quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.3% → 86.6%

Time: 11.8s

Alternatives: 8

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-105}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 10^{-15}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \left|b\_2 \cdot \sqrt{1 - a \cdot \left(c \cdot {b\_2}^{-2}\right)}\right|}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-105)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1e-15)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/
      (- (- b_2) (fabs (* b_2 (sqrt (- 1.0 (* a (* c (pow b_2 -2.0))))))))
      a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-105) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1e-15) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-b_2 - fabs((b_2 * sqrt((1.0 - (a * (c * pow(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 <= (-9d-105)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1d-15) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (-b_2 - abs((b_2 * sqrt((1.0d0 - (a * (c * (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 <= -9e-105) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1e-15) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-b_2 - Math.abs((b_2 * Math.sqrt((1.0 - (a * (c * Math.pow(b_2, -2.0)))))))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-105:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1e-15:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-b_2 - math.fabs((b_2 * math.sqrt((1.0 - (a * (c * math.pow(b_2, -2.0)))))))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-105)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1e-15)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(Float64(-b_2) - abs(Float64(b_2 * sqrt(Float64(1.0 - Float64(a * Float64(c * (b_2 ^ -2.0)))))))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9e-105)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1e-15)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-b_2 - abs((b_2 * sqrt((1.0 - (a * (c * (b_2 ^ -2.0)))))))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-105], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1e-15], 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[((-b$95$2) - N[Abs[N[(b$95$2 * N[Sqrt[N[(1.0 - N[(a * N[(c * N[Power[b$95$2, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.9999999999999995e-105

   1. Initial program 14.6%

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

   3. Taylor expanded in b_2 around -inf 87.7%

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

      1. associate-*r/ 87.8%

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

   5. Simplified 87.8%

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

   if -8.9999999999999995e-105 < b_2 < 1.0000000000000001e-15

   1. Initial program 87.7%

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

   if 1.0000000000000001e-15 < b_2

   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 67.0%

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

      1. mul-1-neg 67.0%

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

      2. unsub-neg 67.0%

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

      3. associate-/l* 67.1%

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

   5. Simplified 67.1%

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

      1. add-sqr-sqrt 67.1%

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

      2. rem-sqrt-square 67.1%

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

      3. sqrt-prod 70.3%

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

      4. sqrt-pow1 97.6%

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

      5. metadata-eval 97.6%

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

      6. pow1 97.6%

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

      7. div-inv 97.6%

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

      8. pow-flip 97.8%

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

      9. metadata-eval 97.8%

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

   7. Applied egg-rr 97.8%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\left|b\_2 \cdot \sqrt{1 - a \cdot \left(c \cdot {b\_2}^{-2}\right)}\right|}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-105}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 10^{-15}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \left|b\_2 \cdot \sqrt{1 - a \cdot \left(c \cdot {b\_2}^{-2}\right)}\right|}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 0.00015:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2 \cdot \sqrt{1 - a \cdot \left(c \cdot {b\_2}^{-2}\right)}}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-102)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 0.00015)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (+ b_2 (* b_2 (sqrt (- 1.0 (* a (* c (pow b_2 -2.0))))))) (- a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-102) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 0.00015) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 + (b_2 * sqrt((1.0 - (a * (c * pow(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 <= (-1d-102)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 0.00015d0) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (b_2 + (b_2 * sqrt((1.0d0 - (a * (c * (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 <= -1e-102) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 0.00015) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 + (b_2 * Math.sqrt((1.0 - (a * (c * Math.pow(b_2, -2.0))))))) / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-102:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 0.00015:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 + (b_2 * math.sqrt((1.0 - (a * (c * math.pow(b_2, -2.0))))))) / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-102)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 0.00015)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 + Float64(b_2 * sqrt(Float64(1.0 - Float64(a * Float64(c * (b_2 ^ -2.0))))))) / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-102)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 0.00015)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 + (b_2 * sqrt((1.0 - (a * (c * (b_2 ^ -2.0))))))) / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-102], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 0.00015], 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[(b$95$2 + N[(b$95$2 * N[Sqrt[N[(1.0 - N[(a * N[(c * N[Power[b$95$2, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.99999999999999933e-103

