quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.7% → 85.1%

Time: 13.0s

Alternatives: 10

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: 51.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.2e-114)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.2e+63)
     (/ (- (- 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 <= -5.2e-114) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.2e+63) {
		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 <= (-5.2d-114)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.2d+63) 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 <= -5.2e-114) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.2e+63) {
		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 <= -5.2e-114:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.2e+63:
		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 <= -5.2e-114)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.2e+63)
		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 <= -5.2e-114)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.2e+63)
		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, -5.2e-114], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.2e+63], 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 < -5.20000000000000026e-114

   1. Initial program 12.8%

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

   3. Taylor expanded in b_2 around -inf 83.8%

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

      1. associate-*r/ 83.9%

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

   5. Simplified 83.9%

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

   if -5.20000000000000026e-114 < b_2 < 2.1999999999999999e63

   1. Initial program 85.8%

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

   if 2.1999999999999999e63 < b_2

   1. Initial program 65.6%

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

   3. Taylor expanded in b_2 around inf 98.5%

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

      1. *-commutative 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-60}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.2e-114)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.3e-60)
     (/ (- (- b_2) (sqrt (* a (- c)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.2e-114) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.3e-60) {
		tmp = (-b_2 - sqrt((a * -c))) / 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 <= (-5.2d-114)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.3d-60) then
        tmp = (-b_2 - sqrt((a * -c))) / 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 <= -5.2e-114) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.3e-60) {
		tmp = (-b_2 - Math.sqrt((a * -c))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.2e-114:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.3e-60:
		tmp = (-b_2 - math.sqrt((a * -c))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.2e-114)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.3e-60)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / 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 <= -5.2e-114)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.3e-60)
		tmp = (-b_2 - sqrt((a * -c))) / 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, -5.2e-114], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.3e-60], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $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 < -5.20000000000000026e-114

   1. Initial program 12.8%

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

   3. Taylor expanded in b_2 around -inf 83.8%

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

      1. associate-*r/ 83.9%

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

   5. Simplified 83.9%

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

   if -5.20000000000000026e-114 < b_2 < 1.2999999999999999e-60

   1. Initial program 83.4%

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

   3. Taylor expanded in b_2 around 0 76.1%

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

      1. mul-1-neg 76.1%

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

      2. distribute-rgt-neg-out 76.1%

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

   5. Simplified 76.1%

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

   if 1.2999999999999999e-60 < b_2

   1. Initial program 73.8%

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

   3. Taylor expanded in b_2 around inf 90.5%

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

      1. *-commutative 90.5%

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

   5. Simplified 90.5%

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

Alternative 3: 68.2% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-310)
   (/ (* -0.5 c) b_2)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))))

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)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	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)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	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)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.999999999999969e-311

   1. Initial program 26.9%

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

   3. Taylor expanded in b_2 around -inf 68.1%

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

      1. associate-*r/ 68.2%

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

   5. Simplified 68.2%

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

   if -9.999999999999969e-311 < b_2

   1. Initial program 80.2%

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

   3. Taylor expanded in c around 0 67.1%

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

Alternative 4: 68.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.5e-299) (/ (* -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 <= -2.5e-299) {
		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 <= (-2.5d-299)) 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 <= -2.5e-299) {
		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 <= -2.5e-299:
		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 <= -2.5e-299)
		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 <= -2.5e-299)
		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, -2.5e-299], 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 < -2.49999999999999978e-299

   1. Initial program 26.3%

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

   3. Taylor expanded in b_2 around -inf 68.7%

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

      1. associate-*r/ 68.7%

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

   5. Simplified 68.7%

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

   if -2.49999999999999978e-299 < b_2

   1. Initial program 80.4%

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

   3. Taylor expanded in b_2 around inf 66.7%

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

      1. *-commutative 66.7%

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

   5. Simplified 66.7%

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

Alternative 5: 67.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.45 \cdot 10^{-299}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.45e-299) (/ (* -0.5 c) b_2) (/ -2.0 (/ a b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.45e-299) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 / (a / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.45d-299)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (-2.0d0) / (a / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.45e-299) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 / (a / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.45e-299:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = -2.0 / (a / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.45e-299)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(-2.0 / Float64(a / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.45e-299)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = -2.0 / (a / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.45e-299], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(-2.0 / N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.45e-299

