quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.6% → 84.3%

Time: 11.8s

Alternatives: 8

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: 51.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{+148}:\\ \;\;\;\;\frac{b\_2 \cdot \left(2 + -0.5 \cdot \left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right)\right)}{0 - a}\\ \mathbf{elif}\;b\_2 \leq 8 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3e+148)
   (/ (* b_2 (+ 2.0 (* -0.5 (* a (/ c (* b_2 b_2)))))) (- 0.0 a))
   (if (<= b_2 8e-160)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e+148) {
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a);
	} else if (b_2 <= 8e-160) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / 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 <= (-3d+148)) then
        tmp = (b_2 * (2.0d0 + ((-0.5d0) * (a * (c / (b_2 * b_2)))))) / (0.0d0 - a)
    else if (b_2 <= 8d-160) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = ((-0.5d0) * c) / 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 <= -3e+148) {
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a);
	} else if (b_2 <= 8e-160) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3e+148:
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a)
	elif b_2 <= 8e-160:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3e+148)
		tmp = Float64(Float64(b_2 * Float64(2.0 + Float64(-0.5 * Float64(a * Float64(c / Float64(b_2 * b_2)))))) / Float64(0.0 - a));
	elseif (b_2 <= 8e-160)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3e+148)
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a);
	elseif (b_2 <= 8e-160)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3e+148], N[(N[(b$95$2 * N[(2.0 + N[(-0.5 * N[(a * N[(c / N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 8e-160], 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[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.00000000000000015e148

   1. Initial program 50.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 50.6%

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

   3. Simplified 50.6%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. clear-num N/A

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

      8. flip-- N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 50.6%

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

   6. Applied egg-rr 50.6%

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

   7. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 95.0%

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

   9. Simplified 95.0%

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

   if -3.00000000000000015e148 < b_2 < 7.9999999999999999e-160

   1. Initial program 86.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 86.8%

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

   3. Simplified 86.8%

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

   if 7.9999999999999999e-160 < b_2

   1. Initial program 21.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 21.8%

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

   3. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   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 83.6%

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

   7. Simplified 83.6%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{+148}:\\ \;\;\;\;\frac{b\_2 \cdot \left(2 + -0.5 \cdot \left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right)\right)}{0 - a}\\ \mathbf{elif}\;b\_2 \leq 8 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{b\_2 \cdot \left(2 + -0.5 \cdot \left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right)\right)}{0 - a}\\ \mathbf{elif}\;b\_2 \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{0 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.2e-139)
   (/ (* b_2 (+ 2.0 (* -0.5 (* a (/ c (* b_2 b_2)))))) (- 0.0 a))
   (if (<= b_2 4.2e-160)
     (/ (- (sqrt (- 0.0 (* a c))) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-139) {
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a);
	} else if (b_2 <= 4.2e-160) {
		tmp = (sqrt((0.0 - (a * c))) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / 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 <= (-4.2d-139)) then
        tmp = (b_2 * (2.0d0 + ((-0.5d0) * (a * (c / (b_2 * b_2)))))) / (0.0d0 - a)
    else if (b_2 <= 4.2d-160) then
        tmp = (sqrt((0.0d0 - (a * c))) - b_2) / a
    else
        tmp = ((-0.5d0) * c) / 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 <= -4.2e-139) {
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a);
	} else if (b_2 <= 4.2e-160) {
		tmp = (Math.sqrt((0.0 - (a * c))) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.2e-139:
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a)
	elif b_2 <= 4.2e-160:
		tmp = (math.sqrt((0.0 - (a * c))) - b_2) / a
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.2e-139)
		tmp = Float64(Float64(b_2 * Float64(2.0 + Float64(-0.5 * Float64(a * Float64(c / Float64(b_2 * b_2)))))) / Float64(0.0 - a));
	elseif (b_2 <= 4.2e-160)
		tmp = Float64(Float64(sqrt(Float64(0.0 - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.2e-139)
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a);
	elseif (b_2 <= 4.2e-160)
		tmp = (sqrt((0.0 - (a * c))) - b_2) / a;
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.2e-139], N[(N[(b$95$2 * N[(2.0 + N[(-0.5 * N[(a * N[(c / N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.2e-160], N[(N[(N[Sqrt[N[(0.0 - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.20000000000000016e-139

   1. Initial program 74.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 74.6%

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

   3. Simplified 74.6%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. clear-num N/A

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

      8. flip-- N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 74.5%

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

   6. Applied egg-rr 74.5%

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

   7. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 79.3%

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

   9. Simplified 79.3%

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

   if -4.20000000000000016e-139 < b_2 < 4.2000000000000001e-160

   1. Initial program 83.0%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in b_2 around 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 81.2%

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

   7. Simplified 81.2%

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

   9. Applied egg-rr 81.2%

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

   if 4.2000000000000001e-160 < b_2

   1. Initial program 21.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 21.8%

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

   3. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   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 83.6%

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

   7. Simplified 83.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{b\_2 \cdot \left(2 + -0.5 \cdot \left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right)\right)}{0 - a}\\ \mathbf{elif}\;b\_2 \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{0 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2 \cdot \left(2 + -0.5 \cdot \left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right)\right)}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-311)
   (/ (* b_2 (+ 2.0 (* -0.5 (* a (/ c (* b_2 b_2)))))) (- 0.0 a))
   (/ (* -0.5 c) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-311) {
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a);
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d-311)) then
        tmp = (b_2 * (2.0d0 + ((-0.5d0) * (a * (c / (b_2 * b_2)))))) / (0.0d0 - a)
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-311) {
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a);
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-311:
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a)
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-311)
		tmp = Float64(Float64(b_2 * Float64(2.0 + Float64(-0.5 * Float64(a * Float64(c / Float64(b_2 * b_2)))))) / Float64(0.0 - a));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-311)
		tmp = (b_2 * (2.0 + (-0.5 * (a * (c / (b_2 * b_2)))))) / (0.0 - a);
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-311], N[(N[(b$95$2 * N[(2.0 + N[(-0.5 * N[(a * N[(c / N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.99999999999948e-312

