quad2m (problem 3.2.1, negative)

Percentage Accurate: 60.7% → 81.6%

Time: 11.4s

Alternatives: 10

Speedup: 0.9×

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: 60.7% 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: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.35e+154)
   (/ (* -0.5 (/ a (/ b_2 c))) a)
   (if (<= b_2 7.5e+112)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.35e+154) {
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	} else if (b_2 <= 7.5e+112) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.35d+154)) then
        tmp = ((-0.5d0) * (a / (b_2 / c))) / a
    else if (b_2 <= 7.5d+112) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.35e+154) {
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	} else if (b_2 <= 7.5e+112) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.35e+154:
		tmp = (-0.5 * (a / (b_2 / c))) / a
	elif b_2 <= 7.5e+112:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.35e+154)
		tmp = Float64(Float64(-0.5 * Float64(a / Float64(b_2 / c))) / a);
	elseif (b_2 <= 7.5e+112)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.35e+154)
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	elseif (b_2 <= 7.5e+112)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.35e+154], N[(N[(-0.5 * N[(a / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 7.5e+112], 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], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.35000000000000003e154

   1. Initial program 1.6%

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

   3. Taylor expanded in b_2 around -inf 63.1%

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

      1. associate-/l* 76.3%

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

   5. Simplified 76.3%

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

   if -1.35000000000000003e154 < b_2 < 7.5e112

   1. Initial program 80.2%

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

   if 7.5e112 < b_2

   1. Initial program 36.2%

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

   3. Taylor expanded in b_2 around inf 98.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.04 \cdot 10^{+210}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq -5.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{b\_2 - \left(b\_2 + a \cdot \left(0.5 \cdot \frac{c}{b\_2}\right)\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 2.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.04e+210)
   (/ (* -0.5 (/ a (/ b_2 c))) a)
   (if (<= b_2 -5.6e-45)
     (/ (- b_2 (+ b_2 (* a (* 0.5 (/ c b_2))))) a)
     (if (<= b_2 2.7e-25)
       (/ (- (- b_2) (sqrt (* a (- c)))) a)
       (* -2.0 (/ b_2 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.04e+210) {
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	} else if (b_2 <= -5.6e-45) {
		tmp = (b_2 - (b_2 + (a * (0.5 * (c / b_2))))) / a;
	} else if (b_2 <= 2.7e-25) {
		tmp = (-b_2 - sqrt((a * -c))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.04d+210)) then
        tmp = ((-0.5d0) * (a / (b_2 / c))) / a
    else if (b_2 <= (-5.6d-45)) then
        tmp = (b_2 - (b_2 + (a * (0.5d0 * (c / b_2))))) / a
    else if (b_2 <= 2.7d-25) then
        tmp = (-b_2 - sqrt((a * -c))) / a
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.04e+210) {
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	} else if (b_2 <= -5.6e-45) {
		tmp = (b_2 - (b_2 + (a * (0.5 * (c / b_2))))) / a;
	} else if (b_2 <= 2.7e-25) {
		tmp = (-b_2 - Math.sqrt((a * -c))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.04e+210:
		tmp = (-0.5 * (a / (b_2 / c))) / a
	elif b_2 <= -5.6e-45:
		tmp = (b_2 - (b_2 + (a * (0.5 * (c / b_2))))) / a
	elif b_2 <= 2.7e-25:
		tmp = (-b_2 - math.sqrt((a * -c))) / a
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.04e+210)
		tmp = Float64(Float64(-0.5 * Float64(a / Float64(b_2 / c))) / a);
	elseif (b_2 <= -5.6e-45)
		tmp = Float64(Float64(b_2 - Float64(b_2 + Float64(a * Float64(0.5 * Float64(c / b_2))))) / a);
	elseif (b_2 <= 2.7e-25)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.04e+210)
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	elseif (b_2 <= -5.6e-45)
		tmp = (b_2 - (b_2 + (a * (0.5 * (c / b_2))))) / a;
	elseif (b_2 <= 2.7e-25)
		tmp = (-b_2 - sqrt((a * -c))) / a;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.04e+210], N[(N[(-0.5 * N[(a / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -5.6e-45], N[(N[(b$95$2 - N[(b$95$2 + N[(a * N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.7e-25], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.04e210