   1. Initial program 26.3%

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

   3. Taylor expanded in b_2 around -inf 68.7%

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

      1. associate-*r/ 68.7%

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

   5. Simplified 68.7%

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

   if -2.45e-299 < b_2

   1. Initial program 80.4%

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

      1. add-cube-cbrt 79.0%

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

      2. pow3 79.0%

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

      3. add-sqr-sqrt 0.7%

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

      4. sqrt-unprod 30.9%

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

      5. sqr-neg 30.9%

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

      6. sqrt-prod 40.4%

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

      7. add-sqr-sqrt 40.4%

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

      8. sub-neg 40.4%

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

      9. add-sqr-sqrt 36.9%

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

      10. hypot-define 30.9%

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

      11. distribute-rgt-neg-in 30.9%

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

   4. Applied egg-rr 30.9%

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

   5. Taylor expanded in b_2 around -inf 1.6%

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

      1. rem-cube-cbrt 1.6%

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

      2. associate-/l* 1.6%

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

   7. Simplified 1.6%

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

      1. frac-2neg 1.6%

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

      2. metadata-eval 1.6%

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

      3. div-inv 1.6%

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

      4. clear-num 1.6%

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

      5. un-div-inv 1.6%

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

      6. add-sqr-sqrt 0.9%

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

      7. sqrt-unprod 27.0%

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

      8. sqr-neg 27.0%

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

      9. sqrt-unprod 34.3%

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

      10. add-sqr-sqrt 66.5%

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

   9. Applied egg-rr 66.5%

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

Alternative 6: 67.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-299}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-299) (* c (/ -0.5 b_2)) (/ -2.0 (/ a b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-299) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = -2.0 / (a / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.8d-299)) then
        tmp = c * ((-0.5d0) / b_2)
    else
        tmp = (-2.0d0) / (a / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-299) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = -2.0 / (a / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-299:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = -2.0 / (a / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-299)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(-2.0 / Float64(a / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-299)
		tmp = c * (-0.5 / b_2);
	else
		tmp = -2.0 / (a / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-299], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.8000000000000001e-299

   1. Initial program 26.3%

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

   3. Taylor expanded in b_2 around -inf 68.7%

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

      1. associate-*r/ 68.7%

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

   5. Simplified 68.7%

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

      1. *-commutative 68.7%

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

      2. *-un-lft-identity 68.7%

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

      3. times-frac 68.6%

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

   7. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{\frac{c}{1} \cdot \frac{-0.5}{b\_2}} \]
   8. Step-by-step derivation

