quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.9% → 85.7%

Time: 12.2s

Alternatives: 9

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: 51.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{+151}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e+151)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 3e-89)
     (/ (- (sqrt (/ 1.0 (/ 1.0 (- (* b_2 b_2) (* a c))))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e+151) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 3e-89) {
		tmp = (sqrt((1.0 / (1.0 / ((b_2 * b_2) - (a * c))))) - b_2) / a;
	} else {
		tmp = (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 <= (-4d+151)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 3d-89) then
        tmp = (sqrt((1.0d0 / (1.0d0 / ((b_2 * b_2) - (a * c))))) - b_2) / a
    else
        tmp = (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 <= -4e+151) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 3e-89) {
		tmp = (Math.sqrt((1.0 / (1.0 / ((b_2 * b_2) - (a * c))))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e+151:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 3e-89:
		tmp = (math.sqrt((1.0 / (1.0 / ((b_2 * b_2) - (a * c))))) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e+151)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 3e-89)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(1.0 / Float64(Float64(b_2 * b_2) - Float64(a * c))))) - b_2) / a);
	else
		tmp = 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 <= -4e+151)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 3e-89)
		tmp = (sqrt((1.0 / (1.0 / ((b_2 * b_2) - (a * c))))) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e+151], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3e-89], N[(N[(N[Sqrt[N[(1.0 / N[(1.0 / N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.00000000000000007e151

   1. Initial program 32.0%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 32.0%

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

   3. Simplified 32.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 96.8%

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

   7. Simplified 96.8%

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

   if -4.00000000000000007e151 < b_2 < 2.9999999999999999e-89

   1. Initial program 83.9%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 83.9%

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

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-- N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 83.9%

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

   6. Applied egg-rr 83.9%

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

   if 2.9999999999999999e-89 < b_2

   1. Initial program 16.5%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 16.5%

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

   3. Simplified 16.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. 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 84.6%

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

   7. Simplified 84.6%

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

Alternative 2: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.55 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.5e+150)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1.55e-92)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.5e+150) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.55e-92) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (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 <= (-5.5d+150)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 1.55d-92) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (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 <= -5.5e+150) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.55e-92) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.5e+150:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 1.55e-92:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.5e+150)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 1.55e-92)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = 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 <= -5.5e+150)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 1.55e-92)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.5e+150], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.55e-92], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.50000000000000017e150

   1. Initial program 32.0%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 32.0%

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

   3. Simplified 32.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 96.8%

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

   7. Simplified 96.8%

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

   if -5.50000000000000017e150 < b_2 < 1.55e-92

   1. Initial program 83.9%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 83.9%

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

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   if 1.55e-92 < b_2

   1. Initial program 16.5%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 16.5%

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

   3. Simplified 16.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. 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 84.6%