   1. Initial program 1.5%

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

   3. Taylor expanded in b_2 around -inf 56.6%

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

      1. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

   if -1.04e210 < b_2 < -5.6000000000000003e-45

   1. Initial program 52.4%

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

   3. Taylor expanded in b_2 around -inf 68.2%

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

      1. +-commutative 68.2%

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

      2. mul-1-neg 68.2%

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

      3. unsub-neg 68.2%

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

      4. associate-*r/ 68.2%

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

      5. *-commutative 68.2%

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

      6. associate-*r* 68.2%

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

      7. *-commutative 68.2%

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

   5. Simplified 68.2%

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

      1. sub-neg 68.2%

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

      2. add-sqr-sqrt 44.3%

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

      3. sqrt-unprod 42.6%

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

      4. sqr-neg 42.6%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 3.0%

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

      7. sub-neg 3.0%

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

      8. associate-/l* 3.0%

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

      9. *-commutative 3.0%

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

      10. add-sqr-sqrt 3.0%

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

      11. sqrt-unprod 2.8%

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

      12. sqr-neg 2.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 68.2%

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

   7. Applied egg-rr 68.2%

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

      1. unsub-neg 68.2%

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

      2. +-commutative 68.2%

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

      3. associate-/r/ 68.2%

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

      4. associate-*r/ 68.2%

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

   9. Simplified 68.2%

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

   if -5.6000000000000003e-45 < b_2 < 2.70000000000000016e-25

   1. Initial program 80.2%

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

   3. Taylor expanded in b_2 around 0 71.2%

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

      1. associate-*r* 71.2%

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

      2. neg-mul-1 71.2%

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

   5. Simplified 71.2%

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

   if 2.70000000000000016e-25 < b_2

   1. Initial program 52.7%

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

   3. Taylor expanded in b_2 around inf 90.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.04 \cdot 10^{+210}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq -5.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{b\_2 - \left(b\_2 + a \cdot \left(0.5 \cdot \frac{c}{b\_2}\right)\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 2.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b\_2 - b\_2}{a}\\ \mathbf{if}\;b\_2 \leq -1.52 \cdot 10^{+210}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq -3.6 \cdot 10^{+99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq -6.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a \cdot c}{b\_2}}{a}\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{-263}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (/ (- b_2 b_2) a)))
   (if (<= b_2 -1.52e+210)
     (/ (* -0.5 (/ a (/ b_2 c))) a)
     (if (<= b_2 -3.6e+99)
       t_0
       (if (<= b_2 -6.6e+18)
         (/ (* -0.5 (/ (* a c) b_2)) a)
         (if (<= b_2 9e-263) t_0 (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))))))))

