quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.1% → 84.0%

Time: 11.5s

Alternatives: 9

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: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.3e+137)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 2.8e+15)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.3e+137) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.8e+15) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.3d+137)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 2.8d+15) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.3e+137) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.8e+15) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.3e+137:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 2.8e+15:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.3e+137)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 2.8e+15)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.3e+137)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 2.8e+15)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.3e+137], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.8e+15], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.29999999999999999e137

   1. Initial program 50.0%

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

      1. +-commutative 50.0%

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

      2. unsub-neg 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in b_2 around -inf 95.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 95.2%

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

   6. Simplified 95.2%

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

   if -2.29999999999999999e137 < b_2 < 2.8e15

   1. Initial program 78.5%

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

      1. +-commutative 78.5%

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

      2. unsub-neg 78.5%

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

   3. Simplified 78.5%

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

   if 2.8e15 < b_2

   1. Initial program 12.1%

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

      1. +-commutative 12.1%

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

      2. unsub-neg 12.1%

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

   3. Simplified 12.1%

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

      1. clear-num 12.1%

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

      2. associate-/r/ 12.1%

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

      3. sub-neg 12.1%

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

      4. add-sqr-sqrt 7.9%

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

      5. hypot-def 25.8%

         \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)} - b_2\right) \]

      6. *-commutative 25.8%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b_2\right) \]

      7. distribute-rgt-neg-in 25.8%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b_2\right) \]

   5. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right)} \]

   6. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 92.8%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 92.8%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 92.8%

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

   8. Simplified 92.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.3 \cdot 10^{+137}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-49}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -5.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \frac{a}{-1}}}{a}\\ \mathbf{elif}\;b_2 \leq -4 \cdot 10^{-136}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-49)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 -5.8e-95)
     (/ (sqrt (* c (/ a -1.0))) a)
     (if (<= b_2 -4e-136)
       (/ (* b_2 -2.0) a)
       (if (<= b_2 2.4e+15)
         (/ (- (sqrt (* a (- c))) b_2) a)
         (/ (* c -0.5) b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-49) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= -5.8e-95) {
		tmp = sqrt((c * (a / -1.0))) / a;
	} else if (b_2 <= -4e-136) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.4e+15) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.8d-49)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= (-5.8d-95)) then
        tmp = sqrt((c * (a / (-1.0d0)))) / a
    else if (b_2 <= (-4d-136)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 2.4d+15) then
        tmp = (sqrt((a * -c)) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-49) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= -5.8e-95) {
		tmp = Math.sqrt((c * (a / -1.0))) / a;
	} else if (b_2 <= -4e-136) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.4e+15) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-49:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= -5.8e-95:
		tmp = math.sqrt((c * (a / -1.0))) / a
	elif b_2 <= -4e-136:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 2.4e+15:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-49)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= -5.8e-95)
		tmp = Float64(sqrt(Float64(c * Float64(a / -1.0))) / a);
	elseif (b_2 <= -4e-136)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 2.4e+15)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-49)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= -5.8e-95)
		tmp = sqrt((c * (a / -1.0))) / a;
	elseif (b_2 <= -4e-136)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 2.4e+15)
		tmp = (sqrt((a * -c)) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-49], 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, -5.8e-95], N[(N[Sqrt[N[(c * N[(a / -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -4e-136], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.4e+15], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b_2 < -2.79999999999999997e-49

   1. Initial program 75.0%

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

      1. +-commutative 75.0%

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

      2. unsub-neg 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in b_2 around -inf 86.3%

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

   if -2.79999999999999997e-49 < b_2 < -5.80000000000000004e-95

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 99.0%

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

      2. pow2 99.0%

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

      3. pow1/2 99.0%

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

      4. sqrt-pow1 99.1%

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

      5. fma-neg 99.1%

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

      6. *-commutative 99.1%

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

      7. distribute-rgt-neg-in 99.1%

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

      8. metadata-eval 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in c around -inf 27.0%

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

      1. unpow2 27.0%

         \[\leadsto \frac{\color{blue}{e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)} \cdot e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)}} - b_2}{a} \]

      2. exp-prod 25.3%

         \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)}} \cdot e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)} - b_2}{a} \]