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

   7. Simplified 84.6%

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

Alternative 3: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-127}:\\ \;\;\;\;b\_2 \cdot \left(\frac{c \cdot -0.5}{0 - b\_2 \cdot b\_2} - \frac{2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.25 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{0 - a \cdot c} - b\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e-127)
   (* b_2 (- (/ (* c -0.5) (- 0.0 (* b_2 b_2))) (/ 2.0 a)))
   (if (<= b_2 1.25e-90)
     (/ 1.0 (/ a (- (sqrt (- 0.0 (* a c))) b_2)))
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e-127) {
		tmp = b_2 * (((c * -0.5) / (0.0 - (b_2 * b_2))) - (2.0 / a));
	} else if (b_2 <= 1.25e-90) {
		tmp = 1.0 / (a / (sqrt((0.0 - (a * c))) - b_2));
	} else {
		tmp = (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 <= (-2.2d-127)) then
        tmp = b_2 * (((c * (-0.5d0)) / (0.0d0 - (b_2 * b_2))) - (2.0d0 / a))
    else if (b_2 <= 1.25d-90) then
        tmp = 1.0d0 / (a / (sqrt((0.0d0 - (a * c))) - b_2))
    else
        tmp = (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 <= -2.2e-127) {
		tmp = b_2 * (((c * -0.5) / (0.0 - (b_2 * b_2))) - (2.0 / a));
	} else if (b_2 <= 1.25e-90) {
		tmp = 1.0 / (a / (Math.sqrt((0.0 - (a * c))) - b_2));
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.2e-127:
		tmp = b_2 * (((c * -0.5) / (0.0 - (b_2 * b_2))) - (2.0 / a))
	elif b_2 <= 1.25e-90:
		tmp = 1.0 / (a / (math.sqrt((0.0 - (a * c))) - b_2))
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e-127)
		tmp = Float64(b_2 * Float64(Float64(Float64(c * -0.5) / Float64(0.0 - Float64(b_2 * b_2))) - Float64(2.0 / a)));
	elseif (b_2 <= 1.25e-90)
		tmp = Float64(1.0 / Float64(a / Float64(sqrt(Float64(0.0 - Float64(a * c))) - b_2)));
	else
		tmp = 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 <= -2.2e-127)
		tmp = b_2 * (((c * -0.5) / (0.0 - (b_2 * b_2))) - (2.0 / a));
	elseif (b_2 <= 1.25e-90)
		tmp = 1.0 / (a / (sqrt((0.0 - (a * c))) - b_2));
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.2e-127], N[(b$95$2 * N[(N[(N[(c * -0.5), $MachinePrecision] / N[(0.0 - N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.25e-90], N[(1.0 / N[(a / N[(N[Sqrt[N[(0.0 - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.2000000000000001e-127

   1. Initial program 67.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 67.2%

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

   3. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)}\right) \]

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

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

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

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

      6. associate-*r/ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 79.3%

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

   7. Simplified 79.3%

      \[\leadsto \color{blue}{0 - b\_2 \cdot \left(\frac{c \cdot -0.5}{b\_2 \cdot b\_2} + \frac{2}{a}\right)} \]

   if -2.2000000000000001e-127 < b_2 < 1.25000000000000005e-90

   1. Initial program 78.5%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 78.5%

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

   3. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 78.5%

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

   6. Applied egg-rr 78.5%

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

   7. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(0 - a \cdot c\right)\right), b\_2\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot c\right)\right)\right), b\_2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(c \cdot a\right)\right)\right), b\_2\right)\right)\right) \]

      5. *-lowering-*.f64 77.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, a\right)\right)\right), b\_2\right)\right)\right) \]

   9. Simplified 77.9%

      \[\leadsto \frac{1}{\frac{a}{\sqrt{\color{blue}{0 - c \cdot a}} - b\_2}} \]

   if 1.25000000000000005e-90 < b_2

   1. Initial program 16.5%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 16.5%

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

   3. Simplified 16.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. 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 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-127}:\\ \;\;\;\;b\_2 \cdot \left(\frac{c \cdot -0.5}{0 - b\_2 \cdot b\_2} - \frac{2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.25 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{0 - a \cdot c} - b\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.4 \cdot 10^{-128}:\\ \;\;\;\;b\_2 \cdot \left(\frac{c \cdot -0.5}{0 - b\_2 \cdot b\_2} - \frac{2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.9 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{0 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.4e-128)
   (* b_2 (- (/ (* c -0.5) (- 0.0 (* b_2 b_2))) (/ 2.0 a)))
   (if (<= b_2 1.9e-89)
     (/ (- (sqrt (- 0.0 (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.4e-128) {
		tmp = b_2 * (((c * -0.5) / (0.0 - (b_2 * b_2))) - (2.0 / a));
	} else if (b_2 <= 1.9e-89) {
		tmp = (sqrt((0.0 - (a * c))) - b_2) / a;
	} else {
		tmp = (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 <= (-6.4d-128)) then
        tmp = b_2 * (((c * (-0.5d0)) / (0.0d0 - (b_2 * b_2))) - (2.0d0 / a))
    else if (b_2 <= 1.9d-89) then
        tmp = (sqrt((0.0d0 - (a * c))) - b_2) / a
    else
        tmp = (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 <= -6.4e-128) {
		tmp = b_2 * (((c * -0.5) / (0.0 - (b_2 * b_2))) - (2.0 / a));
	} else if (b_2 <= 1.9e-89) {
		tmp = (Math.sqrt((0.0 - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.4e-128:
		tmp = b_2 * (((c * -0.5) / (0.0 - (b_2 * b_2))) - (2.0 / a))
	elif b_2 <= 1.9e-89:
		tmp = (math.sqrt((0.0 - (a * c))) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.4e-128)
		tmp = Float64(b_2 * Float64(Float64(Float64(c * -0.5) / Float64(0.0 - Float64(b_2 * b_2))) - Float64(2.0 / a)));
	elseif (b_2 <= 1.9e-89)
		tmp = Float64(Float64(sqrt(Float64(0.0 - Float64(a * c))) - b_2) / a);
	else
		tmp = 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 <= -6.4e-128)
		tmp = b_2 * (((c * -0.5) / (0.0 - (b_2 * b_2))) - (2.0 / a));
	elseif (b_2 <= 1.9e-89)
		tmp = (sqrt((0.0 - (a * c))) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.4e-128], N[(b$95$2 * N[(N[(N[(c * -0.5), $MachinePrecision] / N[(0.0 - N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.9e-89], N[(N[(N[Sqrt[N[(0.0 - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.3999999999999995e-128