C

double code(double a, double b_2, double c) {
	double t_0 = (b_2 - b_2) / a;
	double tmp;
	if (b_2 <= -1.52e+210) {
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	} else if (b_2 <= -3.6e+99) {
		tmp = t_0;
	} else if (b_2 <= -6.6e+18) {
		tmp = (-0.5 * ((a * c) / b_2)) / a;
	} else if (b_2 <= 9e-263) {
		tmp = t_0;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (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) :: t_0
    real(8) :: tmp
    t_0 = (b_2 - b_2) / a
    if (b_2 <= (-1.52d+210)) then
        tmp = ((-0.5d0) * (a / (b_2 / c))) / a
    else if (b_2 <= (-3.6d+99)) then
        tmp = t_0
    else if (b_2 <= (-6.6d+18)) then
        tmp = ((-0.5d0) * ((a * c) / b_2)) / a
    else if (b_2 <= 9d-263) then
        tmp = t_0
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = (b_2 - b_2) / a;
	double tmp;
	if (b_2 <= -1.52e+210) {
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	} else if (b_2 <= -3.6e+99) {
		tmp = t_0;
	} else if (b_2 <= -6.6e+18) {
		tmp = (-0.5 * ((a * c) / b_2)) / a;
	} else if (b_2 <= 9e-263) {
		tmp = t_0;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = (b_2 - b_2) / a
	tmp = 0
	if b_2 <= -1.52e+210:
		tmp = (-0.5 * (a / (b_2 / c))) / a
	elif b_2 <= -3.6e+99:
		tmp = t_0
	elif b_2 <= -6.6e+18:
		tmp = (-0.5 * ((a * c) / b_2)) / a
	elif b_2 <= 9e-263:
		tmp = t_0
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(Float64(b_2 - b_2) / a)
	tmp = 0.0
	if (b_2 <= -1.52e+210)
		tmp = Float64(Float64(-0.5 * Float64(a / Float64(b_2 / c))) / a);
	elseif (b_2 <= -3.6e+99)
		tmp = t_0;
	elseif (b_2 <= -6.6e+18)
		tmp = Float64(Float64(-0.5 * Float64(Float64(a * c) / b_2)) / a);
	elseif (b_2 <= 9e-263)
		tmp = t_0;
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = (b_2 - b_2) / a;
	tmp = 0.0;
	if (b_2 <= -1.52e+210)
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	elseif (b_2 <= -3.6e+99)
		tmp = t_0;
	elseif (b_2 <= -6.6e+18)
		tmp = (-0.5 * ((a * c) / b_2)) / a;
	elseif (b_2 <= 9e-263)
		tmp = t_0;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[(b$95$2 - b$95$2), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[b$95$2, -1.52e+210], N[(N[(-0.5 * N[(a / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -3.6e+99], t$95$0, If[LessEqual[b$95$2, -6.6e+18], N[(N[(-0.5 * N[(N[(a * c), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 9e-263], t$95$0, N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.51999999999999994e210

   1. Initial program 1.5%

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

   3. Taylor expanded in b_2 around -inf 56.6%

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

      1. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

   if -1.51999999999999994e210 < b_2 < -3.6000000000000002e99 or -6.6e18 < b_2 < 8.9999999999999994e-263

   1. Initial program 63.0%

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

   3. Taylor expanded in b_2 around -inf 40.8%

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

      1. mul-1-neg 40.8%

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

   5. Simplified 40.8%

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

   if -3.6000000000000002e99 < b_2 < -6.6e18

   1. Initial program 67.1%

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

   3. Taylor expanded in b_2 around -inf 65.3%

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

   if 8.9999999999999994e-263 < b_2

   1. Initial program 62.0%

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

   3. Taylor expanded in b_2 around inf 67.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.52 \cdot 10^{+210}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq -3.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{b\_2 - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq -6.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a \cdot c}{b\_2}}{a}\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{-263}:\\ \;\;\;\;\frac{b\_2 - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b\_2 - b\_2}{a}\\ t_1 := \frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{if}\;b\_2 \leq -1.12 \cdot 10^{+210}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b\_2 \leq -5.3 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq -3.2 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b\_2 \leq -9 \cdot 10^{-302}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (/ (- b_2 b_2) a)) (t_1 (/ (* -0.5 (/ a (/ b_2 c))) a)))
   (if (<= b_2 -1.12e+210)
     t_1
     (if (<= b_2 -5.3e+98)
       t_0
       (if (<= b_2 -3.2e+19)
         t_1
         (if (<= b_2 -9e-302) t_0 (* -2.0 (/ b_2 a))))))))

