quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.2% → 84.5%

Time: 11.7s

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: 51.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2 \cdot 10^{+115}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.2e-148)
   (/ (* c -0.5) b_2)
   (if (<= b_2 2e+115)
     (/ (- (- 0.0 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 <= -7.2e-148) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2e+115) {
		tmp = ((0.0 - 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 <= (-7.2d-148)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 2d+115) then
        tmp = ((0.0d0 - 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 <= -7.2e-148) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2e+115) {
		tmp = ((0.0 - 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 <= -7.2e-148:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 2e+115:
		tmp = ((0.0 - 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 <= -7.2e-148)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 2e+115)
		tmp = Float64(Float64(Float64(0.0 - 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 <= -7.2e-148)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 2e+115)
		tmp = ((0.0 - 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, -7.2e-148], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2e+115], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - 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 < -7.1999999999999997e-148

   1. Initial program 17.7%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 78.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 78.4%

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

   if -7.1999999999999997e-148 < b_2 < 2e115

   1. Initial program 81.2%

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

   if 2e115 < b_2

   1. Initial program 65.5%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-2, \color{blue}{\left(\frac{b\_2}{a}\right)}\right) \]

      2. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \color{blue}{a}\right)\right) \]

   5. Simplified 100.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2 \cdot 10^{+115}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.25 \cdot 10^{-95}:\\ \;\;\;\;0 - \frac{b\_2 + {\left(\frac{-1}{c \cdot a}\right)}^{-0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.2e-148)
   (/ (* c -0.5) b_2)
   (if (<= b_2 2.25e-95)
     (- 0.0 (/ (+ b_2 (pow (/ -1.0 (* c a)) -0.5)) a))
     (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e-148) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.25e-95) {
		tmp = 0.0 - ((b_2 + pow((-1.0 / (c * a)), -0.5)) / a);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7.2d-148)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 2.25d-95) then
        tmp = 0.0d0 - ((b_2 + (((-1.0d0) / (c * a)) ** (-0.5d0))) / a)
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e-148) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.25e-95) {
		tmp = 0.0 - ((b_2 + Math.pow((-1.0 / (c * a)), -0.5)) / a);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.2e-148:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 2.25e-95:
		tmp = 0.0 - ((b_2 + math.pow((-1.0 / (c * a)), -0.5)) / a)
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.2e-148)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 2.25e-95)
		tmp = Float64(0.0 - Float64(Float64(b_2 + (Float64(-1.0 / Float64(c * a)) ^ -0.5)) / a));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.2e-148)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 2.25e-95)
		tmp = 0.0 - ((b_2 + ((-1.0 / (c * a)) ^ -0.5)) / a);
	else
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.2e-148], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.25e-95], N[(0.0 - N[(N[(b$95$2 + N[Power[N[(-1.0 / N[(c * a), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.1999999999999997e-148

   1. Initial program 17.7%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 78.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 78.4%

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

   if -7.1999999999999997e-148 < b_2 < 2.25e-95

   1. Initial program 77.9%

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

      1. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\sqrt{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}\right)\right), a\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\sqrt{\frac{1}{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}\right)\right), a\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\frac{\sqrt{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}\right)\right), a\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\frac{1}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}\right)\right), a\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{/.f64}\left(1, \left(\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}\right)\right)\right), a\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}\right)\right)\right)\right), a\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}\right)\right)\right)\right), a\right) \]

      8. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{b\_2 \cdot b\_2 - a \cdot c}\right)\right)\right)\right), a\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(b\_2 \cdot b\_2 - a \cdot c\right)\right)\right)\right)\right), a\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right)\right)\right)\right), a\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right)\right)\right)\right), a\right) \]

      12. *-lowering-*.f64 77.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right)\right), a\right) \]

   4. Applied egg-rr 77.8%

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

      1. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left({\left(\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}\right)}^{-1}\right)\right), a\right) \]

      2. sqrt-pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left({\left(\frac{1}{b\_2 \cdot b\_2 - a \cdot c}\right)}^{\left(\frac{-1}{2}\right)}\right)\right), a\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left({\left(\frac{1}{b\_2 \cdot b\_2 - a \cdot c}\right)}^{\frac{-1}{2}}\right)\right), a\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{pow.f64}\left(\left(\frac{1}{b\_2 \cdot b\_2 - a \cdot c}\right), \frac{-1}{2}\right)\right), a\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(b\_2 \cdot b\_2 - a \cdot c\right)\right), \frac{-1}{2}\right)\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), \frac{-1}{2}\right)\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), \frac{-1}{2}\right)\right), a\right) \]

