quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.2% → 84.2%

Time: 8.7s

Alternatives: 10

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% 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: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+42}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.7e-134)
   (/ (* c -0.5) b_2)
   (if (<= b_2 6e+42)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-134) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 6e+42) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	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 <= (-2.7d-134)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 6d+42) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-134) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 6e+42) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.7e-134:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 6e+42:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.7e-134)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 6e+42)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.7e-134)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 6e+42)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.7e-134], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6e+42], 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 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.6999999999999998e-134

   1. Initial program 18.7%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if -2.6999999999999998e-134 < b_2 < 6.00000000000000058e42

   1. Initial program 87.7%

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

   if 6.00000000000000058e42 < b_2

   1. Initial program 66.2%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. *-rgt-identity N/A

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

      2. times-frac N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 97.1

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

   7. Applied rewrites 97.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+42}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{a \cdot 0.5}{b\_2}, b\_2 \cdot -2\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.7e-134)
   (/ (* c -0.5) b_2)
   (if (<= b_2 2.6e-109)
     (/ (- (- b_2) (sqrt (- (* c a)))) a)
     (/ (fma c (/ (* a 0.5) b_2) (* b_2 -2.0)) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-134) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.6e-109) {
		tmp = (-b_2 - sqrt(-(c * a))) / a;
	} else {
		tmp = fma(c, ((a * 0.5) / b_2), (b_2 * -2.0)) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.7e-134)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 2.6e-109)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(-Float64(c * a)))) / a);
	else
		tmp = Float64(fma(c, Float64(Float64(a * 0.5) / b_2), Float64(b_2 * -2.0)) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.7e-134], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.6e-109], N[(N[((-b$95$2) - N[Sqrt[(-N[(c * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * N[(N[(a * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision] + N[(b$95$2 * -2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.6999999999999998e-134

   1. Initial program 18.7%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if -2.6999999999999998e-134 < b_2 < 2.5999999999999998e-109

   1. Initial program 82.2%

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

   3. Taylor expanded in b_2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if 2.5999999999999998e-109 < b_2

   1. Initial program 75.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot b\_2 + \frac{1}{2} \cdot \frac{a \cdot c}{b\_2}}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 89.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{a \cdot 0.5}{b\_2}, b\_2 \cdot -2\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.7e-134)
   (/ (* c -0.5) b_2)
   (if (<= b_2 2.6e-109)
     (/ (- (- b_2) (sqrt (- (* c a)))) a)
     (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-134) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.6e-109) {
		tmp = (-b_2 - sqrt(-(c * a))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	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 <= (-2.7d-134)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 2.6d-109) then
        tmp = (-b_2 - sqrt(-(c * a))) / a
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-134) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.6e-109) {
		tmp = (-b_2 - Math.sqrt(-(c * a))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.7e-134:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 2.6e-109:
		tmp = (-b_2 - math.sqrt(-(c * a))) / a
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.7e-134)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 2.6e-109)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(-Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.7e-134)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 2.6e-109)
		tmp = (-b_2 - sqrt(-(c * a))) / a;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.7e-134], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.6e-109], N[(N[((-b$95$2) - N[Sqrt[(-N[(c * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.6999999999999998e-134

   1. Initial program 18.7%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if -2.6999999999999998e-134 < b_2 < 2.5999999999999998e-109

   1. Initial program 82.2%

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

   3. Taylor expanded in b_2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if 2.5999999999999998e-109 < b_2

   1. Initial program 75.0%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

      1. *-rgt-identity N/A

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

      2. times-frac N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{b\_2 - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.7e-134)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.5e-157) (/ (- b_2 (sqrt (- (* c a)))) a) (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-134) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.5e-157) {
		tmp = (b_2 - sqrt(-(c * a))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	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 <= (-2.7d-134)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 1.5d-157) then
        tmp = (b_2 - sqrt(-(c * a))) / a
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-134) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.5e-157) {
		tmp = (b_2 - Math.sqrt(-(c * a))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.7e-134:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 1.5e-157:
		tmp = (b_2 - math.sqrt(-(c * a))) / a
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.7e-134)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.5e-157)
		tmp = Float64(Float64(b_2 - sqrt(Float64(-Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.7e-134)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 1.5e-157)
		tmp = (b_2 - sqrt(-(c * a))) / a;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.7e-134], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.5e-157], N[(N[(b$95$2 - N[Sqrt[(-N[(c * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.6999999999999998e-134