   1. Initial program 67.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 67.2%

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

   3. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)}\right) \]

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

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

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

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

      6. associate-*r/ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 79.3%

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

   7. Simplified 79.3%

      \[\leadsto \color{blue}{0 - b\_2 \cdot \left(\frac{c \cdot -0.5}{b\_2 \cdot b\_2} + \frac{2}{a}\right)} \]

   if -6.3999999999999995e-128 < b_2 < 1.9000000000000001e-89

   1. Initial program 78.5%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 78.5%

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

   3. Simplified 78.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 77.9%

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

   7. Simplified 77.9%

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

   9. Applied egg-rr 77.9%

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

   if 1.9000000000000001e-89 < b_2

   1. Initial program 16.5%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 16.5%

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

   3. Simplified 16.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. 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 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.4 \cdot 10^{-128}:\\ \;\;\;\;b\_2 \cdot \left(\frac{c \cdot -0.5}{0 - b\_2 \cdot b\_2} - \frac{2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.9 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{0 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.8 \cdot 10^{-299}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.8e-299) (/ (* b_2 -2.0) a) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.8e-299) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (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 <= 2.8d-299) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (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 <= 2.8e-299) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.8e-299:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.8e-299)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = 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 <= 2.8e-299)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.8e-299], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.8000000000000001e-299

   1. Initial program 71.4%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 71.4%

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

   3. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 66.1%

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

   7. Simplified 66.1%

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

   if 2.8000000000000001e-299 < b_2

   1. Initial program 31.6%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 31.6%

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

   3. Simplified 31.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. 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 65.2%

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

   7. Simplified 65.2%

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

Alternative 6: 67.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.8 \cdot 10^{-299}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.8e-299) (/ (* b_2 -2.0) a) (* c (/ -0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.8e-299) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = 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 <= 2.8d-299) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = 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 <= 2.8e-299) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.8e-299:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.8e-299)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.8e-299)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = c * (-0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.8e-299], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.8000000000000001e-299