   1. Initial program 14.6%

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

   3. Taylor expanded in b_2 around -inf 87.7%

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

      1. associate-*r/ 87.8%

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

   5. Simplified 87.8%

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

   if -9.99999999999999933e-103 < b_2 < 1.49999999999999987e-4

   1. Initial program 86.9%

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

   if 1.49999999999999987e-4 < b_2

   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 67.0%

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

      1. mul-1-neg 67.0%

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

      2. unsub-neg 67.0%

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

      3. associate-/l* 67.1%

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

   5. Simplified 67.1%

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

      1. *-commutative 67.1%

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

      2. sqrt-prod 70.4%

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

      3. sqrt-pow1 98.7%

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

      4. metadata-eval 98.7%

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

      5. pow1 98.7%

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

      6. div-inv 98.7%

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

      7. pow-flip 98.8%

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

      8. metadata-eval 98.8%

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

   7. Applied egg-rr 98.8%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt{1 - a \cdot \left(c \cdot {b\_2}^{-2}\right)} \cdot b\_2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 0.00015:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2 \cdot \sqrt{1 - a \cdot \left(c \cdot {b\_2}^{-2}\right)}}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-102}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{-72} \lor \neg \left(b\_2 \leq 1.35 \cdot 10^{-21}\right) \land b\_2 \leq 2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.9e-102)
   (/ (* -0.5 c) b_2)
   (if (or (<= b_2 6.2e-72) (and (not (<= b_2 1.35e-21)) (<= b_2 2.8e-15)))
     (/ (- (- b_2) (sqrt (* c (- a)))) 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 <= -1.9e-102) {
		tmp = (-0.5 * c) / b_2;
	} else if ((b_2 <= 6.2e-72) || (!(b_2 <= 1.35e-21) && (b_2 <= 2.8e-15))) {
		tmp = (-b_2 - sqrt((c * -a))) / 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 <= (-1.9d-102)) then
        tmp = ((-0.5d0) * c) / b_2
    else if ((b_2 <= 6.2d-72) .or. (.not. (b_2 <= 1.35d-21)) .and. (b_2 <= 2.8d-15)) then
        tmp = (-b_2 - sqrt((c * -a))) / 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 <= -1.9e-102) {
		tmp = (-0.5 * c) / b_2;
	} else if ((b_2 <= 6.2e-72) || (!(b_2 <= 1.35e-21) && (b_2 <= 2.8e-15))) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / 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 <= -1.9e-102:
		tmp = (-0.5 * c) / b_2
	elif (b_2 <= 6.2e-72) or (not (b_2 <= 1.35e-21) and (b_2 <= 2.8e-15)):
		tmp = (-b_2 - math.sqrt((c * -a))) / 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 <= -1.9e-102)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif ((b_2 <= 6.2e-72) || (!(b_2 <= 1.35e-21) && (b_2 <= 2.8e-15)))
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / 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 <= -1.9e-102)
		tmp = (-0.5 * c) / b_2;
	elseif ((b_2 <= 6.2e-72) || (~((b_2 <= 1.35e-21)) && (b_2 <= 2.8e-15)))
		tmp = (-b_2 - sqrt((c * -a))) / 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, -1.9e-102], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[Or[LessEqual[b$95$2, 6.2e-72], And[N[Not[LessEqual[b$95$2, 1.35e-21]], $MachinePrecision], LessEqual[b$95$2, 2.8e-15]]], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $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 < -1.90000000000000013e-102

   1. Initial program 14.6%

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

   3. Taylor expanded in b_2 around -inf 87.7%

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

      1. associate-*r/ 87.8%

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

   5. Simplified 87.8%

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

   if -1.90000000000000013e-102 < b_2 < 6.1999999999999996e-72 or 1.3500000000000001e-21 < b_2 < 2.80000000000000014e-15

   1. Initial program 87.8%

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

   3. Taylor expanded in b_2 around 0 82.8%

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

      1. mul-1-neg 82.8%

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

      2. distribute-rgt-neg-out 82.8%

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

   5. Simplified 82.8%

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

   if 6.1999999999999996e-72 < b_2 < 1.3500000000000001e-21 or 2.80000000000000014e-15 < b_2