      3. exp-prod 24.9%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)} \cdot \color{blue}{{\left(e^{0.25}\right)}^{\left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)}} - b_2}{a} \]

      4. pow-sqr 24.9%

         \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)\right)}} - b_2}{a} \]

      5. +-commutative 24.9%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \color{blue}{\left(\log a + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)} - b_2}{a} \]

      6. mul-1-neg 24.9%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log a + \color{blue}{\left(-\log \left(\frac{-1}{c}\right)\right)}\right)\right)} - b_2}{a} \]

      7. unsub-neg 24.9%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \color{blue}{\left(\log a - \log \left(\frac{-1}{c}\right)\right)}\right)} - b_2}{a} \]

   8. Simplified 24.9%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log a - \log \left(\frac{-1}{c}\right)\right)\right)} - b_2}}{a} \]

   9. Taylor expanded in b_2 around 0 27.0%

      \[\leadsto \color{blue}{\frac{e^{0.5 \cdot \left(\log a - \log \left(\frac{-1}{c}\right)\right)}}{a}} \]
   10. Step-by-step derivation

       1. log-div 94.6%

          \[\leadsto \frac{e^{0.5 \cdot \color{blue}{\log \left(\frac{a}{\frac{-1}{c}}\right)}}}{a} \]

       2. *-commutative 94.6%

          \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{a}{\frac{-1}{c}}\right) \cdot 0.5}}}{a} \]

       3. exp-to-pow 99.8%

          \[\leadsto \frac{\color{blue}{{\left(\frac{a}{\frac{-1}{c}}\right)}^{0.5}}}{a} \]

       4. unpow1/2 99.8%

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

       5. associate-/r/ 99.8%

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

   11. Simplified 99.8%

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

   if -5.80000000000000004e-95 < b_2 < -4.00000000000000001e-136

   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 -inf 78.6%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 78.6%

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

   6. Simplified 78.6%

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

   if -4.00000000000000001e-136 < b_2 < 2.4e15

   1. Initial program 68.0%

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

      1. +-commutative 68.0%

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

      2. unsub-neg 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in b_2 around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. distribute-rgt-neg-out 67.2%

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

   6. Simplified 67.2%

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

   if 2.4e15 < b_2

   1. Initial program 12.1%

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

      1. +-commutative 12.1%

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

      2. unsub-neg 12.1%

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

   3. Simplified 12.1%

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

      1. clear-num 12.1%

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

      2. associate-/r/ 12.1%

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

      3. sub-neg 12.1%

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

      4. add-sqr-sqrt 7.9%

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

      5. hypot-def 25.8%

         \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)} - b_2\right) \]

      6. *-commutative 25.8%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b_2\right) \]

      7. distribute-rgt-neg-in 25.8%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b_2\right) \]

   5. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right)} \]

   6. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 92.8%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 92.8%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 92.8%

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

   8. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-49}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -5.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \frac{a}{-1}}}{a}\\ \mathbf{elif}\;b_2 \leq -4 \cdot 10^{-136}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{c \cdot \frac{a}{-1}}}{a}\\ \mathbf{if}\;b_2 \leq -2.7 \cdot 10^{-49}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.6 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -4 \cdot 10^{-136}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (/ (sqrt (* c (/ a -1.0))) a)))
   (if (<= b_2 -2.7e-49)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
     (if (<= b_2 -2.6e-90)
       t_0
       (if (<= b_2 -4e-136)
         (/ (* b_2 -2.0) a)
         (if (<= b_2 2.4e+15) t_0 (/ (* c -0.5) b_2)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt((c * (a / -1.0))) / a;
	double tmp;
	if (b_2 <= -2.7e-49) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= -2.6e-90) {
		tmp = t_0;
	} else if (b_2 <= -4e-136) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.4e+15) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((c * (a / (-1.0d0)))) / a
    if (b_2 <= (-2.7d-49)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= (-2.6d-90)) then
        tmp = t_0
    else if (b_2 <= (-4d-136)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 2.4d+15) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt((c * (a / -1.0))) / a;
	double tmp;
	if (b_2 <= -2.7e-49) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= -2.6e-90) {
		tmp = t_0;
	} else if (b_2 <= -4e-136) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.4e+15) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt((c * (a / -1.0))) / a
	tmp = 0
	if b_2 <= -2.7e-49:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= -2.6e-90:
		tmp = t_0
	elif b_2 <= -4e-136:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 2.4e+15:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(Float64(c * Float64(a / -1.0))) / a)
	tmp = 0.0
	if (b_2 <= -2.7e-49)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= -2.6e-90)
		tmp = t_0;
	elseif (b_2 <= -4e-136)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 2.4e+15)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt((c * (a / -1.0))) / a;
	tmp = 0.0;
	if (b_2 <= -2.7e-49)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= -2.6e-90)
		tmp = t_0;
	elseif (b_2 <= -4e-136)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 2.4e+15)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[(c * N[(a / -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[b$95$2, -2.7e-49], 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, -2.6e-90], t$95$0, If[LessEqual[b$95$2, -4e-136], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.4e+15], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.7e-49