   1. Initial program 71.4%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 71.4%

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

   3. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 66.1%

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

   7. Simplified 66.1%

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

   if 2.8000000000000001e-299 < b_2

   1. Initial program 31.6%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 31.6%

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

   3. Simplified 31.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{8} \cdot \left(a \cdot c\right)}{{b\_2}^{3}} - \color{blue}{\frac{1}{2}} \cdot \frac{1}{b\_2}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\left(\frac{-1}{8} \cdot a\right) \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{8} \cdot a}{{b\_2}^{3}} \cdot c - \color{blue}{\frac{1}{2}} \cdot \frac{1}{b\_2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c - \frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) + \color{blue}{\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{b\_2}\right)\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b\_2}\right)\right), \left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{b\_2}\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right)} \cdot c\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b\_2}\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right)} \cdot c\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\frac{\frac{-1}{8} \cdot a}{{b\_2}^{3}} \cdot c\right)\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\frac{\left(\frac{-1}{8} \cdot a\right) \cdot c}{\color{blue}{{b\_2}^{3}}}\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\frac{\frac{-1}{8} \cdot \left(a \cdot c\right)}{{\color{blue}{b\_2}}^{3}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot c\right)\right), \color{blue}{\left({b\_2}^{3}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot c\right)\right), \left({\color{blue}{b\_2}}^{3}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(c \cdot a\right)\right), \left({b\_2}^{3}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \left({b\_2}^{3}\right)\right)\right)\right) \]

      21. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \left(b\_2 \cdot \color{blue}{\left(b\_2 \cdot b\_2\right)}\right)\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \left(b\_2 \cdot {b\_2}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b\_2, \color{blue}{\left({b\_2}^{2}\right)}\right)\right)\right)\right) \]

   7. Simplified 58.6%

      \[\leadsto \color{blue}{c \cdot \left(\frac{-0.5}{b\_2} + \frac{-0.125 \cdot \left(c \cdot a\right)}{b\_2 \cdot \left(b\_2 \cdot b\_2\right)}\right)} \]

   8. Taylor expanded in b_2 around inf

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{\frac{-1}{2}}{b\_2}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 65.0%

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

   10. Simplified 65.0%

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

Alternative 7: 67.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.8 \cdot 10^{-299}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.8e-299) (* b_2 (/ -2.0 a)) (* c (/ -0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.8e-299) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = 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 <= 2.8d-299) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = 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 <= 2.8e-299) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.8e-299:
		tmp = b_2 * (-2.0 / a)
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.8e-299)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.8e-299)
		tmp = b_2 * (-2.0 / a);
	else
		tmp = c * (-0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.8e-299], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.8000000000000001e-299

   1. Initial program 71.4%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 71.4%

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

   3. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 66.1%

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

   7. Simplified 66.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 65.9%

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

   9. Applied egg-rr 65.9%

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

   if 2.8000000000000001e-299 < b_2

   1. Initial program 31.6%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 31.6%

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

   3. Simplified 31.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{8} \cdot \left(a \cdot c\right)}{{b\_2}^{3}} - \color{blue}{\frac{1}{2}} \cdot \frac{1}{b\_2}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\left(\frac{-1}{8} \cdot a\right) \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{8} \cdot a}{{b\_2}^{3}} \cdot c - \color{blue}{\frac{1}{2}} \cdot \frac{1}{b\_2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c - \frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) + \color{blue}{\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{b\_2}\right)\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b\_2}\right)\right), \left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{b\_2}\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right)} \cdot c\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b\_2}\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right)} \cdot c\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\frac{\frac{-1}{8} \cdot a}{{b\_2}^{3}} \cdot c\right)\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\frac{\left(\frac{-1}{8} \cdot a\right) \cdot c}{\color{blue}{{b\_2}^{3}}}\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\frac{\frac{-1}{8} \cdot \left(a \cdot c\right)}{{\color{blue}{b\_2}}^{3}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot c\right)\right), \color{blue}{\left({b\_2}^{3}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot c\right)\right), \left({\color{blue}{b\_2}}^{3}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(c \cdot a\right)\right), \left({b\_2}^{3}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \left({b\_2}^{3}\right)\right)\right)\right) \]

      21. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \left(b\_2 \cdot \color{blue}{\left(b\_2 \cdot b\_2\right)}\right)\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \left(b\_2 \cdot {b\_2}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b\_2, \color{blue}{\left({b\_2}^{2}\right)}\right)\right)\right)\right) \]

   7. Simplified 58.6%

      \[\leadsto \color{blue}{c \cdot \left(\frac{-0.5}{b\_2} + \frac{-0.125 \cdot \left(c \cdot a\right)}{b\_2 \cdot \left(b\_2 \cdot b\_2\right)}\right)} \]

