quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.3% → 85.2%

Time: 14.0s

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: 52.3% 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: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\frac{b\_2 \cdot \mathsf{fma}\left(-0.5, \frac{c}{b\_2} \cdot \frac{a}{b\_2}, 2\right)}{-a}\\ \mathbf{elif}\;b\_2 \leq 2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+114)
   (/ (* b_2 (fma -0.5 (* (/ c b_2) (/ a b_2)) 2.0)) (- a))
   (if (<= b_2 2e-76)
     (/ (- (sqrt (fma a (- c) (* b_2 b_2))) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+114) {
		tmp = (b_2 * fma(-0.5, ((c / b_2) * (a / b_2)), 2.0)) / -a;
	} else if (b_2 <= 2e-76) {
		tmp = (sqrt(fma(a, -c, (b_2 * b_2))) - 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 <= -2e+114)
		tmp = Float64(Float64(b_2 * fma(-0.5, Float64(Float64(c / b_2) * Float64(a / b_2)), 2.0)) / Float64(-a));
	elseif (b_2 <= 2e-76)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-c), Float64(b_2 * b_2))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e+114], N[(N[(b$95$2 * N[(-0.5 * N[(N[(c / b$95$2), $MachinePrecision] * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], If[LessEqual[b$95$2, 2e-76], N[(N[(N[Sqrt[N[(a * (-c) + N[(b$95$2 * b$95$2), $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 < -2e114

   1. Initial program 47.4%

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

   3. Taylor expanded in a around inf 47.4%

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

      1. unpow2 47.4%

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

   5. Simplified 47.4%

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

   6. Taylor expanded in b_2 around -inf 85.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 \cdot \left(2 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)}}{a} \]
   7. Step-by-step derivation

      1. mul-1-neg 85.9%

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

      2. *-commutative 85.9%

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

      3. distribute-rgt-neg-in 85.9%

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

      4. +-commutative 85.9%

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

      5. fma-define 85.9%

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

      6. *-commutative 85.9%

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

      7. unpow2 85.9%

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

      8. times-frac 92.5%

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

   8. Simplified 92.5%

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

   if -2e114 < b_2 < 1.99999999999999985e-76

   1. Initial program 77.1%

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

      1. remove-double-neg 77.1%

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

      2. distribute-frac-neg2 77.1%

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

      3. distribute-neg-frac 77.1%

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

      4. distribute-neg-out 77.1%

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

      5. remove-double-neg 77.1%

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

      6. sub-neg 77.1%

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

      7. neg-mul-1 77.1%

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

      8. associate-/r* 77.1%

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

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} - b\_2}{a}} \]
   4. Add Preprocessing

   if 1.99999999999999985e-76 < b_2

   1. Initial program 14.8%

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

   3. Taylor expanded in b_2 around inf 89.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 89.7%

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

   5. Applied egg-rr 89.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\frac{b\_2 \cdot \mathsf{fma}\left(-0.5, \frac{c}{b\_2} \cdot \frac{a}{b\_2}, 2\right)}{-a}\\ \mathbf{elif}\;b\_2 \leq 2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.65 \cdot 10^{+114}:\\ \;\;\;\;\frac{b\_2 \cdot \mathsf{fma}\left(-0.5, \frac{c}{b\_2} \cdot \frac{a}{b\_2}, 2\right)}{-a}\\ \mathbf{elif}\;b\_2 \leq 4.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.65e+114)
   (/ (* b_2 (fma -0.5 (* (/ c b_2) (/ a b_2)) 2.0)) (- a))
   (if (<= b_2 4.4e-75)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.65e+114) {
		tmp = (b_2 * fma(-0.5, ((c / b_2) * (a / b_2)), 2.0)) / -a;
	} else if (b_2 <= 4.4e-75) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - 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 <= -2.65e+114)
		tmp = Float64(Float64(b_2 * fma(-0.5, Float64(Float64(c / b_2) * Float64(a / b_2)), 2.0)) / Float64(-a));
	elseif (b_2 <= 4.4e-75)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.65e+114], N[(N[(b$95$2 * N[(-0.5 * N[(N[(c / b$95$2), $MachinePrecision] * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], If[LessEqual[b$95$2, 4.4e-75], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $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 < -2.65e114