   1. Initial program 75.0%

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

      1. +-commutative 75.0%

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

      2. unsub-neg 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in b_2 around -inf 86.3%

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

   if -2.7e-49 < b_2 < -2.6e-90 or -4.00000000000000001e-136 < b_2 < 2.4e15

   1. Initial program 70.3%

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

      1. +-commutative 70.3%

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

      2. unsub-neg 70.3%

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

   3. Simplified 70.3%

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

      1. add-sqr-sqrt 70.0%

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

      2. pow2 70.0%

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

      3. pow1/2 70.0%

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

      4. sqrt-pow1 70.2%

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

      5. fma-neg 70.2%

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

      6. *-commutative 70.2%

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

      7. distribute-rgt-neg-in 70.2%

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

      8. metadata-eval 70.2%

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

   5. Applied egg-rr 70.2%

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

   6. Taylor expanded in c around -inf 34.7%

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

      1. unpow2 34.7%

         \[\leadsto \frac{\color{blue}{e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)} \cdot e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)}} - b_2}{a} \]

      2. exp-prod 33.9%

         \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)}} \cdot e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)} - b_2}{a} \]

      3. exp-prod 33.4%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)} \cdot \color{blue}{{\left(e^{0.25}\right)}^{\left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)}} - b_2}{a} \]

      4. pow-sqr 33.4%

         \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(-1 \cdot \log \left(\frac{-1}{c}\right) + \log a\right)\right)}} - b_2}{a} \]

      5. +-commutative 33.4%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \color{blue}{\left(\log a + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)} - b_2}{a} \]

      6. mul-1-neg 33.4%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log a + \color{blue}{\left(-\log \left(\frac{-1}{c}\right)\right)}\right)\right)} - b_2}{a} \]

      7. unsub-neg 33.4%

         \[\leadsto \frac{{\left(e^{0.25}\right)}^{\left(2 \cdot \color{blue}{\left(\log a - \log \left(\frac{-1}{c}\right)\right)}\right)} - b_2}{a} \]

   8. Simplified 33.4%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log a - \log \left(\frac{-1}{c}\right)\right)\right)} - b_2}}{a} \]

   9. Taylor expanded in b_2 around 0 34.5%

      \[\leadsto \color{blue}{\frac{e^{0.5 \cdot \left(\log a - \log \left(\frac{-1}{c}\right)\right)}}{a}} \]
   10. Step-by-step derivation

       1. log-div 64.7%

          \[\leadsto \frac{e^{0.5 \cdot \color{blue}{\log \left(\frac{a}{\frac{-1}{c}}\right)}}}{a} \]

       2. *-commutative 64.7%

          \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{a}{\frac{-1}{c}}\right) \cdot 0.5}}}{a} \]

       3. exp-to-pow 69.0%

          \[\leadsto \frac{\color{blue}{{\left(\frac{a}{\frac{-1}{c}}\right)}^{0.5}}}{a} \]

       4. unpow1/2 69.0%

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

       5. associate-/r/ 69.0%

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

   11. Simplified 69.0%

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

   if -2.6e-90 < b_2 < -4.00000000000000001e-136

   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 -inf 78.6%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 78.6%

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

   6. Simplified 78.6%

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

   if 2.4e15 < b_2

   1. Initial program 12.1%

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

      1. +-commutative 12.1%

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

      2. unsub-neg 12.1%

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

   3. Simplified 12.1%

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

      1. clear-num 12.1%

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

      2. associate-/r/ 12.1%

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

      3. sub-neg 12.1%

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

      4. add-sqr-sqrt 7.9%

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

      5. hypot-def 25.8%

         \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)} - b_2\right) \]