   1. Initial program 18.7%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if -2.6999999999999998e-134 < b_2 < 1.5e-157

   1. Initial program 81.3%

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

   3. Taylor expanded in b_2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 81.3

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

   5. Applied rewrites 81.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sqrt.f64 53.3

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

   7. Applied rewrites 53.3%

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

   9. Applied rewrites 80.9%

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

   if 1.5e-157 < b_2

   1. Initial program 76.0%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

      1. *-rgt-identity N/A

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

      2. times-frac N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 85.2

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

   7. Applied rewrites 85.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{b\_2 - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310) (/ (* c -0.5) b_2) (* (/ b_2 a) -2.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	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 <= (-2d-310)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-310)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 28.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.3

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

   5. Applied rewrites 72.3%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 76.7%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. *-rgt-identity N/A

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

      2. times-frac N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 72.8

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

   7. Applied rewrites 72.8%

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

Alternative 6: 67.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 28.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.3

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

   5. Applied rewrites 72.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 72.1

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

   7. Applied rewrites 72.1%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 76.7%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. *-rgt-identity N/A

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

      2. times-frac N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 72.8

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

   7. Applied rewrites 72.8%

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

4. Final simplification 72.4%

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

Alternative 7: 47.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 28.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.3

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

   5. Applied rewrites 72.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 72.1

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

   7. Applied rewrites 72.1%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 76.7%

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

   3. Taylor expanded in b_2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\_2\right)}}{a} \]

      2. lower-neg.f64 35.2

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

   8. Applied rewrites 35.2%

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

4. Final simplification 55.4%

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

Alternative 8: 23.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-70}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (if (<= b_2 -2.2e-70) 0.0 (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e-70) {
		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 <= (-2.2d-70)) 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 <= -2.2e-70) {
		tmp = 0.0;
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.2e-70:
		tmp = 0.0
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e-70)
		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 <= -2.2e-70)
		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, -2.2e-70], 0.0, N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.1999999999999999e-70

   1. Initial program 15.6%

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

   4. Applied rewrites 8.9%

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

   5. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.1

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

   7. Applied rewrites 3.1%

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

      1. lift-neg.f64 N/A

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

      2. flip3-- N/A

         \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{{b\_2}^{3} - {\left(\mathsf{neg}\left(b\_2\right)\right)}^{3}}{b\_2 \cdot b\_2 + \left(\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) + b\_2 \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right)}} \]

      3. flip3-- N/A

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

      4. lift--.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. sqr-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. div-sub N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{\color{blue}{0}}{a} \]

      14. div0 34.4

          \[\leadsto \color{blue}{0} \]

   9. Applied rewrites 34.4%

      \[\leadsto \color{blue}{0} \]

   if -2.1999999999999999e-70 < b_2

   1. Initial program 74.6%

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

   3. Taylor expanded in b_2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\_2\right)}}{a} \]

      2. lower-neg.f64 27.8

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

   8. Applied rewrites 27.8%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-70}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 15.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 650000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (if (<= b_2 650000.0) 0.0 (/ c 0.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 650000.0) {
		tmp = 0.0;
	} else {
		tmp = c / 0.0;
	}
	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 <= 650000.0d0) then
        tmp = 0.0d0
    else
        tmp = c / 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 650000.0) {
		tmp = 0.0;
	} else {
		tmp = c / 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 650000.0:
		tmp = 0.0
	else:
		tmp = c / 0.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 650000.0)
		tmp = 0.0;
	else
		tmp = Float64(c / 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 650000.0)
		tmp = 0.0;
	else
		tmp = c / 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 650000.0], 0.0, N[(c / 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 6.5e5