   1. Initial program 47.4%

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

   3. Taylor expanded in a around inf 47.4%

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

      1. unpow2 47.4%

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

   5. Simplified 47.4%

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

   6. Taylor expanded in b_2 around -inf 85.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 \cdot \left(2 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)}}{a} \]
   7. Step-by-step derivation

      1. mul-1-neg 85.9%

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

      2. *-commutative 85.9%

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

      3. distribute-rgt-neg-in 85.9%

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

      4. +-commutative 85.9%

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

      5. fma-define 85.9%

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

      6. *-commutative 85.9%

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

      7. unpow2 85.9%

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

      8. times-frac 92.5%

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

   8. Simplified 92.5%

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

   if -2.65e114 < b_2 < 4.40000000000000011e-75

   1. Initial program 77.1%

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

   if 4.40000000000000011e-75 < b_2

   1. Initial program 14.8%

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

   3. Taylor expanded in b_2 around inf 89.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 89.7%

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

   5. Applied egg-rr 89.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.65 \cdot 10^{+114}:\\ \;\;\;\;\frac{b\_2 \cdot \mathsf{fma}\left(-0.5, \frac{c}{b\_2} \cdot \frac{a}{b\_2}, 2\right)}{-a}\\ \mathbf{elif}\;b\_2 \leq 4.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{+114}:\\ \;\;\;\;\frac{b\_2 \cdot \left(\left(-2\right) - -0.5 \cdot \left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right)\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.8e+114)
   (/ (* b_2 (- (- 2.0) (* -0.5 (* a (/ c (* b_2 b_2)))))) a)
   (if (<= b_2 1.8e-73)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.8e+114) {
		tmp = (b_2 * (-2.0 - (-0.5 * (a * (c / (b_2 * b_2)))))) / a;
	} else if (b_2 <= 1.8e-73) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - 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 <= (-1.8d+114)) then
        tmp = (b_2 * (-2.0d0 - ((-0.5d0) * (a * (c / (b_2 * b_2)))))) / a
    else if (b_2 <= 1.8d-73) then
        tmp = (sqrt(((b_2 * b_2) - (c * a))) - 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 <= -1.8e+114) {
		tmp = (b_2 * (-2.0 - (-0.5 * (a * (c / (b_2 * b_2)))))) / a;
	} else if (b_2 <= 1.8e-73) {
		tmp = (Math.sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.8e+114:
		tmp = (b_2 * (-2.0 - (-0.5 * (a * (c / (b_2 * b_2)))))) / a
	elif b_2 <= 1.8e-73:
		tmp = (math.sqrt(((b_2 * b_2) - (c * a))) - 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 <= -1.8e+114)
		tmp = Float64(Float64(b_2 * Float64(Float64(-2.0) - Float64(-0.5 * Float64(a * Float64(c / Float64(b_2 * b_2)))))) / a);
	elseif (b_2 <= 1.8e-73)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - 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 <= -1.8e+114)
		tmp = (b_2 * (-2.0 - (-0.5 * (a * (c / (b_2 * b_2)))))) / a;
	elseif (b_2 <= 1.8e-73)
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - 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, -1.8e+114], 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] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.8e-73], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $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 < -1.8e114

   1. Initial program 47.4%

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

   3. Taylor expanded in a around inf 47.4%

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

      1. unpow2 47.4%

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

   5. Simplified 47.4%

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

      1. +-commutative 47.4%

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

      2. pow1/2 47.4%

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

      3. neg-sub0 47.4%

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

   7. Applied egg-rr 47.4%

      \[\leadsto \frac{\color{blue}{{\left(a \cdot \left(\frac{b\_2 \cdot b\_2}{a} - c\right)\right)}^{0.5} + \left(0 - b\_2\right)}}{a} \]
   8. Step-by-step derivation

      1. associate-+r- 47.4%

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

      2. +-rgt-identity 47.4%

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

      3. unpow1/2 47.4%

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

      4. associate-*r/ 47.4%

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

   9. Simplified 47.4%

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

   10. Taylor expanded in b_2 around -inf 85.9%

       \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 \cdot \left(2 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)}}{a} \]
   11. Step-by-step derivation