   1. Initial program 68.5%

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

   3. Taylor expanded in c around 0 90.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-102}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{-72} \lor \neg \left(b\_2 \leq 1.35 \cdot 10^{-21}\right) \land b\_2 \leq 2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.06 \cdot 10^{-104}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.06e-104)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6.2e+98)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) 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 <= -1.06e-104) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.2e+98) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / 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 <= (-1.06d-104)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 6.2d+98) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / 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 <= -1.06e-104) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.2e+98) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / 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 <= -1.06e-104:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 6.2e+98:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / 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 <= -1.06e-104)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6.2e+98)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / 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 <= -1.06e-104)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 6.2e+98)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / 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, -1.06e-104], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6.2e+98], 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[(-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 < -1.06e-104

   1. Initial program 14.6%

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

   3. Taylor expanded in b_2 around -inf 87.7%

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

      1. associate-*r/ 87.8%

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

   5. Simplified 87.8%

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

   if -1.06e-104 < b_2 < 6.20000000000000038e98

   1. Initial program 88.5%

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

   if 6.20000000000000038e98 < b_2

   1. Initial program 54.7%

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

   3. Taylor expanded in c around 0 96.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.06 \cdot 10^{-104}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (/ (* -0.5 c) b_2)
   (+ (* -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 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} 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 <= (-5d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    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 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} 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 <= -5e-310:
		tmp = (-0.5 * c) / b_2
	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 <= -5e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	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 <= -5e-310)
		tmp = (-0.5 * c) / b_2;
	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, -5e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $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 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 29.6%

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

   3. Taylor expanded in b_2 around -inf 71.4%

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

      1. associate-*r/ 71.4%

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

   5. Simplified 71.4%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 75.2%

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

   3. Taylor expanded in c around 0 67.9%

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

Alternative 6: 67.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.5 \cdot 10^{-306}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.5e-306) (/ (* -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 <= -2.5e-306) {
		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 <= (-2.5d-306)) 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 <= -2.5e-306) {
		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 <= -2.5e-306:
		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 <= -2.5e-306)
		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 <= -2.5e-306)
		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, -2.5e-306], 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 < -2.49999999999999999e-306

   1. Initial program 29.0%

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

   3. Taylor expanded in b_2 around -inf 72.0%

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

      1. associate-*r/ 72.0%

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

   5. Simplified 72.0%

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

   if -2.49999999999999999e-306 < b_2

   1. Initial program 75.4%

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

   3. Taylor expanded in b_2 around inf 66.4%

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

      1. *-commutative 66.4%

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

   5. Simplified 66.4%

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

Alternative 7: 34.9% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot c}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ (* -0.5 c) b_2))

C

double code(double a, double b_2, double c) {
	return (-0.5 * c) / b_2;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = ((-0.5d0) * c) / b_2
end function

Java

public static double code(double a, double b_2, double c) {
	return (-0.5 * c) / b_2;
}

Python

def code(a, b_2, c):
	return (-0.5 * c) / b_2

Julia

function code(a, b_2, c)
	return Float64(Float64(-0.5 * c) / b_2)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-0.5 * c) / b_2;
end

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in b_2 around -inf 34.1%

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

   1. associate-*r/ 34.1%

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

5. Simplified 34.1%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]
6. Add Preprocessing

Alternative 8: 2.6% accurate, 22.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. clear-num 54.1%

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

   2. associate-/r/ 54.1%

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

   3. add-sqr-sqrt 12.3%

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

   4. sqrt-unprod 28.4%

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

   5. sqr-neg 28.4%

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

   6. sqrt-prod 23.5%

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

   7. add-sqr-sqrt 34.1%

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

   8. sub-neg 34.1%

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

   9. add-sqr-sqrt 31.5%

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

   10. hypot-define 25.7%

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

   11. distribute-rgt-neg-in 25.7%

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

4. Applied egg-rr 25.7%

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

5. Taylor expanded in b_2 around -inf 2.4%

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

   1. associate-*r/ 2.4%

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

   2. *-commutative 2.4%

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

   3. associate-/l* 2.4%

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

7. Simplified 2.4%

   \[\leadsto \color{blue}{b\_2 \cdot \frac{2}{a}} \]
8. Add Preprocessing

Developer target: 99.6% accurate, 0.1× speedup

Math

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

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 2024095 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (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))