      6. *-commutative 25.8%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b_2\right) \]

      7. distribute-rgt-neg-in 25.8%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b_2\right) \]

   5. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right)} \]

   6. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 92.8%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 92.8%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 92.8%

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

   8. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.7 \cdot 10^{-49}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{c \cdot \frac{a}{-1}}}{a}\\ \mathbf{elif}\;b_2 \leq -4 \cdot 10^{-136}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sqrt{c \cdot \frac{a}{-1}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4: 68.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-311)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-311:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-311)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-311)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-311], 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 * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999e-311

   1. Initial program 74.8%

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

      1. +-commutative 74.8%

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

      2. unsub-neg 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in b_2 around -inf 66.3%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 40.3%

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

      1. +-commutative 40.3%

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

      2. unsub-neg 40.3%

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

   3. Simplified 40.3%

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

      1. clear-num 40.3%

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

      2. associate-/r/ 40.2%

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

      3. sub-neg 40.2%

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

      4. add-sqr-sqrt 38.1%

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

      5. hypot-def 47.1%

         \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)} - b_2\right) \]

      6. *-commutative 47.1%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b_2\right) \]

      7. distribute-rgt-neg-in 47.1%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b_2\right) \]

   5. Applied egg-rr 47.1%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right)} \]

   6. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 57.6%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 57.6%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 57.6%

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

   8. Simplified 57.6%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 5: 48.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311) (/ (- b_2) a) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		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 <= (-2d-311)) 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 <= -2e-311) {
		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 <= -2e-311:
		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 <= -2e-311)
		tmp = Float64(Float64(-b_2) / 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 <= -2e-311)
		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, -2e-311], 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 < -1.9999999999999e-311

   1. Initial program 74.8%

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

      1. +-commutative 74.8%

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

      2. unsub-neg 74.8%

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

   3. Simplified 74.8%

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

      1. add-sqr-sqrt 74.5%

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

      2. pow2 74.5%

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

      3. pow1/2 74.5%

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

      4. sqrt-pow1 74.6%

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

      5. fma-neg 74.6%

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

      6. *-commutative 74.6%

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

      7. distribute-rgt-neg-in 74.6%

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

      8. metadata-eval 74.6%

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

   5. Applied egg-rr 74.6%

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

   6. Taylor expanded in b_2 around inf 27.2%

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

      1. associate-*r/ 27.2%

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

      2. neg-mul-1 27.2%

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

   8. Simplified 27.2%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 40.3%

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

      1. +-commutative 40.3%

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

      2. unsub-neg 40.3%

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

   3. Simplified 40.3%

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

   4. Taylor expanded in b_2 around inf 57.5%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6: 48.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311) (/ (- b_2) a) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -b_2 / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-311)) then
        tmp = -b_2 / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -b_2 / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-311:
		tmp = -b_2 / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-311)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-311)
		tmp = -b_2 / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999e-311

   1. Initial program 74.8%

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

      1. +-commutative 74.8%

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

      2. unsub-neg 74.8%

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

   3. Simplified 74.8%

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

      1. add-sqr-sqrt 74.5%

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

      2. pow2 74.5%

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

      3. pow1/2 74.5%

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

      4. sqrt-pow1 74.6%

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

      5. fma-neg 74.6%

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

      6. *-commutative 74.6%

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

      7. distribute-rgt-neg-in 74.6%

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

      8. metadata-eval 74.6%

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

   5. Applied egg-rr 74.6%

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

   6. Taylor expanded in b_2 around inf 27.2%

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

      1. associate-*r/ 27.2%

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

      2. neg-mul-1 27.2%

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

   8. Simplified 27.2%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 40.3%

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

      1. +-commutative 40.3%

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

      2. unsub-neg 40.3%

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

   3. Simplified 40.3%

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

      1. clear-num 40.3%

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

      2. associate-/r/ 40.2%

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

      3. sub-neg 40.2%

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

      4. add-sqr-sqrt 38.1%

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

      5. hypot-def 47.1%

         \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)} - b_2\right) \]

