quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.5% → 89.9%

Time: 15.3s

Alternatives: 9

Speedup: 15.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.5% 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: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -1.35 \cdot 10^{+58}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{-c}{b_2 - t_0}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))))
   (if (<= b_2 -1.35e+58)
     (/ (* c -0.5) b_2)
     (if (<= b_2 5e-309)
       (/ (- c) (- b_2 t_0))
       (if (<= b_2 5e+71) (/ (- (- b_2) t_0) a) (* -2.0 (/ b_2 a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -1.35e+58) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 5e-309) {
		tmp = -c / (b_2 - t_0);
	} else if (b_2 <= 5e+71) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    if (b_2 <= (-1.35d+58)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 5d-309) then
        tmp = -c / (b_2 - t_0)
    else if (b_2 <= 5d+71) then
        tmp = (-b_2 - t_0) / a
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -1.35e+58) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 5e-309) {
		tmp = -c / (b_2 - t_0);
	} else if (b_2 <= 5e+71) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(((b_2 * b_2) - (c * a)))
	tmp = 0
	if b_2 <= -1.35e+58:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 5e-309:
		tmp = -c / (b_2 - t_0)
	elif b_2 <= 5e+71:
		tmp = (-b_2 - t_0) / a
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -1.35e+58)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 5e-309)
		tmp = Float64(Float64(-c) / Float64(b_2 - t_0));
	elseif (b_2 <= 5e+71)
		tmp = Float64(Float64(Float64(-b_2) - t_0) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt(((b_2 * b_2) - (c * a)));
	tmp = 0.0;
	if (b_2 <= -1.35e+58)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 5e-309)
		tmp = -c / (b_2 - t_0);
	elseif (b_2 <= 5e+71)
		tmp = (-b_2 - t_0) / a;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -1.35e+58], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5e-309], N[((-c) / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 5e+71], N[(N[((-b$95$2) - t$95$0), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.3500000000000001e58

   1. Initial program 12.6%

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

   2. Taylor expanded in b_2 around -inf 80.5%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b_2}}}{a} \]
   3. Step-by-step derivation

      1. associate-/l* 84.0%

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

   4. Simplified 84.0%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a}{\frac{b_2}{c}}}}{a} \]
   5. Step-by-step derivation

      1. associate-/l* 84.0%

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

      2. associate-/r/ 83.9%

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

      3. div-inv 83.9%

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

      4. clear-num 84.4%

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

   6. Applied egg-rr 84.4%

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

      1. associate-*r* 94.8%

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

      2. associate-*r/ 94.8%

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

      3. *-commutative 94.8%

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

      4. div-inv 94.8%

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

      5. associate-*l* 94.8%

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

      6. lft-mult-inverse 95.0%

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

      7. metadata-eval 95.0%

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

   8. Applied egg-rr 95.0%

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

   if -1.3500000000000001e58 < b_2 < 4.9999999999999995e-309

   1. Initial program 65.7%

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

   3. Applied egg-rr 65.9%

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

   4. Taylor expanded in b_2 around 0 79.8%

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

      1. mul-1-neg 79.8%

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

      2. distribute-rgt-neg-out 79.8%

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

   6. Simplified 79.8%

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

      1. div-inv 79.5%

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

      2. associate-/l* 82.8%

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

      3. associate-*l/ 88.7%

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

      4. rgt-mult-inverse 88.7%

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

      5. clear-num 88.8%

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

      6. remove-double-div 88.9%

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

      7. div-inv 88.8%

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

   8. Applied egg-rr 88.8%

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

      1. associate-*r/ 88.9%

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

      2. *-rgt-identity 88.9%

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

   10. Simplified 88.9%

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

   if 4.9999999999999995e-309 < b_2 < 4.99999999999999972e71

   1. Initial program 87.9%

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

   if 4.99999999999999972e71 < b_2

   1. Initial program 57.5%

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

   2. Taylor expanded in b_2 around inf 96.7%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.35 \cdot 10^{+58}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{-c}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 2: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 7 \cdot 10^{-70}:\\ \;\;\;\;\frac{-c}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.8e+58)
   (/ (* c -0.5) b_2)
   (if (<= b_2 7e-70)
     (/ (- c) (- b_2 (sqrt (- (* b_2 b_2) (* c a)))))
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.8e+58) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 7e-70) {
		tmp = -c / (b_2 - sqrt(((b_2 * b_2) - (c * a))));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.8d+58)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 7d-70) then
        tmp = -c / (b_2 - sqrt(((b_2 * b_2) - (c * a))))
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.8e+58) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 7e-70) {
		tmp = -c / (b_2 - Math.sqrt(((b_2 * b_2) - (c * a))));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.8e+58:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 7e-70:
		tmp = -c / (b_2 - math.sqrt(((b_2 * b_2) - (c * a))))
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.8e+58)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 7e-70)
		tmp = Float64(Float64(-c) / Float64(b_2 - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.8e+58)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 7e-70)
		tmp = -c / (b_2 - sqrt(((b_2 * b_2) - (c * a))));
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.8e+58], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7e-70], N[((-c) / N[(b$95$2 - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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 < -1.79999999999999998e58

   1. Initial program 12.6%

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

   2. Taylor expanded in b_2 around -inf 80.5%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b_2}}}{a} \]
   3. Step-by-step derivation

