quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.5% → 85.3%

Time: 14.7s

Alternatives: 11

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-98) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 8.8e+94) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((a * 0.5) * (c / (b_2 * a))) - ((b_2 / a) * 2.0);
	}
	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) :: tmp
    if (b_2 <= (-2.6d-98)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 8.8d+94) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((a * 0.5d0) * (c / (b_2 * a))) - ((b_2 / a) * 2.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.60000000000000013e-98

   1. Initial program 12.7%

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

   3. Taylor expanded in b_2 around -inf 90.0%

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

      1. associate-*r/ 90.0%

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

   5. Simplified 90.0%

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

   if -2.60000000000000013e-98 < b_2 < 8.80000000000000047e94

   1. Initial program 82.9%

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

   if 8.80000000000000047e94 < b_2

   1. Initial program 57.9%

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

      1. sub-neg 57.9%

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

      2. distribute-neg-out 57.9%

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

      3. sub-neg 57.9%

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

      4. add-sqr-sqrt 36.9%

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

      5. hypot-define 55.7%

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

      6. distribute-rgt-neg-in 55.7%

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

   4. Applied egg-rr 55.7%

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

      1. +-commutative 55.7%

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

      2. distribute-neg-in 55.7%

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

      3. sub-neg 55.7%

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

      4. distribute-rgt-neg-out 55.7%

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

      5. *-commutative 55.7%

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

      6. distribute-rgt-neg-in 55.7%

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

   6. Simplified 55.7%

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

      1. div-inv 55.6%

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

      2. *-commutative 55.6%

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

   8. Applied egg-rr 55.6%

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

   9. Taylor expanded in a around 0 0.0%

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

       1. associate-*r/ 0.0%

          \[\leadsto \left(\left(\color{blue}{\frac{-0.5 \cdot \left(a \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}{b\_2}} - b\_2\right) - b\_2\right) \cdot \frac{1}{a} \]

       2. metadata-eval 0.0%

          \[\leadsto \left(\left(\frac{\color{blue}{\left(-0.5\right)} \cdot \left(a \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}{b\_2} - b\_2\right) - b\_2\right) \cdot \frac{1}{a} \]

       3. distribute-lft-neg-in 0.0%

          \[\leadsto \left(\left(\frac{\color{blue}{-0.5 \cdot \left(a \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{b\_2} - b\_2\right) - b\_2\right) \cdot \frac{1}{a} \]

       4. distribute-rgt-neg-in 0.0%

          \[\leadsto \left(\left(\frac{\color{blue}{0.5 \cdot \left(-a \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}{b\_2} - b\_2\right) - b\_2\right) \cdot \frac{1}{a} \]

       5. *-commutative 0.0%

          \[\leadsto \left(\left(\frac{0.5 \cdot \left(-a \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}\right)}{b\_2} - b\_2\right) - b\_2\right) \cdot \frac{1}{a} \]

       6. unpow2 0.0%

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

       7. rem-square-sqrt 91.1%

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

       8. neg-mul-1 91.1%

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

       9. distribute-rgt-neg-in 91.1%

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

       10. rem-square-sqrt 48.8%

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

       11. distribute-lft-neg-in 48.8%

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

       12. distribute-rgt-neg-out 48.8%

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

       13. sqr-neg 48.8%

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

       14. rem-square-sqrt 91.1%

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

       15. associate-*r/ 91.1%

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

       16. associate-*r/ 97.7%

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

   11. Simplified 97.7%

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

       1. un-div-inv 97.9%

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

       2. associate--l- 97.9%

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

       3. div-sub 97.9%

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

       4. associate-*r* 97.9%

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

   13. Applied egg-rr 97.9%

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

       1. associate-/l* 97.9%

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

       2. *-commutative 97.9%

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

       3. associate-/l/ 97.9%

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

       4. count-2 97.9%

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

       5. associate-*r/ 97.9%

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

       6. *-commutative 97.9%

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

   15. Simplified 97.9%

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

4. Final simplification 87.9%

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

Alternative 2: 80.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e-98)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 4.8e-59)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + 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 <= -2.6e-98)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 4.8e-59)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.60000000000000013e-98

