quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.5% → 85.5%

Time: 24.7s

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: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 8.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2 - \frac{c}{\frac{b\_2}{-0.5 \cdot a}}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-98)
   (/ (* c -0.5) b_2)
   (if (<= b_2 8.8e+94)
     (/ (- (- 0.0 b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (- (* b_2 -2.0) (/ c (/ b_2 (* -0.5 a)))) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-98) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 8.8e+94) {
		tmp = ((0.0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((b_2 * -2.0) - (c / (b_2 / (-0.5 * a)))) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.6d-98)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 8.8d+94) then
        tmp = ((0.0d0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((b_2 * (-2.0d0)) - (c / (b_2 / ((-0.5d0) * a)))) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-98) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 8.8e+94) {
		tmp = ((0.0 - b_2) - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((b_2 * -2.0) - (c / (b_2 / (-0.5 * a)))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.6e-98:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 8.8e+94:
		tmp = ((0.0 - b_2) - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = ((b_2 * -2.0) - (c / (b_2 / (-0.5 * a)))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e-98)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 8.8e+94)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(Float64(b_2 * -2.0) - Float64(c / Float64(b_2 / Float64(-0.5 * a)))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.6e-98)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 8.8e+94)
		tmp = ((0.0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = ((b_2 * -2.0) - (c / (b_2 / (-0.5 * a)))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e-98], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 8.8e+94], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(b$95$2 * -2.0), $MachinePrecision] - N[(c / N[(b$95$2 / N[(-0.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.60000000000000013e-98

   1. Initial program 12.7%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 90.0%

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

   5. Simplified 90.0%

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

   if -2.60000000000000013e-98 < b_2 < 8.80000000000000047e94

   1. Initial program 82.9%

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

   if 8.80000000000000047e94 < b_2

   1. Initial program 57.9%

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

   3. Taylor expanded in a around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-lft-neg-in N/A

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

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

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

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

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

      23. *-lowering-*.f64 91.2%

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

   5. Simplified 91.2%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2 + \frac{\left(a \cdot c\right) \cdot 0.5}{b\_2}}}{a} \]
   6. Step-by-step derivation

      1. fma-define N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

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

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

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

      8. associate-*r/ N/A

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

      9. frac-2neg N/A

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

      10. fmm-undef N/A

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

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

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

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

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

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

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

      16. /-lowering-/.f64 N/A

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

      17. *-lowering-*.f64 97.9%

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

   7. Applied egg-rr 97.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 8.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2 - \frac{c}{\frac{b\_2}{-0.5 \cdot a}}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{a \cdot \left(0 - c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-98)
   (/ (* c -0.5) b_2)
   (if (<= b_2 4.8e-59)
     (/ (- (- 0.0 b_2) (sqrt (* a (- 0.0 c)))) a)
     (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-98) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 4.8e-59) {
		tmp = ((0.0 - b_2) - sqrt((a * (0.0 - c)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.6d-98)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 4.8d-59) then
        tmp = ((0.0d0 - b_2) - sqrt((a * (0.0d0 - c)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-98) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 4.8e-59) {
		tmp = ((0.0 - b_2) - Math.sqrt((a * (0.0 - c)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.6e-98:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 4.8e-59:
		tmp = ((0.0 - b_2) - math.sqrt((a * (0.0 - c)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e-98)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 4.8e-59)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(a * Float64(0.0 - c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.6e-98)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 4.8e-59)
		tmp = ((0.0 - b_2) - sqrt((a * (0.0 - c)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e-98], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 4.8e-59], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - N[Sqrt[N[(a * N[(0.0 - c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.60000000000000013e-98

   1. Initial program 12.7%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 90.0%

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

   5. Simplified 90.0%

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

   if -2.60000000000000013e-98 < b_2 < 4.8000000000000003e-59

   1. Initial program 79.8%

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

   3. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 71.4%

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

   5. Simplified 71.4%

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

   if 4.8000000000000003e-59 < b_2

   1. Initial program 72.3%

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

   3. Taylor expanded in c around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 87.4%

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

   5. Simplified 87.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{a \cdot \left(0 - c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (/ (* c -0.5) b_2)
   (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 33.1%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 64.0%