      6. *-commutative 47.1%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b_2\right) \]

      7. distribute-rgt-neg-in 47.1%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b_2\right) \]

   5. Applied egg-rr 47.1%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right)} \]

   6. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 57.6%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 57.6%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 57.6%

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

   8. Simplified 57.6%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 7: 67.9% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2e-309) (/ (* b_2 -2.0) a) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2e-309) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 2d-309) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2e-309) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2e-309:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2e-309)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2e-309)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2e-309], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.9999999999999988e-309

   1. Initial program 74.8%

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

      1. +-commutative 74.8%

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

      2. unsub-neg 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in b_2 around -inf 66.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 66.1%

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

   6. Simplified 66.1%

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

   if 1.9999999999999988e-309 < b_2

   1. Initial program 40.3%

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

      1. +-commutative 40.3%

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

      2. unsub-neg 40.3%

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

   3. Simplified 40.3%

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

      1. clear-num 40.3%

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

      2. associate-/r/ 40.2%

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

      3. sub-neg 40.2%

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

      4. add-sqr-sqrt 38.1%

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

      5. hypot-def 47.1%

         \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)} - b_2\right) \]

      6. *-commutative 47.1%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b_2\right) \]

      7. distribute-rgt-neg-in 47.1%

         \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b_2\right) \]

   5. Applied egg-rr 47.1%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2\right)} \]

   6. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 57.6%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 57.6%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 57.6%

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

   8. Simplified 57.6%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 8: 23.4% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-299}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e-299) (/ (- b_2) a) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e-299) {
		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 <= (-2.2d-299)) 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 <= -2.2e-299) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.2e-299:
		tmp = -b_2 / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e-299)
		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 <= -2.2e-299)
		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, -2.2e-299], N[((-b$95$2) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.2e-299

   1. Initial program 74.4%

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

      1. +-commutative 74.4%

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

      2. unsub-neg 74.4%

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

   3. Simplified 74.4%

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

      1. add-sqr-sqrt 74.1%

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

      2. pow2 74.1%

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

      3. pow1/2 74.1%

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

      4. sqrt-pow1 74.2%

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

      5. fma-neg 74.2%

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

      6. *-commutative 74.2%

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

      7. distribute-rgt-neg-in 74.2%

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

      8. metadata-eval 74.2%

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

   5. Applied egg-rr 74.2%

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

   6. Taylor expanded in b_2 around inf 27.6%

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

      1. associate-*r/ 27.6%

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

      2. neg-mul-1 27.6%

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

   8. Simplified 27.6%

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

   if -2.2e-299 < b_2

   1. Initial program 41.2%

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

      1. +-commutative 41.2%

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

      2. unsub-neg 41.2%

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

   3. Simplified 41.2%

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

      1. add-sqr-sqrt 38.7%

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

      2. pow2 38.7%

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

      3. pow1/2 38.7%

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

      4. sqrt-pow1 38.8%

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

      5. fma-neg 38.8%

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

      6. *-commutative 38.8%

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

      7. distribute-rgt-neg-in 38.8%

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

      8. metadata-eval 38.8%

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

   5. Applied egg-rr 38.8%

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

   6. Taylor expanded in b_2 around inf 16.3%

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

      1. distribute-lft1-in 16.3%

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

      2. metadata-eval 16.3%

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

      3. mul0-lft 16.3%

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

   8. Simplified 16.3%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-299}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 9: 10.9% 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 57.7%

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

   1. +-commutative 57.7%

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

   2. unsub-neg 57.7%

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

3. Simplified 57.7%

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

   1. add-sqr-sqrt 56.3%

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

   2. pow2 56.3%

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

   3. pow1/2 56.3%

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

   4. sqrt-pow1 56.4%

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

   5. fma-neg 56.4%

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

   6. *-commutative 56.4%

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

   7. distribute-rgt-neg-in 56.4%

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

   8. metadata-eval 56.4%

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

5. Applied egg-rr 56.4%

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

6. Taylor expanded in b_2 around inf 9.5%

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

   1. distribute-lft1-in 9.5%

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

   2. metadata-eval 9.5%

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

   3. mul0-lft 9.5%

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

8. Simplified 9.5%

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

9. Final simplification 9.5%

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

Reproduce

herbie shell --seed 2023258 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))