   8. Taylor expanded in b_2 around inf

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{\frac{-1}{2}}{b\_2}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 65.0%

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

   10. Simplified 65.0%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.8 \cdot 10^{-299}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.8 \cdot 10^{-299}:\\ \;\;\;\;0 - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.8e-299) (- 0.0 (/ b_2 a)) (* c (/ -0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.8e-299) {
		tmp = 0.0 - (b_2 / a);
	} else {
		tmp = 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 <= 2.8d-299) then
        tmp = 0.0d0 - (b_2 / a)
    else
        tmp = 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 <= 2.8e-299) {
		tmp = 0.0 - (b_2 / a);
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.8e-299:
		tmp = 0.0 - (b_2 / a)
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.8e-299)
		tmp = Float64(0.0 - Float64(b_2 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.8e-299)
		tmp = 0.0 - (b_2 / a);
	else
		tmp = c * (-0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.8e-299], N[(0.0 - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.8000000000000001e-299

   1. Initial program 71.4%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 71.4%

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

   3. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 41.4%

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

   7. Simplified 41.4%

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

   8. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 27.1%

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

   10. Simplified 27.1%

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

   if 2.8000000000000001e-299 < b_2

   1. Initial program 31.6%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 31.6%

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

   3. Simplified 31.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{8} \cdot \left(a \cdot c\right)}{{b\_2}^{3}} - \color{blue}{\frac{1}{2}} \cdot \frac{1}{b\_2}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\left(\frac{-1}{8} \cdot a\right) \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{8} \cdot a}{{b\_2}^{3}} \cdot c - \color{blue}{\frac{1}{2}} \cdot \frac{1}{b\_2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c - \frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) + \color{blue}{\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{b\_2}\right)\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b\_2}\right)\right), \left(\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{b\_2}\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right)} \cdot c\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b\_2}\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{a}{{b\_2}^{3}}\right) \cdot c\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right)} \cdot c\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\frac{\frac{-1}{8} \cdot a}{{b\_2}^{3}} \cdot c\right)\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\frac{\left(\frac{-1}{8} \cdot a\right) \cdot c}{\color{blue}{{b\_2}^{3}}}\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \left(\frac{\frac{-1}{8} \cdot \left(a \cdot c\right)}{{\color{blue}{b\_2}}^{3}}\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot c\right)\right), \color{blue}{\left({b\_2}^{3}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot c\right)\right), \left({\color{blue}{b\_2}}^{3}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(c \cdot a\right)\right), \left({b\_2}^{3}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \left({b\_2}^{3}\right)\right)\right)\right) \]

      21. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \left(b\_2 \cdot \color{blue}{\left(b\_2 \cdot b\_2\right)}\right)\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \left(b\_2 \cdot {b\_2}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b\_2, \color{blue}{\left({b\_2}^{2}\right)}\right)\right)\right)\right) \]

   7. Simplified 58.6%

      \[\leadsto \color{blue}{c \cdot \left(\frac{-0.5}{b\_2} + \frac{-0.125 \cdot \left(c \cdot a\right)}{b\_2 \cdot \left(b\_2 \cdot b\_2\right)}\right)} \]

   8. Taylor expanded in b_2 around inf

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{\frac{-1}{2}}{b\_2}\right)}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 65.0%

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

   10. Simplified 65.0%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.8 \cdot 10^{-299}:\\ \;\;\;\;0 - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 15.2% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{b\_2}{a} \end{array} \]

FPCore

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

C

double code(double a, double b_2, double c) {
	return 0.0 - (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 = 0.0d0 - (b_2 / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

   8. *-lowering-*.f64 52.1%

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

3. Simplified 52.1%

   \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
4. Add Preprocessing

5. Taylor expanded in b_2 around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 34.3%

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

7. Simplified 34.3%

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

8. Taylor expanded in b_2 around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. neg-lowering-neg.f64 15.2%

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

10. Simplified 15.2%

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

11. Final simplification 15.2%

    \[\leadsto 0 - \frac{b\_2}{a} \]
12. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

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) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	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(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	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 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	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[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024163 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :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) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

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