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

   5. Simplified 64.0%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 77.0%

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

   3. Taylor expanded in c around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 60.9%

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

   5. Simplified 60.9%

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

Alternative 4: 68.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.5 \cdot 10^{-292}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.5e-292) (/ (* c -0.5) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.5e-292) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.5d-292)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.5e-292) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.5e-292:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.5e-292)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.5e-292)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.5e-292], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.5e-292

   1. Initial program 30.9%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 66.0%

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

   5. Simplified 66.0%

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

   if -3.5e-292 < b_2

   1. Initial program 77.6%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 58.6%

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

   5. Simplified 58.6%

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

Alternative 5: 43.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.05 \cdot 10^{-31}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.05e-31) (/ (* c 0.5) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.05e-31) {
		tmp = (c * 0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.05d-31)) then
        tmp = (c * 0.5d0) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.05e-31) {
		tmp = (c * 0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.05e-31:
		tmp = (c * 0.5) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.05e-31)
		tmp = Float64(Float64(c * 0.5) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.05e-31)
		tmp = (c * 0.5) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.05e-31], N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.04999999999999996e-31

   1. Initial program 8.7%

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

   3. Taylor expanded in a around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-lft-neg-in N/A

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

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

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

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

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

      23. *-lowering-*.f64 2.1%

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

   5. Simplified 2.1%

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

   6. Taylor expanded in b_2 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2}} \]
   7. 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. *-lowering-*.f64 30.5%

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

   8. Simplified 30.5%

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

   if -1.04999999999999996e-31 < b_2

   1. Initial program 74.4%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 44.0%

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

   5. Simplified 44.0%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.05 \cdot 10^{-31}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-34}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-34) (/ (* c 0.5) b_2) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-34) {
		tmp = (c * 0.5) / b_2;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.9d-34)) then
        tmp = (c * 0.5d0) / b_2
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-34) {
		tmp = (c * 0.5) / b_2;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.9e-34:
		tmp = (c * 0.5) / b_2
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e-34)
		tmp = Float64(Float64(c * 0.5) / b_2);
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.9e-34)
		tmp = (c * 0.5) / b_2;
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e-34], N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.9000000000000002e-34

   1. Initial program 8.7%

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

   3. Taylor expanded in a around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-lft-neg-in N/A

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

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

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

      19. *-commutative N/A

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

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

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

      21. metadata-eval N/A

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

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

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

      23. *-lowering-*.f64 2.1%

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

   5. Simplified 2.1%

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

   6. Taylor expanded in b_2 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2}} \]
   7. 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. *-lowering-*.f64 30.5%

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

   8. Simplified 30.5%

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

   if -2.9000000000000002e-34 < b_2

   1. Initial program 74.4%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 44.0%

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

   5. Simplified 44.0%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]
   6. 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 43.9%

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

   7. Applied egg-rr 43.9%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-34}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.2% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ b\_2 \cdot \frac{-2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* b_2 (/ -2.0 a)))

C

double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / 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 * ((-2.0d0) / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / a);
}

Python

def code(a, b_2, c):
	return b_2 * (-2.0 / a)

Julia

function code(a, b_2, c)
	return Float64(b_2 * Float64(-2.0 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = b_2 * (-2.0 / a);
end

Wolfram

code[a_, b$95$2_, c_] := N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in b_2 around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 31.8%

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

5. Simplified 31.8%

   \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]
6. 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 31.7%

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

7. Applied egg-rr 31.7%

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

8. Final simplification 31.7%

   \[\leadsto b\_2 \cdot \frac{-2}{a} \]
9. Add Preprocessing

Alternative 8: 2.5% accurate, 37.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

4. Applied egg-rr 36.5%

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

5. Taylor expanded in b_2 around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}\right)\right)\right), \mathsf{\_.f64}\left(0, a\right)\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

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

   4. *-lowering-*.f64 29.8%

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

7. Simplified 29.8%

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

8. Taylor expanded in b_2 around inf

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

   1. /-lowering-/.f64 2.5%

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

10. Simplified 2.5%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024141 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

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