quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.4% → 84.8%

Time: 11.0s

Alternatives: 7

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: 52.4% 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: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.3 \cdot 10^{-124}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.25 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.3e-124)
   (* -0.5 (/ c b_2))
   (if (<= b_2 4.25e+70)
     (/ (- (- 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 <= -1.3e-124) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 4.25e+70) {
		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 <= (-1.3d-124)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 4.25d+70) 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 <= -1.3e-124) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 4.25e+70) {
		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 <= -1.3e-124:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 4.25e+70:
		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 <= -1.3e-124)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 4.25e+70)
		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 <= -1.3e-124)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 4.25e+70)
		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, -1.3e-124], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.25e+70], 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 < -1.3e-124

   1. Initial program 14.5%

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

   3. Taylor expanded in b_2 around -inf 87.5%

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

   if -1.3e-124 < b_2 < 4.2499999999999998e70

   1. Initial program 80.9%

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

   if 4.2499999999999998e70 < b_2

   1. Initial program 54.6%

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

   3. Taylor expanded in b_2 around inf 92.0%

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

      1. *-commutative 92.0%

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

   5. Simplified 92.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.3 \cdot 10^{-124}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.25 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.65 \cdot 10^{-124}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.65e-124)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.95e-25)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (+ (* (/ b_2 a) -2.0) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.65e-124) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.95e-25) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = ((b_2 / a) * -2.0) + ((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 <= (-2.65d-124)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 1.95d-25) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = ((b_2 / a) * (-2.0d0)) + ((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 <= -2.65e-124) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.95e-25) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.65e-124:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 1.95e-25:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.65e-124)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.95e-25)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + 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 <= -2.65e-124)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 1.95e-25)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.65e-124], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.95e-25], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.6499999999999999e-124

   1. Initial program 14.5%

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

   3. Taylor expanded in b_2 around -inf 87.5%

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

   if -2.6499999999999999e-124 < b_2 < 1.95e-25

   1. Initial program 79.5%

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

   3. Taylor expanded in b_2 around 0 70.0%

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

      1. associate-*r* 70.0%

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

      2. neg-mul-1 70.0%

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

   5. Simplified 70.0%

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

   if 1.95e-25 < b_2

   1. Initial program 62.5%

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

   3. Taylor expanded in c around 0 88.0%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.65 \cdot 10^{-124}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-311)
   (* -0.5 (/ c b_2))
   (+ (* (/ b_2 a) -2.0) (* (/ c b_2) 0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-311) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = ((b_2 / a) * -2.0) + ((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 <= (-4d-311)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = ((b_2 / a) * (-2.0d0)) + ((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 <= -4e-311) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-311:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-311)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + 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 <= -4e-311)
		tmp = -0.5 * (c / b_2);
	else
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-311], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.99999999999979e-311

   1. Initial program 24.4%

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

   3. Taylor expanded in b_2 around -inf 74.5%

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

   if -3.99999999999979e-311 < b_2

   1. Initial program 69.3%

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

   3. Taylor expanded in c around 0 68.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-307}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-307) (* -0.5 (/ c b_2)) (* (/ b_2 a) -2.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-307) {
		tmp = -0.5 * (c / 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 <= (-7d-307)) then
        tmp = (-0.5d0) * (c / 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 <= -7e-307) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7e-307:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-307)
		tmp = Float64(-0.5 * Float64(c / 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 <= -7e-307)
		tmp = -0.5 * (c / 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, -7e-307], N[(-0.5 * N[(c / 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 < -7.0000000000000004e-307

   1. Initial program 23.7%

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

   3. Taylor expanded in b_2 around -inf 75.1%

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

   if -7.0000000000000004e-307 < b_2

   1. Initial program 69.5%

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

   3. Taylor expanded in b_2 around inf 67.5%

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

      1. *-commutative 67.5%

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

   5. Simplified 67.5%

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

Alternative 5: 47.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-307}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-307) (* -0.5 (/ c b_2)) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-307) {
		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 <= (-7d-307)) 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 <= -7e-307) {
		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 <= -7e-307:
		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 <= -7e-307)
		tmp = Float64(-0.5 * Float64(c / 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 <= -7e-307)
		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, -7e-307], 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 < -7.0000000000000004e-307

   1. Initial program 23.7%

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

   3. Taylor expanded in b_2 around -inf 75.1%

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

   if -7.0000000000000004e-307 < b_2

   1. Initial program 69.5%

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

      1. flip-- 22.6%

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

      2. div-inv 22.6%

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

      3. sqr-neg 22.6%

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

      4. add-sqr-sqrt 22.6%

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

      5. associate--r- 24.2%

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

      6. fmm-def 25.6%

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

      7. add-sqr-sqrt 24.6%

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

      8. sqrt-prod 25.6%

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

      9. sqr-neg 25.6%

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

      10. sqrt-unprod 0.7%

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

      11. add-sqr-sqrt 22.1%

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

      12. distribute-rgt-neg-out 22.1%

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

      13. sqr-neg 22.1%

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

      14. pow2 22.1%

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

   4. Applied egg-rr 21.9%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(b\_2, b\_2, {b\_2}^{2}\right) + a \cdot c\right) \cdot \frac{1}{b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}}{a} \]
   5. Step-by-step derivation

      1. associate-*r/ 21.9%

         \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left(b\_2, b\_2, {b\_2}^{2}\right) + a \cdot c\right) \cdot 1}{b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}}{a} \]