       1. mul-1-neg 85.9%

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

       2. distribute-rgt-neg-in 85.9%

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

       3. associate-/l* 92.5%

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

       4. unpow2 92.5%

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

   12. Simplified 92.5%

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

   if -1.8e114 < b_2 < 1.8e-73

   1. Initial program 77.1%

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

   if 1.8e-73 < b_2

   1. Initial program 14.8%

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

   3. Taylor expanded in b_2 around inf 89.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 89.7%

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

   5. Applied egg-rr 89.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{+114}:\\ \;\;\;\;\frac{b\_2 \cdot \left(\left(-2\right) - -0.5 \cdot \left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right)\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.15 \cdot 10^{-67}:\\ \;\;\;\;\frac{b\_2 \cdot \left(\left(-2\right) - -0.5 \cdot \left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right)\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.15e-67)
   (/ (* b_2 (- (- 2.0) (* -0.5 (* a (/ c (* b_2 b_2)))))) a)
   (if (<= b_2 7.5e-76) (/ (- (sqrt (* c (- a))) b_2) a) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.15e-67) {
		tmp = (b_2 * (-2.0 - (-0.5 * (a * (c / (b_2 * b_2)))))) / a;
	} else if (b_2 <= 7.5e-76) {
		tmp = (sqrt((c * -a)) - 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 <= (-2.15d-67)) then
        tmp = (b_2 * (-2.0d0 - ((-0.5d0) * (a * (c / (b_2 * b_2)))))) / a
    else if (b_2 <= 7.5d-76) then
        tmp = (sqrt((c * -a)) - 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 <= -2.15e-67) {
		tmp = (b_2 * (-2.0 - (-0.5 * (a * (c / (b_2 * b_2)))))) / a;
	} else if (b_2 <= 7.5e-76) {
		tmp = (Math.sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.15e-67:
		tmp = (b_2 * (-2.0 - (-0.5 * (a * (c / (b_2 * b_2)))))) / a
	elif b_2 <= 7.5e-76:
		tmp = (math.sqrt((c * -a)) - 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 <= -2.15e-67)
		tmp = Float64(Float64(b_2 * Float64(Float64(-2.0) - Float64(-0.5 * Float64(a * Float64(c / Float64(b_2 * b_2)))))) / a);
	elseif (b_2 <= 7.5e-76)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(-a))) - 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 <= -2.15e-67)
		tmp = (b_2 * (-2.0 - (-0.5 * (a * (c / (b_2 * b_2)))))) / a;
	elseif (b_2 <= 7.5e-76)
		tmp = (sqrt((c * -a)) - 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, -2.15e-67], 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] / a), $MachinePrecision], If[LessEqual[b$95$2, 7.5e-76], N[(N[(N[Sqrt[N[(c * (-a)), $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 < -2.15000000000000013e-67

   1. Initial program 67.7%

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

   3. Taylor expanded in a around inf 64.7%

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

      1. unpow2 64.7%

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

   5. Simplified 64.7%

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

      1. +-commutative 64.7%

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

      2. pow1/2 64.7%

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

      3. neg-sub0 64.7%

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

   7. Applied egg-rr 64.7%

      \[\leadsto \frac{\color{blue}{{\left(a \cdot \left(\frac{b\_2 \cdot b\_2}{a} - c\right)\right)}^{0.5} + \left(0 - b\_2\right)}}{a} \]
   8. Step-by-step derivation

      1. associate-+r- 64.7%

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

      2. +-rgt-identity 64.7%

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

      3. unpow1/2 64.7%

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

      4. associate-*r/ 64.7%

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

   9. Simplified 64.7%

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

   10. Taylor expanded in b_2 around -inf 82.7%

       \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 \cdot \left(2 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)}}{a} \]
   11. Step-by-step derivation

       1. mul-1-neg 82.7%

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

       2. distribute-rgt-neg-in 82.7%

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

       3. associate-/l* 86.2%

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

       4. unpow2 86.2%

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

   12. Simplified 86.2%

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

   if -2.15000000000000013e-67 < b_2 < 7.4999999999999997e-76

   1. Initial program 68.5%

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

   3. Taylor expanded in b_2 around 0 62.0%

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

      1. mul-1-neg 62.0%

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

      2. distribute-rgt-neg-in 62.0%

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

   5. Simplified 62.0%

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

      1. div-inv 61.9%

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

      2. neg-sub0 61.9%

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

      3. pow1/2 61.9%

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

      4. neg-sub0 61.9%

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

   7. Applied egg-rr 61.9%

      \[\leadsto \color{blue}{\left(\left(0 - b\_2\right) + {\left(a \cdot \left(0 - c\right)\right)}^{0.5}\right) \cdot \frac{1}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/ 62.0%