C

double code(double a, double b_2, double c) {
	double t_0 = (b_2 - b_2) / a;
	double t_1 = (-0.5 * (a / (b_2 / c))) / a;
	double tmp;
	if (b_2 <= -1.12e+210) {
		tmp = t_1;
	} else if (b_2 <= -5.3e+98) {
		tmp = t_0;
	} else if (b_2 <= -3.2e+19) {
		tmp = t_1;
	} else if (b_2 <= -9e-302) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (b_2 / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b_2 - b_2) / a
    t_1 = ((-0.5d0) * (a / (b_2 / c))) / a
    if (b_2 <= (-1.12d+210)) then
        tmp = t_1
    else if (b_2 <= (-5.3d+98)) then
        tmp = t_0
    else if (b_2 <= (-3.2d+19)) then
        tmp = t_1
    else if (b_2 <= (-9d-302)) then
        tmp = t_0
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = (b_2 - b_2) / a;
	double t_1 = (-0.5 * (a / (b_2 / c))) / a;
	double tmp;
	if (b_2 <= -1.12e+210) {
		tmp = t_1;
	} else if (b_2 <= -5.3e+98) {
		tmp = t_0;
	} else if (b_2 <= -3.2e+19) {
		tmp = t_1;
	} else if (b_2 <= -9e-302) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = (b_2 - b_2) / a
	t_1 = (-0.5 * (a / (b_2 / c))) / a
	tmp = 0
	if b_2 <= -1.12e+210:
		tmp = t_1
	elif b_2 <= -5.3e+98:
		tmp = t_0
	elif b_2 <= -3.2e+19:
		tmp = t_1
	elif b_2 <= -9e-302:
		tmp = t_0
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(Float64(b_2 - b_2) / a)
	t_1 = Float64(Float64(-0.5 * Float64(a / Float64(b_2 / c))) / a)
	tmp = 0.0
	if (b_2 <= -1.12e+210)
		tmp = t_1;
	elseif (b_2 <= -5.3e+98)
		tmp = t_0;
	elseif (b_2 <= -3.2e+19)
		tmp = t_1;
	elseif (b_2 <= -9e-302)
		tmp = t_0;
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = (b_2 - b_2) / a;
	t_1 = (-0.5 * (a / (b_2 / c))) / a;
	tmp = 0.0;
	if (b_2 <= -1.12e+210)
		tmp = t_1;
	elseif (b_2 <= -5.3e+98)
		tmp = t_0;
	elseif (b_2 <= -3.2e+19)
		tmp = t_1;
	elseif (b_2 <= -9e-302)
		tmp = t_0;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[(b$95$2 - b$95$2), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.5 * N[(a / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[b$95$2, -1.12e+210], t$95$1, If[LessEqual[b$95$2, -5.3e+98], t$95$0, If[LessEqual[b$95$2, -3.2e+19], t$95$1, If[LessEqual[b$95$2, -9e-302], t$95$0, N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.12000000000000005e210 or -5.29999999999999997e98 < b_2 < -3.2e19

   1. Initial program 28.0%

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

   3. Taylor expanded in b_2 around -inf 60.1%

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

      1. associate-/l* 73.2%

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

   5. Simplified 73.2%

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

   if -1.12000000000000005e210 < b_2 < -5.29999999999999997e98 or -3.2e19 < b_2 < -9.00000000000000018e-302

   1. Initial program 61.5%

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

   3. Taylor expanded in b_2 around -inf 43.4%

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

      1. mul-1-neg 43.4%

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

   5. Simplified 43.4%

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

   if -9.00000000000000018e-302 < b_2

   1. Initial program 63.1%

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

   3. Taylor expanded in b_2 around inf 63.9%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.12 \cdot 10^{+210}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq -5.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{b\_2 - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq -3.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq -9 \cdot 10^{-302}:\\ \;\;\;\;\frac{b\_2 - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b\_2 - b\_2}{a}\\ \mathbf{if}\;b\_2 \leq -6.5 \cdot 10^{+210}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq -3.6 \cdot 10^{+99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a \cdot c}{b\_2}}{a}\\ \mathbf{elif}\;b\_2 \leq -1.42 \cdot 10^{-307}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (/ (- b_2 b_2) a)))
   (if (<= b_2 -6.5e+210)
     (/ (* -0.5 (/ a (/ b_2 c))) a)
     (if (<= b_2 -3.6e+99)
       t_0
       (if (<= b_2 -1.35e+19)
         (/ (* -0.5 (/ (* a c) b_2)) a)
         (if (<= b_2 -1.42e-307) t_0 (* -2.0 (/ b_2 a))))))))