      8. *-lowering-*.f64 77.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), \frac{-1}{2}\right)\right), a\right) \]

   6. Applied egg-rr 77.9%

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

   7. Taylor expanded in b_2 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{pow.f64}\left(\color{blue}{\left(\frac{-1}{a \cdot c}\right)}, \frac{-1}{2}\right)\right), a\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, \left(a \cdot c\right)\right), \frac{-1}{2}\right)\right), a\right) \]

      2. *-lowering-*.f64 75.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(a, c\right)\right), \frac{-1}{2}\right)\right), a\right) \]

   9. Simplified 75.2%

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

   if 2.25e-95 < b_2

   1. Initial program 74.1%

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

   3. Taylor expanded in c around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(-2 \cdot \frac{b\_2}{a}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \left(\frac{b\_2}{a}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \left(\frac{\frac{1}{2} \cdot c}{\color{blue}{b\_2}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot c\right), \color{blue}{b\_2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \mathsf{/.f64}\left(\left(c \cdot \frac{1}{2}\right), b\_2\right)\right) \]

      7. *-lowering-*.f64 83.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\_2\right)\right) \]

   5. Simplified 83.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.25 \cdot 10^{-95}:\\ \;\;\;\;0 - \frac{b\_2 + {\left(\frac{-1}{c \cdot a}\right)}^{-0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{0 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.2e-148)
   (/ (* c -0.5) b_2)
   (if (<= b_2 3.7e-96)
     (/ (- (- 0.0 b_2) (sqrt (- 0.0 (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e-148) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 3.7e-96) {
		tmp = ((0.0 - b_2) - sqrt((0.0 - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7.2d-148)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 3.7d-96) then
        tmp = ((0.0d0 - b_2) - sqrt((0.0d0 - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e-148) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 3.7e-96) {
		tmp = ((0.0 - b_2) - Math.sqrt((0.0 - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.2e-148:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 3.7e-96:
		tmp = ((0.0 - b_2) - math.sqrt((0.0 - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.2e-148)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 3.7e-96)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(0.0 - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.2e-148)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 3.7e-96)
		tmp = ((0.0 - b_2) - sqrt((0.0 - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.2e-148], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.7e-96], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - N[Sqrt[N[(0.0 - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.1999999999999997e-148

   1. Initial program 17.7%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 78.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 78.4%

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

   if -7.1999999999999997e-148 < b_2 < 3.69999999999999986e-96

   1. Initial program 77.9%

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

   3. Taylor expanded in b_2 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}\right)\right), a\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot c\right)\right)\right)\right), a\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(0 - a \cdot c\right)\right)\right), a\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot c\right)\right)\right)\right), a\right) \]

      4. *-lowering-*.f64 75.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, c\right)\right)\right)\right), a\right) \]

   5. Simplified 75.2%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{0 - a \cdot c}}}{a} \]

   if 3.69999999999999986e-96 < b_2

   1. Initial program 74.1%

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

   3. Taylor expanded in c around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(-2 \cdot \frac{b\_2}{a}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \left(\frac{b\_2}{a}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \left(\frac{\frac{1}{2} \cdot c}{\color{blue}{b\_2}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot c\right), \color{blue}{b\_2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \mathsf{/.f64}\left(\left(c \cdot \frac{1}{2}\right), b\_2\right)\right) \]

      7. *-lowering-*.f64 83.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\_2\right)\right) \]

   5. Simplified 83.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-96}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{0 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309)
   (/ (* c -0.5) b_2)
   (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d-309)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 27.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 65.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 65.8%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 76.5%

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

   3. Taylor expanded in c around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(-2 \cdot \frac{b\_2}{a}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \left(\frac{b\_2}{a}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \left(\frac{\frac{1}{2} \cdot c}{\color{blue}{b\_2}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot c\right), \color{blue}{b\_2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \mathsf{/.f64}\left(\left(c \cdot \frac{1}{2}\right), b\_2\right)\right) \]

      7. *-lowering-*.f64 62.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\_2\right)\right) \]