   1. Initial program 44.5%

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

   4. Applied rewrites 30.8%

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

   5. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.3

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

   7. Applied rewrites 3.3%

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

      1. lift-neg.f64 N/A

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

      2. flip3-- N/A

         \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{{b\_2}^{3} - {\left(\mathsf{neg}\left(b\_2\right)\right)}^{3}}{b\_2 \cdot b\_2 + \left(\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) + b\_2 \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right)}} \]

      3. flip3-- N/A

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

      4. lift--.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-lft-identity N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. sqr-neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. div-sub N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{\color{blue}{0}}{a} \]

      14. div0 20.6

          \[\leadsto \color{blue}{0} \]

   9. Applied rewrites 20.6%

      \[\leadsto \color{blue}{0} \]

   if 6.5e5 < b_2

   1. Initial program 67.2%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 2.0

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

   5. Applied rewrites 2.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 2.0

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

   7. Applied rewrites 2.0%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{c}{\frac{b\_2}{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(\frac{b\_2}{\frac{1}{2}}\right)}} \]

      10. div-inv N/A

          \[\leadsto \frac{c}{\mathsf{neg}\left(\color{blue}{b\_2 \cdot \frac{1}{\frac{1}{2}}}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{c}{\mathsf{neg}\left(b\_2 \cdot \color{blue}{2}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{c}{\mathsf{neg}\left(b\_2 \cdot \color{blue}{\left(1 + 1\right)}\right)} \]

      13. distribute-lft-out N/A

          \[\leadsto \frac{c}{\mathsf{neg}\left(\color{blue}{\left(b\_2 \cdot 1 + b\_2 \cdot 1\right)}\right)} \]

      14. *-rgt-identity N/A

          \[\leadsto \frac{c}{\mathsf{neg}\left(\left(\color{blue}{b\_2} + b\_2 \cdot 1\right)\right)} \]

      15. *-rgt-identity N/A

          \[\leadsto \frac{c}{\mathsf{neg}\left(\left(b\_2 + \color{blue}{b\_2}\right)\right)} \]

      16. remove-double-neg N/A

          \[\leadsto \frac{c}{\mathsf{neg}\left(\left(b\_2 + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b\_2\right)\right)\right)\right)}\right)\right)} \]

      17. lift-neg.f64 N/A

          \[\leadsto \frac{c}{\mathsf{neg}\left(\left(b\_2 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)}\right)\right)\right)\right)} \]

      18. sub-neg N/A

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

      19. flip-- N/A

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

      20. sqr-neg N/A

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

      21. lift-neg.f64 N/A

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

      22. lift-neg.f64 N/A

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

      23. div-sub N/A

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

      24. +-inverses N/A

          \[\leadsto \frac{c}{\mathsf{neg}\left(\color{blue}{0}\right)} \]

      25. metadata-eval N/A

          \[\leadsto \frac{c}{\color{blue}{0}} \]

   9. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\frac{c}{0}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 11.1% accurate, 40.0× 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 50.4%

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

4. Applied rewrites 33.0%

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

5. Taylor expanded in b_2 around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 2.6

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

7. Applied rewrites 2.6%

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

   1. lift-neg.f64 N/A

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

   2. flip3-- N/A

      \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{{b\_2}^{3} - {\left(\mathsf{neg}\left(b\_2\right)\right)}^{3}}{b\_2 \cdot b\_2 + \left(\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) + b\_2 \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right)}} \]

   3. flip3-- N/A

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

   4. lift--.f64 N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

   7. lift--.f64 N/A

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

   8. flip-- N/A

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

   9. sqr-neg N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. div-sub N/A

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

   13. +-inverses N/A

       \[\leadsto \frac{\color{blue}{0}}{a} \]

   14. div0 15.8

       \[\leadsto \color{blue}{0} \]

9. Applied rewrites 15.8%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.2× 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 2024219 
(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))