C

double code(double a, double b_2, double c) {
	double t_0 = (b_2 - b_2) / a;
	double tmp;
	if (b_2 <= -6.5e+210) {
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	} else if (b_2 <= -3.6e+99) {
		tmp = t_0;
	} else if (b_2 <= -1.35e+19) {
		tmp = (-0.5 * ((a * c) / b_2)) / a;
	} else if (b_2 <= -1.42e-307) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (b_2 / 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) :: t_0
    real(8) :: tmp
    t_0 = (b_2 - b_2) / a
    if (b_2 <= (-6.5d+210)) then
        tmp = ((-0.5d0) * (a / (b_2 / c))) / a
    else if (b_2 <= (-3.6d+99)) then
        tmp = t_0
    else if (b_2 <= (-1.35d+19)) then
        tmp = ((-0.5d0) * ((a * c) / b_2)) / a
    else if (b_2 <= (-1.42d-307)) then
        tmp = t_0
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = (b_2 - b_2) / a;
	double tmp;
	if (b_2 <= -6.5e+210) {
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	} else if (b_2 <= -3.6e+99) {
		tmp = t_0;
	} else if (b_2 <= -1.35e+19) {
		tmp = (-0.5 * ((a * c) / b_2)) / a;
	} else if (b_2 <= -1.42e-307) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = (b_2 - b_2) / a
	tmp = 0
	if b_2 <= -6.5e+210:
		tmp = (-0.5 * (a / (b_2 / c))) / a
	elif b_2 <= -3.6e+99:
		tmp = t_0
	elif b_2 <= -1.35e+19:
		tmp = (-0.5 * ((a * c) / b_2)) / a
	elif b_2 <= -1.42e-307:
		tmp = t_0
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(Float64(b_2 - b_2) / a)
	tmp = 0.0
	if (b_2 <= -6.5e+210)
		tmp = Float64(Float64(-0.5 * Float64(a / Float64(b_2 / c))) / a);
	elseif (b_2 <= -3.6e+99)
		tmp = t_0;
	elseif (b_2 <= -1.35e+19)
		tmp = Float64(Float64(-0.5 * Float64(Float64(a * c) / b_2)) / a);
	elseif (b_2 <= -1.42e-307)
		tmp = t_0;
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = (b_2 - b_2) / a;
	tmp = 0.0;
	if (b_2 <= -6.5e+210)
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	elseif (b_2 <= -3.6e+99)
		tmp = t_0;
	elseif (b_2 <= -1.35e+19)
		tmp = (-0.5 * ((a * c) / b_2)) / a;
	elseif (b_2 <= -1.42e-307)
		tmp = t_0;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[(b$95$2 - b$95$2), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[b$95$2, -6.5e+210], N[(N[(-0.5 * N[(a / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -3.6e+99], t$95$0, If[LessEqual[b$95$2, -1.35e+19], N[(N[(-0.5 * N[(N[(a * c), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -1.42e-307], t$95$0, N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -6.4999999999999996e210

   1. Initial program 1.5%

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

   3. Taylor expanded in b_2 around -inf 56.6%

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

      1. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

   if -6.4999999999999996e210 < b_2 < -3.6000000000000002e99 or -1.35e19 < b_2 < -1.42000000000000001e-307