   1. Initial program 12.7%

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

   3. Taylor expanded in b_2 around -inf 90.0%

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

      1. associate-*r/ 90.0%

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

   5. Simplified 90.0%

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

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

   1. Initial program 79.8%

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

   3. Taylor expanded in b_2 around 0 71.4%

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

      1. associate-*r* 71.4%

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

      2. neg-mul-1 71.4%

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

   5. Simplified 71.4%

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

   if 4.8000000000000003e-59 < b_2

   1. Initial program 72.3%

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

   3. Taylor expanded in c around 0 87.4%

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

4. Final simplification 82.5%

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

Alternative 3: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-98)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3e-59)
     (/ (sqrt (* c (- a))) (- a))
     (+ (* (/ b_2 a) -2.0) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-98) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3e-59) {
		tmp = sqrt((c * -a)) / -a;
	} else {
		tmp = ((b_2 / a) * -2.0) + (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-98)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 3d-59) then
        tmp = sqrt((c * -a)) / -a
    else
        tmp = ((b_2 / a) * (-2.0d0)) + (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-98) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3e-59) {
		tmp = Math.sqrt((c * -a)) / -a;
	} else {
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-98:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 3e-59:
		tmp = math.sqrt((c * -a)) / -a
	else:
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-98)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3e-59)
		tmp = Float64(sqrt(Float64(c * Float64(-a))) / Float64(-a));
	else
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + 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-98)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 3e-59)
		tmp = sqrt((c * -a)) / -a;
	else
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-98], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3e-59], N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.99999999999999988e-98

   1. Initial program 12.7%

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

   3. Taylor expanded in b_2 around -inf 90.0%

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

      1. associate-*r/ 90.0%

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

   5. Simplified 90.0%

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

   if -1.99999999999999988e-98 < b_2 < 3.0000000000000001e-59

   1. Initial program 79.8%

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

      1. sub-neg 79.8%

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

      2. +-commutative 79.8%

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

      3. distribute-lft-neg-in 79.8%

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

      4. add-cube-cbrt 79.2%

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

      5. associate-*r* 79.2%

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

      6. fma-define 79.2%

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

      7. pow2 79.2%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(\left(-a\right) \cdot \color{blue}{{\left(\sqrt[3]{c}\right)}^{2}}, \sqrt[3]{c}, b\_2 \cdot b\_2\right)}}{a} \]

      8. pow2 79.2%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(\left(-a\right) \cdot {\left(\sqrt[3]{c}\right)}^{2}, \sqrt[3]{c}, \color{blue}{{b\_2}^{2}}\right)}}{a} \]

   4. Applied egg-rr 79.2%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(\left(-a\right) \cdot {\left(\sqrt[3]{c}\right)}^{2}, \sqrt[3]{c}, {b\_2}^{2}\right)}}}{a} \]

   5. Taylor expanded in c around -inf 0.0%

      \[\leadsto \frac{\color{blue}{\sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)} \cdot {\left(\sqrt{-1}\right)}^{2}}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)}}}{a} \]

      2. unpow2 0.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)}}{a} \]

      3. rem-square-sqrt 70.5%

         \[\leadsto \frac{\color{blue}{-1} \cdot \sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-1}\right)}^{3}\right)}}{a} \]

      4. rem-cube-cbrt 70.5%

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

      5. associate-*r* 70.5%

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

      6. *-commutative 70.5%

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

      7. associate-*r* 70.5%

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

      8. neg-mul-1 70.5%

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

      9. mul-1-neg 70.5%

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

      10. *-commutative 70.5%

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

   7. Simplified 70.5%

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

   if 3.0000000000000001e-59 < b_2

   1. Initial program 72.3%

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

   3. Taylor expanded in c around 0 87.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.15 \cdot 10^{-101}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.4 \cdot 10^{-163}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.15e-101)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6.4e-163)
     (- (sqrt (/ c (- a))))
     (+ (* (/ b_2 a) -2.0) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.15e-101) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.4e-163) {
		tmp = -sqrt((c / -a));
	} else {
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.15d-101)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 6.4d-163) then
        tmp = -sqrt((c / -a))
    else
        tmp = ((b_2 / a) * (-2.0d0)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.15e-101) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.4e-163) {
		tmp = -Math.sqrt((c / -a));
	} else {
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.15e-101:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 6.4e-163:
		tmp = -math.sqrt((c / -a))
	else:
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.15e-101)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6.4e-163)
		tmp = Float64(-sqrt(Float64(c / Float64(-a))));
	else
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + 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 <= -2.15e-101)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 6.4e-163)
		tmp = -sqrt((c / -a));
	else
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.15e-101], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6.4e-163], (-N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision]), N[(N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.1499999999999999e-101

