.8e-87) {
		tmp = -Math.sqrt((c / -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 <= -1.3e-187:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 1.8e-87:
		tmp = -math.sqrt((c / -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 <= -1.3e-187)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.8e-87)
		tmp = Float64(-sqrt(Float64(c / Float64(-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 <= -1.3e-187)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 1.8e-87)
		tmp = -sqrt((c / -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, -1.3e-187], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.8e-87], (-N[Sqrt[N[(c / (-a)), $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 3 regimes

2. if b_2 < -1.3e-187

   1. Initial program 25.1%

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

   3. Taylor expanded in b_2 around -inf 73.7%

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

   if -1.3e-187 < b_2 < 1.79999999999999996e-87

   1. Initial program 74.4%

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

      1. prod-diff 73.8%

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

      2. *-commutative 73.8%

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

      3. fma-define 73.8%

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

      4. associate-+l+ 73.8%

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

      5. pow2 73.8%

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

      6. distribute-rgt-neg-in 73.8%

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

      7. fma-define 73.9%

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

      8. *-commutative 73.9%

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

      9. fma-undefine 73.8%

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

      10. distribute-lft-neg-in 73.8%

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

      11. *-commutative 73.8%

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

      12. distribute-rgt-neg-in 73.8%

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

      13. fma-define 73.9%

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

   4. Applied egg-rr 73.9%

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

      1. unpow2 73.9%

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

      2. fma-undefine 73.9%

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

      3. *-commutative 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in a around inf 49.7%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{c + -2 \cdot c}{a}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 49.7%

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

      2. distribute-rgt1-in 49.7%

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

      3. metadata-eval 49.7%

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

      4. mul-1-neg 49.7%

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

   9. Simplified 49.7%

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

   if 1.79999999999999996e-87 < b_2

   1. Initial program 66.8%

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

   3. Taylor expanded in c around 0 86.5%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.3 \cdot 10^{-187}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.8 \cdot 10^{-87}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-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 <= -1e-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 <= (-1d-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 <= -1e-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 <= -1e-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 <= -1e-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 <= -1e-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, -1e-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 < -9.999999999999969e-311

   1. Initial program 33.6%

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

   3. Taylor expanded in b_2 around -inf 60.3%

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

   if -9.999999999999969e-311 < b_2

   1. Initial program 70.4%

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

   3. Taylor expanded in c around 0 66.7%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-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 <= -1e-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 <= (-1d-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 <= -1e-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 <= -1e-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 <= -1e-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 <= -1e-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, -1e-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 < -9.999999999999969e-311

   1. Initial program 33.6%

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

   3. Taylor expanded in b_2 around -inf 60.3%

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

   if -9.999999999999969e-311 < b_2

   1. Initial program 70.4%

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

   3. Taylor expanded in b_2 around inf 66.6%

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

      1. associate-*r/ 66.6%

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

      2. *-commutative 66.6%

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

   5. Simplified 66.6%

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

Alternative 7: 48.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-310) (* -0.5 (/ c b_2)) (- (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-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 <= (-1d-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 <= -1e-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 <= -1e-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 <= -1e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(-Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-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, -1e-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 < -9.999999999999969e-311

   1. Initial program 33.6%

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

   3. Taylor expanded in b_2 around -inf 60.3%

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

   if -9.999999999999969e-311 < b_2

   1. Initial program 70.4%

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

   3. Taylor expanded in b_2 around 0 39.3%

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

      1. associate-*r* 39.3%

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

      2. neg-mul-1 39.3%

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

   5. Simplified 39.3%

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

   6. Taylor expanded in b_2 around inf 24.1%

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

      1. mul-1-neg 24.1%

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

   8. Simplified 24.1%

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

Alternative 8: 15.6% 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(-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 51.1%

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

3. Taylor expanded in b_2 around 0 34.3%

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

   1. associate-*r* 34.3%

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

   2. neg-mul-1 34.3%

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

5. Simplified 34.3%

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

6. Taylor expanded in b_2 around inf 13.1%

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

   1. mul-1-neg 13.1%

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

8. Simplified 13.1%

   \[\leadsto \color{blue}{-\frac{b\_2}{a}} \]
9. 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 51.1%

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

3. Taylor expanded in b_2 around 0 34.3%

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

   1. associate-*r* 34.3%

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

   2. neg-mul-1 34.3%

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

5. Simplified 34.3%

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

6. Taylor expanded in b_2 around inf 13.1%

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

   1. mul-1-neg 13.1%

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

8. Simplified 13.1%

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

   1. neg-sub0 13.1%

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

   2. sub-neg 13.1%

      \[\leadsto \color{blue}{0 + \left(-\frac{b\_2}{a}\right)} \]

   3. add-sqr-sqrt 6.3%

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

   4. sqrt-unprod 6.6%

      \[\leadsto 0 + \color{blue}{\sqrt{\left(-\frac{b\_2}{a}\right) \cdot \left(-\frac{b\_2}{a}\right)}} \]

   5. sqr-neg 6.6%

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

   6. sqrt-unprod 1.5%

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

   7. add-sqr-sqrt 2.6%

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

10. Applied egg-rr 2.6%

    \[\leadsto \color{blue}{0 + \frac{b\_2}{a}} \]
11. Step-by-step derivation

    1. +-lft-identity 2.6%

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

12. Simplified 2.6%

    \[\leadsto \color{blue}{\frac{b\_2}{a}} \]
13. 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 2024180 
(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))