quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.8% → 86.6%

Time: 11.0s

Alternatives: 8

Speedup: 15.9×

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: 51.8% 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: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{+140}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+140)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 2.2e-52)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+140) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.2e-52) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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+140)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 2.2d-52) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+140) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.2e-52) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e+140:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 2.2e-52:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+140)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 2.2e-52)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e+140)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 2.2e-52)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+140], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.2e-52], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.00000000000000008e140

   1. Initial program 46.8%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 46.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 46.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 95.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -5.00000000000000008e140 < b_2 < 2.20000000000000009e-52

   1. Initial program 80.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 80.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 80.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   if 2.20000000000000009e-52 < b_2

   1. Initial program 16.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 16.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 16.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 16.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 91.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{+140}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 2: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.9 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-8)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.9e-53)
     (- (/ (sqrt (* a (- c))) a) (/ b_2 a))
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-8) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.9e-53) {
		tmp = (sqrt((a * -c)) / a) - (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.8d-8)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 1.9d-53) then
        tmp = (sqrt((a * -c)) / a) - (b_2 / a)
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-8) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.9e-53) {
		tmp = (Math.sqrt((a * -c)) / a) - (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-8:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 1.9e-53:
		tmp = (math.sqrt((a * -c)) / a) - (b_2 / a)
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-8)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.9e-53)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) / a) - Float64(b_2 / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-8)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 1.9e-53)
		tmp = (sqrt((a * -c)) / a) - (b_2 / a);
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-8], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.9e-53], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.7999999999999999e-8

   1. Initial program 64.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 64.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 64.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 89.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -2.7999999999999999e-8 < b_2 < 1.8999999999999999e-53

   1. Initial program 78.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 78.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 78.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 78.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow2 78.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}} - b_2}{a} \]

      3. pow1/2 78.0%

         \[\leadsto \frac{{\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b_2}{a} \]

      4. sqrt-pow1 78.1%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b_2}{a} \]

      5. fma-neg 78.1%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      6. *-commutative 78.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      7. distribute-rgt-neg-in 78.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      8. metadata-eval 78.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b_2}{a} \]

   5. Applied egg-rr 78.1%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}} - b_2}{a} \]

   6. Taylor expanded in b_2 around 0 69.6%

      \[\leadsto \frac{{\color{blue}{\left({\left(-1 \cdot \left(c \cdot a\right)\right)}^{0.25}\right)}}^{2} - b_2}{a} \]
   7. Step-by-step derivation

      1. mul-1-neg 69.6%

         \[\leadsto \frac{{\left({\color{blue}{\left(-c \cdot a\right)}}^{0.25}\right)}^{2} - b_2}{a} \]

      2. distribute-rgt-neg-out 69.6%

         \[\leadsto \frac{{\left({\color{blue}{\left(c \cdot \left(-a\right)\right)}}^{0.25}\right)}^{2} - b_2}{a} \]

   8. Simplified 69.6%

      \[\leadsto \frac{{\color{blue}{\left({\left(c \cdot \left(-a\right)\right)}^{0.25}\right)}}^{2} - b_2}{a} \]
   9. Step-by-step derivation

      1. pow-pow 69.8%

         \[\leadsto \frac{\color{blue}{{\left(c \cdot \left(-a\right)\right)}^{\left(0.25 \cdot 2\right)}} - b_2}{a} \]

      2. metadata-eval 69.8%

         \[\leadsto \frac{{\left(c \cdot \left(-a\right)\right)}^{\color{blue}{0.5}} - b_2}{a} \]

      3. pow1/2 69.8%

         \[\leadsto \frac{\color{blue}{\sqrt{c \cdot \left(-a\right)}} - b_2}{a} \]

      4. div-sub 69.8%

         \[\leadsto \color{blue}{\frac{\sqrt{c \cdot \left(-a\right)}}{a} - \frac{b_2}{a}} \]

      5. distribute-rgt-neg-out 69.8%

         \[\leadsto \frac{\sqrt{\color{blue}{-c \cdot a}}}{a} - \frac{b_2}{a} \]

   10. Applied egg-rr 69.8%

       \[\leadsto \color{blue}{\frac{\sqrt{-c \cdot a}}{a} - \frac{b_2}{a}} \]

   if 1.8999999999999999e-53 < b_2

   1. Initial program 16.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 16.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 16.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 16.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 91.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.9 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-8)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 4.1e-54) (/ (- (sqrt (* a (- c))) b_2) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-8) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4.1e-54) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.8d-8)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 4.1d-54) then
        tmp = (sqrt((a * -c)) - b_2) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-8) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4.1e-54) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-8:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 4.1e-54:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-8)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 4.1e-54)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-8)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 4.1e-54)
		tmp = (sqrt((a * -c)) - b_2) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-8], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.1e-54], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.7999999999999999e-8

   1. Initial program 64.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 64.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 64.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 89.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -2.7999999999999999e-8 < b_2 < 4.1000000000000001e-54

   1. Initial program 78.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 78.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 78.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around 0 69.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}} - b_2}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 69.8%

         \[\leadsto \frac{\sqrt{\color{blue}{-c \cdot a}} - b_2}{a} \]

