quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.6% → 83.7%

Time: 11.2s

Alternatives: 6

Speedup: 15.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{+30}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+30)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.45e+46)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+30) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.45e+46) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = -2.0 * (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 <= (-2d+30)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 1.45d+46) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (-2.0d0) * (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 <= -2e+30) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.45e+46) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e+30:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 1.45e+46:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e+30)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.45e+46)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e+30)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 1.45e+46)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e+30], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.45e+46], 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[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2e30

   1. Initial program 17.9%

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

   2. Taylor expanded in b_2 around -inf 91.3%

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

   if -2e30 < b_2 < 1.4500000000000001e46

   1. Initial program 71.1%

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

   if 1.4500000000000001e46 < b_2

   1. Initial program 59.7%

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

   2. Taylor expanded in b_2 around inf 95.2%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{+30}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 2: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{-71}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.65 \cdot 10^{-96}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.4e-71)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.65e-96)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.4e-71) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.65e-96) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.4d-71)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 1.65d-96) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.4e-71) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.65e-96) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.4e-71:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 1.65e-96:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.4e-71)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.65e-96)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.4e-71)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 1.65e-96)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.4e-71], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.65e-96], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.4e-71

   1. Initial program 24.1%

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

   2. Taylor expanded in b_2 around -inf 83.8%

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

   if -1.4e-71 < b_2 < 1.64999999999999995e-96

   1. Initial program 76.3%

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

   2. Taylor expanded in b_2 around 0 72.4%

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

      1. associate-*r* 72.4%

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

      2. neg-mul-1 72.4%

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

   4. Simplified 72.4%

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

   if 1.64999999999999995e-96 < b_2

   1. Initial program 63.0%

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

   2. Taylor expanded in b_2 around inf 85.7%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{-71}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.65 \cdot 10^{-96}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3: 68.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-312}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-312)
   (* -0.5 (/ c b_2))
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-312) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-312)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-312) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-312:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-312)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-312)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-312], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.0000000000019e-312

   1. Initial program 35.5%

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

   2. Taylor expanded in b_2 around -inf 69.4%

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

   if -2.0000000000019e-312 < b_2

   1. Initial program 66.2%

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

   2. Taylor expanded in b_2 around inf 69.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-312}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4: 44.4% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4.1 \cdot 10^{-129}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.1e-129) 0.0 (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.1e-129) {
		tmp = 0.0;
	} else {
		tmp = -2.0 * (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 <= (-4.1d-129)) then
        tmp = 0.0d0
    else
        tmp = (-2.0d0) * (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 <= -4.1e-129) {
		tmp = 0.0;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.1e-129:
		tmp = 0.0
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.1e-129)
		tmp = 0.0;
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.1e-129)
		tmp = 0.0;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.1e-129], 0.0, N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.1e-129

   1. Initial program 27.5%

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

   2. Taylor expanded in b_2 around -inf 24.8%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(-1 \cdot b_2 + 0.5 \cdot \frac{a \cdot c}{b_2}\right)}}{a} \]
   3. Step-by-step derivation

      1. +-commutative 24.8%

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

      2. mul-1-neg 24.8%

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

      3. unsub-neg 24.8%

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

      4. associate-/l* 24.9%

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

      5. associate-*r/ 24.9%

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

   4. Simplified 24.9%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(\frac{0.5 \cdot a}{\frac{b_2}{c}} - b_2\right)}}{a} \]
   5. Step-by-step derivation

      1. div-sub 19.2%

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

      2. div-inv 19.2%

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

      3. *-commutative 19.2%

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

      4. clear-num 19.2%

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

   6. Applied egg-rr 19.2%

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

   7. Taylor expanded in c around 0 15.9%

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

      1. distribute-lft1-in 15.9%

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

      2. metadata-eval 15.9%

         \[\leadsto \color{blue}{0} \cdot \frac{b_2}{a} \]

      3. mul0-lft 21.5%

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

   9. Simplified 21.5%

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

   if -4.1e-129 < b_2

   1. Initial program 67.4%

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

   2. Taylor expanded in b_2 around inf 60.4%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.1 \cdot 10^{-129}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 5: 68.1% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-300}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-300) (* -0.5 (/ c b_2)) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-300) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -2.0 * (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 <= (-2.9d-300)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (-2.0d0) * (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 <= -2.9e-300) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.9e-300:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e-300)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.9e-300)
		tmp = -0.5 * (c / b_2);
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e-300], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.89999999999999992e-300

   1. Initial program 34.4%

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

   2. Taylor expanded in b_2 around -inf 70.5%

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

   if -2.89999999999999992e-300 < b_2

   1. Initial program 66.7%

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

   2. Taylor expanded in b_2 around inf 67.9%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-300}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 6: 11.5% accurate, 112.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

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

2. Taylor expanded in b_2 around -inf 11.0%

   \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(-1 \cdot b_2 + 0.5 \cdot \frac{a \cdot c}{b_2}\right)}}{a} \]
3. Step-by-step derivation

   1. +-commutative 11.0%

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

   2. mul-1-neg 11.0%

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

   3. unsub-neg 11.0%

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

   4. associate-/l* 11.0%

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

   5. associate-*r/ 11.0%

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

4. Simplified 11.0%

   \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left(\frac{0.5 \cdot a}{\frac{b_2}{c}} - b_2\right)}}{a} \]
5. Step-by-step derivation

   1. div-sub 8.6%

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

   2. div-inv 8.6%

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

   3. *-commutative 8.6%

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

   4. clear-num 8.6%

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

6. Applied egg-rr 8.6%

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

7. Taylor expanded in c around 0 7.7%

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

   1. distribute-lft1-in 7.7%

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

   2. metadata-eval 7.7%

      \[\leadsto \color{blue}{0} \cdot \frac{b_2}{a} \]

   3. mul0-lft 10.1%

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

9. Simplified 10.1%

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

10. Final simplification 10.1%

    \[\leadsto 0 \]

Developer target: 99.7% 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 2023310 
(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))