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

      2. *-rgt-identity 62.0%

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

      3. +-commutative 62.0%

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

      4. associate-+r- 62.0%

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

      5. +-rgt-identity 62.0%

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

      6. unpow1/2 62.0%

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

      7. sub0-neg 62.0%

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

   9. Simplified 62.0%

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

   if 7.4999999999999997e-76 < b_2

   1. Initial program 14.8%

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

   3. Taylor expanded in b_2 around inf 89.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 89.7%

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

   5. Applied egg-rr 89.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.15 \cdot 10^{-67}:\\ \;\;\;\;\frac{b\_2 \cdot \left(\left(-2\right) - -0.5 \cdot \left(a \cdot \frac{c}{b\_2 \cdot b\_2}\right)\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.15 \cdot 10^{-277}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.15e-277) (/ (* 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 <= 1.15e-277) {
		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 <= 1.15d-277) 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 <= 1.15e-277) {
		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 <= 1.15e-277:
		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 <= 1.15e-277)
		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 <= 1.15e-277)
		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, 1.15e-277], 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 < 1.15e-277

   1. Initial program 68.2%

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

   3. Taylor expanded in b_2 around -inf 67.3%

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

      1. *-commutative 67.3%

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

   5. Simplified 67.3%

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

   if 1.15e-277 < b_2

   1. Initial program 27.5%

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

   3. Taylor expanded in b_2 around inf 73.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 73.8%

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

   5. Applied egg-rr 73.8%

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

Alternative 6: 47.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.15 \cdot 10^{-277}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.15e-277) (/ b_2 (- a)) (/ (* -0.5 c) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.15e-277) {
		tmp = 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 <= 1.15d-277) then
        tmp = 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 <= 1.15e-277) {
		tmp = b_2 / -a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.15e-277:
		tmp = 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 <= 1.15e-277)
		tmp = Float64(b_2 / Float64(-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 <= 1.15e-277)
		tmp = 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, 1.15e-277], N[(b$95$2 / (-a)), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.15e-277

   1. Initial program 68.2%

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

   3. Taylor expanded in b_2 around 0 37.3%

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

      1. mul-1-neg 37.3%

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

      2. distribute-rgt-neg-in 37.3%

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

   5. Simplified 37.3%

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

   6. Taylor expanded in b_2 around inf 26.6%

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

      1. mul-1-neg 26.6%

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

      2. distribute-neg-frac2 26.6%

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

   8. Simplified 26.6%

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

   if 1.15e-277 < b_2

   1. Initial program 27.5%

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

   3. Taylor expanded in b_2 around inf 73.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ 73.8%

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

   5. Applied egg-rr 73.8%

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

Alternative 7: 47.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.15 \cdot 10^{-277}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.15e-277) (/ b_2 (- a)) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.15e-277) {
		tmp = 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 <= 1.15d-277) then
        tmp = 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 <= 1.15e-277) {
		tmp = b_2 / -a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.15e-277:
		tmp = 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 <= 1.15e-277)
		tmp = Float64(b_2 / Float64(-a));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.15e-277)
		tmp = 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, 1.15e-277], N[(b$95$2 / (-a)), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.15e-277

   1. Initial program 68.2%

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

   3. Taylor expanded in b_2 around 0 37.3%

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

      1. mul-1-neg 37.3%

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

      2. distribute-rgt-neg-in 37.3%

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

   5. Simplified 37.3%

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

   6. Taylor expanded in b_2 around inf 26.6%

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

      1. mul-1-neg 26.6%

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

      2. distribute-neg-frac2 26.6%

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

   8. Simplified 26.6%

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

   if 1.15e-277 < b_2

   1. Initial program 27.5%

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

   3. Taylor expanded in b_2 around inf 73.7%

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

Alternative 8: 15.2% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.3%

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

3. Taylor expanded in b_2 around 0 30.0%

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

   1. mul-1-neg 30.0%

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

   2. distribute-rgt-neg-in 30.0%

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

5. Simplified 30.0%

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

6. Taylor expanded in b_2 around inf 14.9%

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

   1. mul-1-neg 14.9%

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

   2. distribute-neg-frac2 14.9%

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

8. Simplified 14.9%

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

Developer target: 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 2024097 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b_2 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))) b_2) a) (/ (- c) (+ b_2 (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))))))

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