      2. distribute-rgt-neg-out 69.8%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   6. Simplified 69.8%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   if 4.1000000000000001e-54 < b_2

   1. Initial program 16.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 16.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 16.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 16.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 91.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 4: 68.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d-310)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 71.5%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 71.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 71.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 64.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -3.999999999999988e-310 < b_2

   1. Initial program 35.7%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 35.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 35.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 35.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 66.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 5: 44.4% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310) (* -2.0 (/ b_2 a)) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = 0.0 / 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 <= (-4d-310)) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 71.5%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 71.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 71.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 64.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

   if -3.999999999999988e-310 < b_2

   1. Initial program 35.7%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 35.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 35.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 35.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 32.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow2 32.8%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}} - b_2}{a} \]

      3. pow1/2 32.8%

         \[\leadsto \frac{{\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b_2}{a} \]

      4. sqrt-pow1 32.9%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b_2}{a} \]

      5. fma-neg 32.9%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      6. *-commutative 32.9%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      7. distribute-rgt-neg-in 32.9%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      8. metadata-eval 32.9%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b_2}{a} \]

   5. Applied egg-rr 32.9%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}} - b_2}{a} \]

   6. Taylor expanded in b_2 around inf 22.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot b_2 + b_2}}{a} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 22.9%

         \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

      2. metadata-eval 22.9%

         \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

      3. mul0-lft 22.9%

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

   8. Simplified 22.9%

      \[\leadsto \frac{\color{blue}{0}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 6: 68.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 1.8 \cdot 10^{-300}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.8e-300) (* -2.0 (/ b_2 a)) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.8e-300) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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-300) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = (c / b_2) * (-0.5d0)
    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-300) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.8e-300:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.8e-300)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.8e-300)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.8e-300], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.80000000000000008e-300

   1. Initial program 71.7%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 71.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 71.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 63.7%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

   if 1.80000000000000008e-300 < b_2

   1. Initial program 35.2%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 35.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 35.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 35.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 66.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 1.8 \cdot 10^{-300}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 7: 23.4% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310) (/ (- b_2) a) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0 / 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 <= (-4d-310)) then
        tmp = -b_2 / a
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = -b_2 / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = -b_2 / a;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[((-b$95$2) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 71.5%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 71.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 71.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 71.3%

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow2 71.3%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}} - b_2}{a} \]

      3. pow1/2 71.3%

         \[\leadsto \frac{{\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b_2}{a} \]

      4. sqrt-pow1 71.3%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b_2}{a} \]

      5. fma-neg 71.4%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      6. *-commutative 71.4%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      7. distribute-rgt-neg-in 71.4%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      8. metadata-eval 71.4%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b_2}{a} \]

   5. Applied egg-rr 71.4%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}} - b_2}{a} \]

   6. Taylor expanded in b_2 around inf 25.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
   7. Step-by-step derivation

      1. associate-*r/ 25.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

      2. neg-mul-1 25.6%

         \[\leadsto \frac{\color{blue}{-b_2}}{a} \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

   if -3.999999999999988e-310 < b_2

   1. Initial program 35.7%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 35.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 35.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 35.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 32.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow2 32.8%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}} - b_2}{a} \]

      3. pow1/2 32.8%

         \[\leadsto \frac{{\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b_2}{a} \]

      4. sqrt-pow1 32.9%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b_2}{a} \]

      5. fma-neg 32.9%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      6. *-commutative 32.9%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      7. distribute-rgt-neg-in 32.9%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      8. metadata-eval 32.9%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b_2}{a} \]

   5. Applied egg-rr 32.9%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}} - b_2}{a} \]

   6. Taylor expanded in b_2 around inf 22.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot b_2 + b_2}}{a} \]
   7. Step-by-step derivation

      1. distribute-lft1-in 22.9%

         \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

      2. metadata-eval 22.9%

         \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

      3. mul0-lft 22.9%

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

   8. Simplified 22.9%

      \[\leadsto \frac{\color{blue}{0}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 8: 11.2% accurate, 37.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

   \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 54.1%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

   2. unsub-neg 54.1%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

3. Simplified 54.1%

   \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 52.6%

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

   2. pow2 52.6%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}} - b_2}{a} \]

   3. pow1/2 52.6%

      \[\leadsto \frac{{\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b_2}{a} \]

   4. sqrt-pow1 52.7%

      \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b_2}{a} \]

   5. fma-neg 52.7%

      \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

   6. *-commutative 52.7%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

   7. distribute-rgt-neg-in 52.7%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

   8. metadata-eval 52.7%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b_2}{a} \]

5. Applied egg-rr 52.7%

   \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}} - b_2}{a} \]

6. Taylor expanded in b_2 around inf 12.5%

   \[\leadsto \frac{\color{blue}{-1 \cdot b_2 + b_2}}{a} \]
7. Step-by-step derivation

   1. distribute-lft1-in 12.5%

      \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

   2. metadata-eval 12.5%

      \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

   3. mul0-lft 12.5%

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

8. Simplified 12.5%

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

9. Final simplification 12.5%

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

Reproduce

herbie shell --seed 2023187 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))