   1. Initial program 13.7%

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

   3. Taylor expanded in b_2 around -inf 89.1%

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

      1. associate-*r/ 89.1%

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

   5. Simplified 89.1%

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

   if -2.1499999999999999e-101 < b_2 < 6.39999999999999976e-163

   1. Initial program 74.7%

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

      1. sub-neg 74.7%

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

      2. +-commutative 74.7%

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

      3. distribute-lft-neg-in 74.7%

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

      4. add-cube-cbrt 74.1%

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

      5. associate-*r* 74.0%

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

      6. fma-define 74.0%

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

      7. pow2 74.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(\left(-a\right) \cdot \color{blue}{{\left(\sqrt[3]{c}\right)}^{2}}, \sqrt[3]{c}, b\_2 \cdot b\_2\right)}}{a} \]

      8. pow2 74.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(\left(-a\right) \cdot {\left(\sqrt[3]{c}\right)}^{2}, \sqrt[3]{c}, \color{blue}{{b\_2}^{2}}\right)}}{a} \]

   4. Applied egg-rr 74.0%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(\left(-a\right) \cdot {\left(\sqrt[3]{c}\right)}^{2}, \sqrt[3]{c}, {b\_2}^{2}\right)}}}{a} \]

   5. Taylor expanded in c around -inf 0.0%

      \[\leadsto \color{blue}{\sqrt{\frac{c \cdot {\left(\sqrt[3]{-1}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-1}\right)}^{3}}{a}}} \]

      2. unpow2 0.0%

         \[\leadsto \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-1}\right)}^{3}}{a}} \]

      3. rem-square-sqrt 34.7%

         \[\leadsto \color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-1}\right)}^{3}}{a}} \]

      4. rem-cube-cbrt 34.7%

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

      5. *-commutative 34.7%

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

      6. mul-1-neg 34.7%

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

   7. Simplified 34.7%

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

   if 6.39999999999999976e-163 < b_2

   1. Initial program 77.3%

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

   3. Taylor expanded in c around 0 79.5%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.15 \cdot 10^{-101}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.4 \cdot 10^{-163}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.2% accurate, 7.0× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = ((b_2 / a) * -2.0) + (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 <= (-5d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((b_2 / a) * (-2.0d0)) + (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 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

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

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + 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 <= -5e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 33.1%

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

   3. Taylor expanded in b_2 around -inf 64.0%

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

      1. associate-*r/ 64.0%

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

   5. Simplified 64.0%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 77.0%

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

   3. Taylor expanded in c around 0 60.9%

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

4. Final simplification 62.4%

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

Alternative 6: 68.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.3 \cdot 10^{-292}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.3e-292) (/ (* -0.5 c) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.3e-292) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.3d-292)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.3e-292) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.3e-292:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.3e-292)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.3e-292)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.3e-292], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.29999999999999995e-292

   1. Initial program 30.9%

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

   3. Taylor expanded in b_2 around -inf 66.0%

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

      1. associate-*r/ 66.0%

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

   5. Simplified 66.0%

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

   if -3.29999999999999995e-292 < b_2

   1. Initial program 77.6%

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

   3. Taylor expanded in b_2 around inf 58.6%

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

      1. associate-*r/ 58.6%

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

      2. *-commutative 58.6%

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

   5. Simplified 58.6%

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

Alternative 7: 47.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.3 \cdot 10^{-292}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.3e-292) (/ (* -0.5 c) b_2) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.3e-292) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = 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 <= (-3.3d-292)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = 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 <= -3.3e-292) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.3e-292:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.3e-292)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.3e-292)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.3e-292], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.29999999999999995e-292

   1. Initial program 30.9%

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

   3. Taylor expanded in b_2 around -inf 66.0%

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

      1. associate-*r/ 66.0%

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

   5. Simplified 66.0%

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

   if -3.29999999999999995e-292 < b_2

   1. Initial program 77.6%

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

   3. Taylor expanded in b_2 around 0 47.6%

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

      1. associate-*r* 47.6%

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

      2. neg-mul-1 47.6%

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

   5. Simplified 47.6%

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

   6. Taylor expanded in b_2 around inf 26.0%

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

      1. associate-*r/ 26.0%

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

      2. mul-1-neg 26.0%

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

   8. Simplified 26.0%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.3 \cdot 10^{-292}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.3 \cdot 10^{-292}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.3e-292) (* c (/ -0.5 b_2)) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.3e-292) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = 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 <= (-3.3d-292)) then
        tmp = c * ((-0.5d0) / b_2)
    else
        tmp = 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 <= -3.3e-292) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.3e-292:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.3e-292)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.3e-292)
		tmp = c * (-0.5 / b_2);
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.3e-292], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.29999999999999995e-292