      1. /-rgt-identity 68.6%

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

   9. Simplified 68.6%

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

   if -2.8000000000000001e-299 < b_2

   1. Initial program 80.4%

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

      1. add-cube-cbrt 79.0%

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

      2. pow3 79.0%

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

      3. add-sqr-sqrt 0.7%

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

      4. sqrt-unprod 30.9%

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

      5. sqr-neg 30.9%

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

      6. sqrt-prod 40.4%

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

      7. add-sqr-sqrt 40.4%

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

      8. sub-neg 40.4%

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

      9. add-sqr-sqrt 36.9%

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

      10. hypot-define 30.9%

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

      11. distribute-rgt-neg-in 30.9%

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

   4. Applied egg-rr 30.9%

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

   5. Taylor expanded in b_2 around -inf 1.6%

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

      1. rem-cube-cbrt 1.6%

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

      2. associate-/l* 1.6%

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

   7. Simplified 1.6%

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

      1. frac-2neg 1.6%

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

      2. metadata-eval 1.6%

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

      3. div-inv 1.6%

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

      4. clear-num 1.6%

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

      5. un-div-inv 1.6%

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

      6. add-sqr-sqrt 0.9%

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

      7. sqrt-unprod 27.0%

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

      8. sqr-neg 27.0%

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

      9. sqrt-unprod 34.3%

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

      10. add-sqr-sqrt 66.5%

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

   9. Applied egg-rr 66.5%

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

Alternative 7: 67.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.45 \cdot 10^{-299}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.45e-299) (* 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 <= -2.45e-299) {
		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 <= (-2.45d-299)) 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 <= -2.45e-299) {
		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 <= -2.45e-299:
		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 <= -2.45e-299)
		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 <= -2.45e-299)
		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, -2.45e-299], 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 < -2.45e-299

   1. Initial program 26.3%

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

   3. Taylor expanded in b_2 around -inf 68.7%

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

      1. associate-*r/ 68.7%

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

   5. Simplified 68.7%

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

      1. *-commutative 68.7%

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

      2. *-un-lft-identity 68.7%

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

      3. times-frac 68.6%

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

   7. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{\frac{c}{1} \cdot \frac{-0.5}{b\_2}} \]
   8. Step-by-step derivation

      1. /-rgt-identity 68.6%

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

   9. Simplified 68.6%

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

   if -2.45e-299 < b_2

   1. Initial program 80.4%

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

      1. div-sub 80.4%

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

      2. neg-mul-1 80.4%

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

      3. associate-/l* 80.4%

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

      4. add-sqr-sqrt 79.5%

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

      5. sqrt-prod 80.1%

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

      6. sqr-neg 80.1%

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

      7. sqrt-unprod 0.7%

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

      8. add-sqr-sqrt 32.1%

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

      9. fma-neg 32.1%

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

      10. add-sqr-sqrt 0.7%

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

      11. sqrt-unprod 80.1%

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

      12. sqr-neg 80.1%

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

      13. sqrt-prod 79.5%

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

      14. add-sqr-sqrt 80.4%

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

   4. Applied egg-rr 72.8%

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

      1. fma-undefine 72.8%

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

      2. unsub-neg 72.8%

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

      3. mul-1-neg 72.8%

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

      4. distribute-frac-neg2 72.8%

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

   6. Simplified 72.8%

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

   7. Taylor expanded in b_2 around inf 66.7%

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

      1. associate-*r/ 66.7%

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

      2. *-commutative 66.7%

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

      3. *-lft-identity 66.7%

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

      4. times-frac 66.4%

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

      5. /-rgt-identity 66.4%

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

   9. Simplified 66.4%

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

Alternative 8: 43.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.15 \cdot 10^{-288}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.15e-288) 0.0 (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.15e-288) {
		tmp = 0.0;
	} 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 <= (-3.15d-288)) then
        tmp = 0.0d0
    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 <= -3.15e-288) {
		tmp = 0.0;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.15e-288:
		tmp = 0.0
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.15e-288)
		tmp = 0.0;
	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 <= -3.15e-288)
		tmp = 0.0;
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.15e-288], 0.0, N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.1500000000000001e-288

   1. Initial program 25.7%

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

   3. Taylor expanded in b_2 around -inf 69.2%

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

      1. associate-*r/ 69.3%

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

   5. Simplified 69.3%

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

      1. expm1-log1p-u 61.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5 \cdot c}{b\_2}\right)\right)} \]

      2. expm1-undefine 28.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5 \cdot c}{b\_2}\right)} - 1} \]

      3. associate-/l* 28.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-0.5 \cdot \frac{c}{b\_2}}\right)} - 1 \]