      2. *-rgt-identity 21.9%

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

      3. +-commutative 21.9%

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

      4. *-commutative 21.9%

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

      5. fma-define 21.9%

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

      6. fma-undefine 21.9%

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

      7. unpow2 21.9%

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

      8. count-2 21.9%

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

      9. distribute-rgt-neg-out 21.9%

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

      10. *-commutative 21.9%

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

      11. distribute-rgt-neg-in 21.9%

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

   6. Simplified 21.9%

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

   7. Taylor expanded in c around 0 1.4%

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

      1. frac-2neg 1.4%

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

      2. neg-mul-1 1.4%

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

      3. *-un-lft-identity 1.4%

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

      4. times-frac 1.4%

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

      5. metadata-eval 1.4%

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

      6. add-sqr-sqrt 0.7%

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

      7. sqrt-unprod 13.0%

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

      8. sqr-neg 13.0%

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

      9. sqrt-unprod 13.9%

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

      10. add-sqr-sqrt 28.7%

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

   9. Applied egg-rr 28.7%

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

       1. mul-1-neg 28.7%

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

   11. Simplified 28.7%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-307}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 15.6% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{b\_2}{-a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.9%

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

   1. flip-- 22.7%

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

   2. div-inv 22.7%

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

   3. sqr-neg 22.7%

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

   4. add-sqr-sqrt 22.7%

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

   5. associate--r- 32.7%

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

   6. fmm-def 22.2%

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

   7. add-sqr-sqrt 13.0%

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

   8. sqrt-prod 22.1%

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

   9. sqr-neg 22.1%

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

   10. sqrt-unprod 9.0%

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

   11. add-sqr-sqrt 20.2%

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

   12. distribute-rgt-neg-out 20.2%

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

   13. sqr-neg 20.2%

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

   14. pow2 20.2%

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

4. Applied egg-rr 19.7%

   \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(b\_2, b\_2, {b\_2}^{2}\right) + a \cdot c\right) \cdot \frac{1}{b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}}{a} \]
5. Step-by-step derivation

   1. associate-*r/ 19.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left(b\_2, b\_2, {b\_2}^{2}\right) + a \cdot c\right) \cdot 1}{b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}}{a} \]

   2. *-rgt-identity 19.7%

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

   3. +-commutative 19.7%

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

   4. *-commutative 19.7%

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

   5. fma-define 19.7%

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

   6. fma-undefine 19.7%

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

   7. unpow2 19.7%

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

   8. count-2 19.7%

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

   9. distribute-rgt-neg-out 19.7%

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

   10. *-commutative 19.7%

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

   11. distribute-rgt-neg-in 19.7%

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

6. Simplified 19.7%

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

7. Taylor expanded in c around 0 2.3%

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

   1. frac-2neg 2.3%

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

   2. neg-mul-1 2.3%

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

   3. *-un-lft-identity 2.3%

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

   4. times-frac 2.3%

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

   5. metadata-eval 2.3%

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

   6. add-sqr-sqrt 1.1%

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

   7. sqrt-unprod 9.3%

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

   8. sqr-neg 9.3%

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

   9. sqrt-unprod 8.0%

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

   10. add-sqr-sqrt 16.4%

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

9. Applied egg-rr 16.4%

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

    1. mul-1-neg 16.4%

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

11. Simplified 16.4%

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

12. Final simplification 16.4%

    \[\leadsto \frac{b\_2}{-a} \]
13. Add Preprocessing

Alternative 7: 2.5% accurate, 37.3× speedup

Math

\[\begin{array}{l} \\ \frac{b\_2}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.9%

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

   1. flip-- 22.7%

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

   2. div-inv 22.7%

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

   3. sqr-neg 22.7%

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

   4. add-sqr-sqrt 22.7%

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

   5. associate--r- 32.7%

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

   6. fmm-def 22.2%

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

   7. add-sqr-sqrt 13.0%

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

   8. sqrt-prod 22.1%

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

   9. sqr-neg 22.1%

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

   10. sqrt-unprod 9.0%

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

   11. add-sqr-sqrt 20.2%

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

   12. distribute-rgt-neg-out 20.2%

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

   13. sqr-neg 20.2%

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

   14. pow2 20.2%

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

4. Applied egg-rr 19.7%

   \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(b\_2, b\_2, {b\_2}^{2}\right) + a \cdot c\right) \cdot \frac{1}{b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}}{a} \]
5. Step-by-step derivation

   1. associate-*r/ 19.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left(b\_2, b\_2, {b\_2}^{2}\right) + a \cdot c\right) \cdot 1}{b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}}{a} \]

   2. *-rgt-identity 19.7%

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

   3. +-commutative 19.7%

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

   4. *-commutative 19.7%

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

   5. fma-define 19.7%

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

   6. fma-undefine 19.7%

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

   7. unpow2 19.7%

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

   8. count-2 19.7%

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

   9. distribute-rgt-neg-out 19.7%

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

   10. *-commutative 19.7%

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

   11. distribute-rgt-neg-in 19.7%

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

6. Simplified 19.7%

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

7. Taylor expanded in c around 0 2.3%

   \[\leadsto \color{blue}{\frac{b\_2}{a}} \]
8. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b\_2\right| - t\_0} \cdot \sqrt{\left|b\_2\right| + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b\_2, t\_0\right)\\ \end{array}\\ \mathbf{if}\;b\_2 < 0:\\ \;\;\;\;\frac{c}{t\_1 - b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + t\_1}{-a}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024151 
(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))