   1. Initial program 61.9%

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

   3. Taylor expanded in b_2 around -inf 43.0%

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

      1. mul-1-neg 43.0%

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

   5. Simplified 43.0%

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

   if -3.6000000000000002e99 < b_2 < -1.35e19

   1. Initial program 67.1%

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

   3. Taylor expanded in b_2 around -inf 65.3%

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

   if -1.42000000000000001e-307 < b_2

   1. Initial program 62.8%

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

   3. Taylor expanded in b_2 around inf 64.5%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.5 \cdot 10^{+210}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq -3.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{b\_2 - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a \cdot c}{b\_2}}{a}\\ \mathbf{elif}\;b\_2 \leq -1.42 \cdot 10^{-307}:\\ \;\;\;\;\frac{b\_2 - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{if}\;b\_2 \leq -1.7 \cdot 10^{+210}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b\_2 - \left(b\_2 + a \cdot t\_0\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ c b_2))))
   (if (<= b_2 -1.7e+210)
     (/ (* -0.5 (/ a (/ b_2 c))) a)
     (if (<= b_2 -2e-310)
       (/ (- b_2 (+ b_2 (* a t_0))) a)
       (+ (* -2.0 (/ b_2 a)) t_0)))))

C

double code(double a, double b_2, double c) {
	double t_0 = 0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -1.7e+210) {
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	} else if (b_2 <= -2e-310) {
		tmp = (b_2 - (b_2 + (a * t_0))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + t_0;
	}
	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 = 0.5d0 * (c / b_2)
    if (b_2 <= (-1.7d+210)) then
        tmp = ((-0.5d0) * (a / (b_2 / c))) / a
    else if (b_2 <= (-2d-310)) then
        tmp = (b_2 - (b_2 + (a * t_0))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = 0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -1.7e+210) {
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	} else if (b_2 <= -2e-310) {
		tmp = (b_2 - (b_2 + (a * t_0))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + t_0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = 0.5 * (c / b_2)
	tmp = 0
	if b_2 <= -1.7e+210:
		tmp = (-0.5 * (a / (b_2 / c))) / a
	elif b_2 <= -2e-310:
		tmp = (b_2 - (b_2 + (a * t_0))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + t_0
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(0.5 * Float64(c / b_2))
	tmp = 0.0
	if (b_2 <= -1.7e+210)
		tmp = Float64(Float64(-0.5 * Float64(a / Float64(b_2 / c))) / a);
	elseif (b_2 <= -2e-310)
		tmp = Float64(Float64(b_2 - Float64(b_2 + Float64(a * t_0))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = 0.5 * (c / b_2);
	tmp = 0.0;
	if (b_2 <= -1.7e+210)
		tmp = (-0.5 * (a / (b_2 / c))) / a;
	elseif (b_2 <= -2e-310)
		tmp = (b_2 - (b_2 + (a * t_0))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -1.7e+210], N[(N[(-0.5 * N[(a / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -2e-310], N[(N[(b$95$2 - N[(b$95$2 + N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.70000000000000012e210