   7. Applied egg-rr 28.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \frac{c}{b\_2}\right)} - 1} \]

   8. Taylor expanded in c around 0 19.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -3.1500000000000001e-288 < b_2

   1. Initial program 80.5%

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

      1. div-sub 80.5%

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

      2. neg-mul-1 80.5%

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

      3. associate-/l* 80.5%

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

      4. add-sqr-sqrt 78.9%

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

      5. sqrt-prod 80.3%

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

      6. sqr-neg 80.3%

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

      7. sqrt-unprod 1.5%

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

      8. add-sqr-sqrt 32.6%

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

      9. fma-neg 32.6%

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

      10. add-sqr-sqrt 1.5%

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

      11. sqrt-unprod 80.3%

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

      12. sqr-neg 80.3%

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

      13. sqrt-prod 78.9%

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

      14. add-sqr-sqrt 80.5%

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

   4. Applied egg-rr 73.0%

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

      1. fma-undefine 73.0%

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

      2. unsub-neg 73.0%

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

      3. mul-1-neg 73.0%

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

      4. distribute-frac-neg2 73.0%

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

   6. Simplified 73.0%

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

   7. Taylor expanded in b_2 around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. *-commutative 66.2%

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

      3. *-lft-identity 66.2%

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

      4. times-frac 65.9%

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

      5. /-rgt-identity 65.9%

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

   9. Simplified 65.9%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.15 \cdot 10^{-288}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.4% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.9 \cdot 10^{-288}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.9e-288) 0.0 (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.9e-288) {
		tmp = 0.0;
	} 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 <= (-3.9d-288)) then
        tmp = 0.0d0
    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 <= -3.9e-288) {
		tmp = 0.0;
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.9e-288:
		tmp = 0.0
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.9e-288)
		tmp = 0.0;
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.9e-288)
		tmp = 0.0;
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.9e-288], 0.0, N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.8999999999999997e-288

   1. Initial program 25.7%

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

   3. Taylor expanded in b_2 around -inf 69.2%

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

      1. associate-*r/ 69.3%

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

   5. Simplified 69.3%

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

      1. expm1-log1p-u 61.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5 \cdot c}{b\_2}\right)\right)} \]

      2. expm1-undefine 28.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5 \cdot c}{b\_2}\right)} - 1} \]

      3. associate-/l* 28.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-0.5 \cdot \frac{c}{b\_2}}\right)} - 1 \]

   7. Applied egg-rr 28.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \frac{c}{b\_2}\right)} - 1} \]

   8. Taylor expanded in c around 0 19.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -3.8999999999999997e-288 < b_2

   1. Initial program 80.5%

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

   3. Taylor expanded in b_2 around 0 45.3%

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

      1. mul-1-neg 45.3%

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

      2. distribute-rgt-neg-out 45.3%

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

   5. Simplified 45.3%

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

   6. Taylor expanded in b_2 around inf 23.5%

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

      1. associate-*r/ 23.5%

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

      2. mul-1-neg 23.5%

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

   8. Simplified 23.5%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.9 \cdot 10^{-288}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 11.2% accurate, 112.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 0.0)

C

double code(double a, double b_2, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0;
}

Python

def code(a, b_2, c):
	return 0.0

Julia

function code(a, b_2, c)
	return 0.0
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0;
end

Wolfram

code[a_, b$95$2_, c_] := 0.0

Derivation

1. Initial program 55.0%

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

3. Taylor expanded in b_2 around -inf 33.3%

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

   1. associate-*r/ 33.4%

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

5. Simplified 33.4%

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

   1. expm1-log1p-u 29.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5 \cdot c}{b\_2}\right)\right)} \]

   2. expm1-undefine 14.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5 \cdot c}{b\_2}\right)} - 1} \]

   3. associate-/l* 14.1%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-0.5 \cdot \frac{c}{b\_2}}\right)} - 1 \]

7. Applied egg-rr 14.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \frac{c}{b\_2}\right)} - 1} \]

8. Taylor expanded in c around 0 10.3%

   \[\leadsto \color{blue}{1} - 1 \]

9. Final simplification 10.3%

   \[\leadsto 0 \]
10. 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 2024137 
(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))