   1. Initial program 30.9%

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

      1. sub-neg 30.9%

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

      2. +-commutative 30.9%

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

      3. distribute-lft-neg-in 30.9%

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

      4. add-cube-cbrt 30.6%

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

      5. associate-*r* 30.6%

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

      6. fma-define 30.6%

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

      7. pow2 30.6%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(\left(-a\right) \cdot \color{blue}{{\left(\sqrt[3]{c}\right)}^{2}}, \sqrt[3]{c}, b\_2 \cdot b\_2\right)}}{a} \]

      8. pow2 30.6%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(\left(-a\right) \cdot {\left(\sqrt[3]{c}\right)}^{2}, \sqrt[3]{c}, \color{blue}{{b\_2}^{2}}\right)}}{a} \]

   4. Applied egg-rr 30.6%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(\left(-a\right) \cdot {\left(\sqrt[3]{c}\right)}^{2}, \sqrt[3]{c}, {b\_2}^{2}\right)}}}{a} \]

   5. Taylor expanded in b_2 around -inf 51.2%

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

      1. associate-*r/ 51.2%

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

      2. associate-*r* 51.2%

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

   7. Simplified 51.2%

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

   8. Taylor expanded in a around 0 66.0%

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

      1. associate-*r/ 66.0%

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

      2. *-rgt-identity 66.0%

         \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2 \cdot 1}} \]

      3. times-frac 65.8%

         \[\leadsto \color{blue}{\frac{-0.5}{b\_2} \cdot \frac{c}{1}} \]

      4. /-rgt-identity 65.8%

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

   10. Simplified 65.8%

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

   if -3.29999999999999995e-292 < b_2

   1. Initial program 77.6%

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

   3. Taylor expanded in b_2 around 0 47.6%

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

      1. associate-*r* 47.6%

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

      2. neg-mul-1 47.6%

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

   5. Simplified 47.6%

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

   6. Taylor expanded in b_2 around inf 26.0%

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

      1. associate-*r/ 26.0%

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

      2. mul-1-neg 26.0%

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

   8. Simplified 26.0%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.3 \cdot 10^{-292}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-34}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-34) (* 0.5 (/ c b_2)) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-34) {
		tmp = 0.5 * (c / b_2);
	} else {
		tmp = 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 <= (-7d-34)) then
        tmp = 0.5d0 * (c / b_2)
    else
        tmp = 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 <= -7e-34) {
		tmp = 0.5 * (c / b_2);
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7e-34:
		tmp = 0.5 * (c / b_2)
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-34)
		tmp = Float64(0.5 * Float64(c / b_2));
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7e-34)
		tmp = 0.5 * (c / b_2);
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7e-34], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -7e-34

   1. Initial program 8.7%

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

   3. Taylor expanded in a around 0 2.1%

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

      1. associate-*r/ 2.1%

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

      2. *-commutative 2.1%

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

   5. Simplified 2.1%

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

   6. Taylor expanded in b_2 around 0 30.5%

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

   if -7e-34 < b_2

   1. Initial program 74.4%

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

   3. Taylor expanded in b_2 around 0 52.3%

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

      1. associate-*r* 52.3%

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

      2. neg-mul-1 52.3%

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

   5. Simplified 52.3%

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

   6. Taylor expanded in b_2 around inf 20.0%

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

      1. associate-*r/ 20.0%

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

      2. mul-1-neg 20.0%

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

   8. Simplified 20.0%

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

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-34}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 15.3% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in b_2 around 0 38.1%

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

   1. associate-*r* 38.1%

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

   2. neg-mul-1 38.1%

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

5. Simplified 38.1%

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

6. Taylor expanded in b_2 around inf 14.9%

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

   1. associate-*r/ 14.9%

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

   2. mul-1-neg 14.9%

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

8. Simplified 14.9%

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

9. Final simplification 14.9%

   \[\leadsto \frac{b\_2}{-a} \]
10. Add Preprocessing

Alternative 11: 2.5% accurate, 37.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in b_2 around 0 38.1%

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

   1. associate-*r* 38.1%

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

   2. neg-mul-1 38.1%

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

5. Simplified 38.1%

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

6. Taylor expanded in b_2 around inf 14.9%

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

   1. associate-*r/ 14.9%

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

   2. mul-1-neg 14.9%

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

8. Simplified 14.9%

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

   1. div-inv 14.9%

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

   2. add-sqr-sqrt 1.4%

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

   3. sqrt-unprod 2.1%

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

   4. sqr-neg 2.1%

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

   5. sqrt-unprod 0.7%

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

   6. add-sqr-sqrt 2.5%

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

10. Applied egg-rr 2.5%

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

    1. associate-*r/ 2.5%

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

    2. *-rgt-identity 2.5%

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

12. Simplified 2.5%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

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

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

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