   1. Initial program 1.5%

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

   3. Taylor expanded in b_2 around -inf 56.6%

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

      1. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

   if -1.70000000000000012e210 < b_2 < -1.999999999999994e-310

   1. Initial program 63.3%

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

   3. Taylor expanded in b_2 around -inf 44.5%

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

      1. +-commutative 44.5%

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

      2. mul-1-neg 44.5%

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

      3. unsub-neg 44.5%

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

      4. associate-*r/ 44.5%

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

      5. *-commutative 44.5%

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

      6. associate-*r* 44.5%

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

      7. *-commutative 44.5%

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

   5. Simplified 44.5%

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

      1. sub-neg 44.5%

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

      2. add-sqr-sqrt 29.3%

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

      3. sqrt-unprod 29.2%

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

      4. sqr-neg 29.2%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 3.7%

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

      7. sub-neg 3.7%

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

      8. associate-/l* 3.6%

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

      9. *-commutative 3.6%

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

      10. add-sqr-sqrt 3.6%

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

      11. sqrt-unprod 3.5%

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

      12. sqr-neg 3.5%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 44.5%

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

   7. Applied egg-rr 44.5%

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

      1. unsub-neg 44.5%

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

      2. +-commutative 44.5%

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

      3. associate-/r/ 44.6%

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

      4. associate-*r/ 44.6%

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

   9. Simplified 44.6%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 62.5%

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

   3. Taylor expanded in b_2 around inf 65.2%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.7 \cdot 10^{+210}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{a}{\frac{b\_2}{c}}}{a}\\ \mathbf{elif}\;b\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b\_2 - \left(b\_2 + a \cdot \left(0.5 \cdot \frac{c}{b\_2}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.5% accurate, 11.2× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310) (* -0.5 (/ c b_2)) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -2.0 * (b_2 / 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 <= (-2d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 49.7%

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

   3. Taylor expanded in b_2 around -inf 36.4%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 62.5%

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

   3. Taylor expanded in b_2 around inf 65.0%

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

4. Final simplification 49.3%

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

Alternative 8: 54.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{-0.5}{\frac{b\_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-309) (/ -0.5 (/ b_2 c)) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-309) {
		tmp = -0.5 / (b_2 / c);
	} else {
		tmp = -2.0 * (b_2 / 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 <= (-5d-309)) then
        tmp = (-0.5d0) / (b_2 / c)
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-309) {
		tmp = -0.5 / (b_2 / c);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-309:
		tmp = -0.5 / (b_2 / c)
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-309)
		tmp = Float64(-0.5 / Float64(b_2 / c));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-309)
		tmp = -0.5 / (b_2 / c);
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-309], N[(-0.5 / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.9999999999999995e-309

   1. Initial program 49.7%

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

   3. Taylor expanded in b_2 around -inf 36.4%

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

      1. add-cube-cbrt 36.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{-0.5 \cdot \frac{c}{b\_2}} \cdot \sqrt[3]{-0.5 \cdot \frac{c}{b\_2}}\right) \cdot \sqrt[3]{-0.5 \cdot \frac{c}{b\_2}}} \]

      2. pow3 36.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{-0.5 \cdot \frac{c}{b\_2}}\right)}^{3}} \]

      3. associate-*r/ 36.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{-0.5 \cdot c}{b\_2}}}\right)}^{3} \]

   5. Applied egg-rr 36.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-0.5 \cdot c}{b\_2}}\right)}^{3}} \]
   6. Step-by-step derivation

      1. rem-cube-cbrt 36.4%

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

      2. associate-/l* 36.6%

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

   7. Applied egg-rr 36.6%

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

   if -4.9999999999999995e-309 < b_2

   1. Initial program 62.5%

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

   3. Taylor expanded in b_2 around inf 65.0%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{-0.5}{\frac{b\_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.42 \cdot 10^{-307}:\\ \;\;\;\;\frac{b\_2 - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.42e-307) (/ (- b_2 b_2) a) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.42e-307) {
		tmp = (b_2 - b_2) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.42d-307)) then
        tmp = (b_2 - b_2) / a
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.42e-307) {
		tmp = (b_2 - b_2) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.42e-307:
		tmp = (b_2 - b_2) / a
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.42e-307)
		tmp = Float64(Float64(b_2 - b_2) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.42e-307)
		tmp = (b_2 - b_2) / a;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.42e-307], N[(N[(b$95$2 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.42000000000000001e-307

   1. Initial program 49.3%

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

   3. Taylor expanded in b_2 around -inf 47.7%

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

      1. mul-1-neg 47.7%

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

   5. Simplified 47.7%

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

   if -1.42000000000000001e-307 < b_2

   1. Initial program 62.8%

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

   3. Taylor expanded in b_2 around inf 64.5%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.42 \cdot 10^{-307}:\\ \;\;\;\;\frac{b\_2 - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.1% accurate, 22.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-2.0d0) * (b_2 / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in b_2 around inf 30.6%

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

4. Final simplification 30.6%

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

Developer target: 86.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024031 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

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

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