   1. Initial program 76.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 76.2%

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

   3. Simplified 76.2%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. clear-num N/A

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

      8. flip-- N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 76.2%

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

   6. Applied egg-rr 76.2%

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

   7. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 67.5%

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

   9. Simplified 67.5%

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

   if -9.99999999999948e-312 < b_2

   1. Initial program 33.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 33.3%

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

   3. Simplified 33.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   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 69.3%

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

   7. Simplified 69.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2 \cdot \left(2 + -0.5 \cdot \left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right)\right)}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-311}:\\ \;\;\;\;b\_2 \cdot \left(\frac{-0.5 \cdot c}{0 - b\_2 \cdot b\_2} - \frac{2}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-311)
   (* b_2 (- (/ (* -0.5 c) (- 0.0 (* b_2 b_2))) (/ 2.0 a)))
   (/ (* -0.5 c) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-311) {
		tmp = b_2 * (((-0.5 * c) / (0.0 - (b_2 * b_2))) - (2.0 / a));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d-311)) then
        tmp = b_2 * ((((-0.5d0) * c) / (0.0d0 - (b_2 * b_2))) - (2.0d0 / a))
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-311) {
		tmp = b_2 * (((-0.5 * c) / (0.0 - (b_2 * b_2))) - (2.0 / a));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-311:
		tmp = b_2 * (((-0.5 * c) / (0.0 - (b_2 * b_2))) - (2.0 / a))
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-311)
		tmp = Float64(b_2 * Float64(Float64(Float64(-0.5 * c) / Float64(0.0 - Float64(b_2 * b_2))) - Float64(2.0 / a)));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-311)
		tmp = b_2 * (((-0.5 * c) / (0.0 - (b_2 * b_2))) - (2.0 / a));
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-311], N[(b$95$2 * N[(N[(N[(-0.5 * c), $MachinePrecision] / N[(0.0 - N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.99999999999948e-312

   1. Initial program 76.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 76.2%

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

   3. Simplified 76.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   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 67.4%

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

   7. Simplified 67.4%

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

   if -9.99999999999948e-312 < b_2

   1. Initial program 33.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 33.3%

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

   3. Simplified 33.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   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 69.3%

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

   7. Simplified 69.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-311}:\\ \;\;\;\;b\_2 \cdot \left(\frac{-0.5 \cdot c}{0 - b\_2 \cdot b\_2} - \frac{2}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 4.1 \cdot 10^{-285}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 4.1e-285) (/ (* b_2 -2.0) a) (/ (* -0.5 c) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 4.1e-285) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 * c) / 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 <= 4.1d-285) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = ((-0.5d0) * c) / 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 <= 4.1e-285) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 4.1e-285:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 4.1e-285)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 4.1e-285)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 4.1e-285], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 4.1e-285

   1. Initial program 76.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   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 63.1%

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

   7. Simplified 63.1%

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

   if 4.1e-285 < b_2

   1. Initial program 30.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 30.7%

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

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   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 73.2%

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

   7. Simplified 73.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 4.1 \cdot 10^{-285}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 4.1 \cdot 10^{-285}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 4.1e-285) (/ (* 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 <= 4.1e-285) {
		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 <= 4.1d-285) 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 <= 4.1e-285) {
		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 <= 4.1e-285:
		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 <= 4.1e-285)
		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 <= 4.1e-285)
		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, 4.1e-285], 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 < 4.1e-285

   1. Initial program 76.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   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 63.1%

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

   7. Simplified 63.1%

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

   if 4.1e-285 < b_2

   1. Initial program 30.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 30.7%

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

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   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 63.0%

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

   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 73.0%

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

   10. Simplified 73.0%

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

Alternative 7: 67.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 4.1 \cdot 10^{-285}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 4.1e-285) (* 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 <= 4.1e-285) {
		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 <= 4.1d-285) 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 <= 4.1e-285) {
		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 <= 4.1e-285:
		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 <= 4.1e-285)
		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 <= 4.1e-285)
		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, 4.1e-285], 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 < 4.1e-285

   1. Initial program 76.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   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 63.1%

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

   7. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{b\_2 \cdot -2}{a}} \]
   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 62.9%

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

   9. Applied egg-rr 62.9%

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

   if 4.1e-285 < b_2

   1. Initial program 30.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 30.7%

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

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   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 63.0%

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

   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 73.0%

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

   10. Simplified 73.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 4.1 \cdot 10^{-285}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.0% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* c (/ -0.5 b_2)))

C

double code(double a, double b_2, double c) {
	return c * (-0.5 / b_2);
}

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 = c * ((-0.5d0) / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return c * (-0.5 / b_2);
}

Python

def code(a, b_2, c):
	return c * (-0.5 / b_2)

Julia

function code(a, b_2, c)
	return Float64(c * Float64(-0.5 / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = c * (-0.5 / b_2);
end

Wolfram

code[a_, b$95$2_, c_] := N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.4%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
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 52.4%

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

3. Simplified 52.4%

   \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
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 34.4%

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

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 39.3%

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

10. Simplified 39.3%

    \[\leadsto c \cdot \color{blue}{\frac{-0.5}{b\_2}} \]
11. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

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

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 2024155 
(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))