   5. Simplified 62.5%

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

Alternative 5: 68.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.25 \cdot 10^{-295}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.25e-295) (/ (* c -0.5) b_2) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e-295) {
		tmp = (c * -0.5) / 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 <= (-2.25d-295)) then
        tmp = (c * (-0.5d0)) / 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 <= -2.25e-295) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.25e-295:
		tmp = (c * -0.5) / b_2
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.25e-295)
		tmp = Float64(Float64(c * -0.5) / 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 <= -2.25e-295)
		tmp = (c * -0.5) / 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, -2.25e-295], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.2500000000000001e-295

   1. Initial program 27.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 66.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 66.8%

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

   if -2.2500000000000001e-295 < b_2

   1. Initial program 76.1%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-2, \color{blue}{\left(\frac{b\_2}{a}\right)}\right) \]

      2. /-lowering-/.f64 61.4%

         \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \color{blue}{a}\right)\right) \]

   5. Simplified 61.4%

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

Alternative 6: 43.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.1 \cdot 10^{-284}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.1e-284) 0.0 (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.1e-284) {
		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-284)) 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-284) {
		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-284:
		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-284)
		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-284)
		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-284], 0.0, N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.09999999999999999e-284

   1. Initial program 26.4%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)}\right), a\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\mathsf{neg}\left(b\_2 \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\mathsf{neg}\left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot b\_2\right)\right)\right), a\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot \left(a \cdot c\right)}{{b\_2}^{2}}\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \left(a \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot c\right) \cdot \frac{-1}{2}\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot \left(c \cdot \frac{-1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot \left(\frac{-1}{2} \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{2} \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot \frac{-1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{-1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{-1}{2}\right)\right), \left(b\_2 \cdot b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

      16. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right), \left(0 - b\_2\right)\right)\right), a\right) \]

      17. --lowering--.f64 18.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right), \mathsf{\_.f64}\left(0, b\_2\right)\right)\right), a\right) \]

   5. Simplified 18.9%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(1 + \frac{a \cdot \left(c \cdot -0.5\right)}{b\_2 \cdot b\_2}\right) \cdot \left(0 - b\_2\right)}}{a} \]

   6. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{-1 \cdot \left(0 \cdot b\_2\right)}{a} \]

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 18.2%

         \[\leadsto \mathsf{/.f64}\left(0, \color{blue}{a}\right) \]

   8. Simplified 18.2%

      \[\leadsto \color{blue}{\frac{0}{a}} \]
   9. Step-by-step derivation

      1. div0 18.2%

         \[\leadsto 0 \]

   10. Applied egg-rr 18.2%

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

   if -4.09999999999999999e-284 < b_2

   1. Initial program 76.3%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-2, \color{blue}{\left(\frac{b\_2}{a}\right)}\right) \]

      2. /-lowering-/.f64 61.0%

         \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \color{blue}{a}\right)\right) \]

   5. Simplified 61.0%

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

Alternative 7: 10.9% 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 52.3%

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

3. Taylor expanded in b_2 around -inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)}\right), a\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\mathsf{neg}\left(b\_2 \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\mathsf{neg}\left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot b\_2\right)\right)\right), a\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot \left(a \cdot c\right)}{{b\_2}^{2}}\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \left(a \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot c\right) \cdot \frac{-1}{2}\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot \left(c \cdot \frac{-1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot \left(\frac{-1}{2} \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{-1}{2} \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot \frac{-1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{-1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{-1}{2}\right)\right), \left(b\_2 \cdot b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right), a\right) \]

   16. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right), \left(0 - b\_2\right)\right)\right), a\right) \]

   17. --lowering--.f64 10.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right), \mathsf{\_.f64}\left(0, b\_2\right)\right)\right), a\right) \]

5. Simplified 10.0%

   \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left(1 + \frac{a \cdot \left(c \cdot -0.5\right)}{b\_2 \cdot b\_2}\right) \cdot \left(0 - b\_2\right)}}{a} \]

6. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{-1 \cdot \left(0 \cdot b\_2\right)}{a} \]

   4. mul0-lft N/A

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

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 10.2%

      \[\leadsto \mathsf{/.f64}\left(0, \color{blue}{a}\right) \]

8. Simplified 10.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]
9. Step-by-step derivation

   1. div0 10.2%

      \[\leadsto 0 \]

10. Applied egg-rr 10.2%

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

Developer Target 1: 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 2024145 
(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))