quad2m (problem 3.2.1, negative)

Percentage Accurate: 53.1% → 85.5%

Time: 9.9s

Alternatives: 6

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: 53.1% 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: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-61}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.9 \cdot 10^{+59}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-61)
   (* -0.5 (/ c b_2))
   (if (<= b_2 2.9e+59)
     (/ (- (- 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 <= -5e-61) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.9e+59) {
		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 <= (-5d-61)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 2.9d+59) 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 <= -5e-61) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.9e+59) {
		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 <= -5e-61:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 2.9e+59:
		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 <= -5e-61)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 2.9e+59)
		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 <= -5e-61)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 2.9e+59)
		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, -5e-61], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.9e+59], 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 < -4.9999999999999999e-61

   1. Initial program 15.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 89.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -4.9999999999999999e-61 < b_2 < 2.89999999999999991e59

   1. Initial program 79.6%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 2.89999999999999991e59 < b_2

   1. Initial program 58.1%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 96.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 96.1%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   5. Simplified 96.1%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-61}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.9 \cdot 10^{+59}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-61}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.3 \cdot 10^{-114}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-61)
   (* -0.5 (/ c b_2))
   (if (<= b_2 4.3e-114)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-61) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 4.3e-114) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-9d-61)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 4.3d-114) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-61) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 4.3e-114) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-61:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 4.3e-114:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-61)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 4.3e-114)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-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 <= -9e-61)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 4.3e-114)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-61], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.3e-114], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9e-61

   1. Initial program 15.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 89.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -9e-61 < b_2 < 4.3e-114

   1. Initial program 74.8%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0 70.4%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* 70.4%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. neg-mul-1 70.4%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

   5. Simplified 70.4%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]

   if 4.3e-114 < b_2

   1. Initial program 67.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 86.7%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 86.7%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   5. Simplified 86.7%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-61}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.3 \cdot 10^{-114}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310) (* -0.5 (/ c b_2)) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 28.8%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 73.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -3.999999999999988e-310 < b_2

   1. Initial program 70.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0 37.9%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* 37.9%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. neg-mul-1 37.9%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

   5. Simplified 37.9%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]

   6. Taylor expanded in b_2 around inf 32.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
   7. Step-by-step derivation

      1. associate-*r/ 32.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

      2. mul-1-neg 32.8%

         \[\leadsto \frac{\color{blue}{-b_2}}{a} \]

   8. Simplified 32.8%

      \[\leadsto \color{blue}{\frac{-b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310) (* -0.5 (/ c b_2)) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 28.8%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 73.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -3.999999999999988e-310 < b_2

   1. Initial program 70.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 73.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 73.1%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   5. Simplified 73.1%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 23.8% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310) (/ 0.0 a) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = 0.0 / a;
	} 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 <= (-4d-310)) then
        tmp = 0.0d0 / a
    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 <= -4e-310) {
		tmp = 0.0 / a;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = 0.0 / a
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(0.0 / a);
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = 0.0 / a;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(0.0 / a), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 28.8%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 17.9%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(-1 \cdot b_2 + 0.5 \cdot \frac{a \cdot c}{b_2}\right)}}{a} \]

   4. Taylor expanded in a around 0 16.8%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b_2 + -1 \cdot b_2\right)}}{a} \]
   5. Step-by-step derivation

      1. distribute-rgt1-in 16.8%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b_2\right)}}{a} \]

      2. metadata-eval 16.8%

         \[\leadsto \frac{-1 \cdot \left(\color{blue}{0} \cdot b_2\right)}{a} \]

      3. mul0-lft 16.8%

         \[\leadsto \frac{-1 \cdot \color{blue}{0}}{a} \]

      4. metadata-eval 16.8%

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

   6. Simplified 16.8%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 70.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0 37.9%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* 37.9%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. neg-mul-1 37.9%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

   5. Simplified 37.9%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]

   6. Taylor expanded in b_2 around inf 32.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
   7. Step-by-step derivation

      1. associate-*r/ 32.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

      2. mul-1-neg 32.8%

         \[\leadsto \frac{\color{blue}{-b_2}}{a} \]

   8. Simplified 32.8%

      \[\leadsto \color{blue}{\frac{-b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 11.3% accurate, 37.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b_2, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b$95$2_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 48.7%

   \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in b_2 around -inf 10.2%

   \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(-1 \cdot b_2 + 0.5 \cdot \frac{a \cdot c}{b_2}\right)}}{a} \]

4. Taylor expanded in a around 0 9.9%

   \[\leadsto \frac{\color{blue}{-1 \cdot \left(b_2 + -1 \cdot b_2\right)}}{a} \]
5. Step-by-step derivation

   1. distribute-rgt1-in 9.9%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b_2\right)}}{a} \]

   2. metadata-eval 9.9%

      \[\leadsto \frac{-1 \cdot \left(\color{blue}{0} \cdot b_2\right)}{a} \]

   3. mul0-lft 9.9%

      \[\leadsto \frac{-1 \cdot \color{blue}{0}}{a} \]

   4. metadata-eval 9.9%

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

6. Simplified 9.9%

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

7. Final simplification 9.9%

   \[\leadsto \frac{0}{a} \]
8. Add Preprocessing

Developer target: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b_2\right| - t_0} \cdot \sqrt{\left|b_2\right| + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b_2, t_0\right)\\ \end{array}\\ \mathbf{if}\;b_2 < 0:\\ \;\;\;\;\frac{c}{t_1 - b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 + t_1}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(c / N[(t$95$1 - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + t$95$1), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024020 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

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

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