      1. associate-/l* 84.0%

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

   4. Simplified 84.0%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a}{\frac{b_2}{c}}}}{a} \]
   5. Step-by-step derivation

      1. associate-/l* 84.0%

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

      2. associate-/r/ 83.9%

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

      3. div-inv 83.9%

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

      4. clear-num 84.4%

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

   6. Applied egg-rr 84.4%

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

      1. associate-*r* 94.8%

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

      2. associate-*r/ 94.8%

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

      3. *-commutative 94.8%

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

      4. div-inv 94.8%

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

      5. associate-*l* 94.8%

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

      6. lft-mult-inverse 95.0%

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

      7. metadata-eval 95.0%

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

   8. Applied egg-rr 95.0%

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

   if -1.79999999999999998e58 < b_2 < 6.99999999999999949e-70

   1. Initial program 71.5%

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

   3. Applied egg-rr 67.4%

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

   4. Taylor expanded in b_2 around 0 77.4%

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

      1. mul-1-neg 77.4%

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

      2. distribute-rgt-neg-out 77.4%

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

   6. Simplified 77.4%

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

      1. div-inv 77.2%

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

      2. associate-/l* 79.7%

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

      3. associate-*l/ 84.0%

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

      4. rgt-mult-inverse 84.1%

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

      5. clear-num 84.1%

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

      6. remove-double-div 84.2%

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

      7. div-inv 84.0%

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

   8. Applied egg-rr 84.0%

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

      1. associate-*r/ 84.2%

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

      2. *-rgt-identity 84.2%

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

   10. Simplified 84.2%

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

   if 6.99999999999999949e-70 < b_2

   1. Initial program 68.5%

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

   2. Taylor expanded in b_2 around inf 85.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 7 \cdot 10^{-70}:\\ \;\;\;\;\frac{-c}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.36 \cdot 10^{-22}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.36e-22)
   (/ (* c -0.5) b_2)
   (if (<= b_2 3.2e-5)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.36e-22) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 3.2e-5) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.36d-22)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 3.2d-5) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.36e-22) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 3.2e-5) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.36e-22:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 3.2e-5:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.36e-22)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 3.2e-5)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.36e-22)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 3.2e-5)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.36e-22], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.2e-5], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.36e-22

   1. Initial program 16.2%

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

   2. Taylor expanded in b_2 around -inf 75.8%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b_2}}}{a} \]
   3. Step-by-step derivation

      1. associate-/l* 78.7%

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

   4. Simplified 78.7%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a}{\frac{b_2}{c}}}}{a} \]
   5. Step-by-step derivation

      1. associate-/l* 78.6%

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

      2. associate-/r/ 78.6%

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

      3. div-inv 78.6%

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

      4. clear-num 79.0%

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

   6. Applied egg-rr 79.0%

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

      1. associate-*r* 89.9%

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

      2. associate-*r/ 89.9%

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

      3. *-commutative 89.9%

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

      4. div-inv 89.9%

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

      5. associate-*l* 89.9%

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

      6. lft-mult-inverse 90.1%

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

      7. metadata-eval 90.1%

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

   8. Applied egg-rr 90.1%

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

   if -1.36e-22 < b_2 < 3.19999999999999986e-5

   1. Initial program 77.1%

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

   2. Taylor expanded in b_2 around 0 67.8%

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

      1. mul-1-neg 67.8%

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

      2. distribute-rgt-neg-out 67.8%

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

   4. Simplified 67.8%

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

   if 3.19999999999999986e-5 < b_2

   1. Initial program 67.8%

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

   2. Taylor expanded in b_2 around inf 92.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.36 \cdot 10^{-22}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 4: 67.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (/ (* c -0.5) 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 = (c * -0.5) / 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 = (c * (-0.5d0)) / 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 = (c * -0.5) / 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 = (c * -0.5) / 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(c * -0.5) / 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 = (c * -0.5) / 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[(c * -0.5), $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 37.7%

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

   2. Taylor expanded in b_2 around -inf 51.9%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b_2}}}{a} \]
   3. Step-by-step derivation

      1. associate-/l* 55.7%

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

   4. Simplified 55.7%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a}{\frac{b_2}{c}}}}{a} \]
   5. Step-by-step derivation

      1. associate-/l* 55.7%

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

      2. associate-/r/ 55.7%

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

      3. div-inv 55.7%

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

      4. clear-num 56.0%

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

   6. Applied egg-rr 56.0%

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

      1. associate-*r* 64.8%

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

      2. associate-*r/ 64.8%

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

      3. *-commutative 64.8%

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

      4. div-inv 64.8%

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

      5. associate-*l* 64.8%

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

      6. lft-mult-inverse 64.9%

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

      7. metadata-eval 64.9%

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

   8. Applied egg-rr 64.9%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 72.7%

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

   2. Taylor expanded in b_2 around inf 70.5%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5: 67.3% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 37.7%

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

   2. Taylor expanded in b_2 around -inf 64.9%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 72.7%

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

   2. Taylor expanded in b_2 around inf 70.3%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 6: 67.3% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 37.7%

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

   2. Taylor expanded in b_2 around -inf 51.9%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b_2}}}{a} \]
   3. Step-by-step derivation

      1. associate-/l* 55.7%

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

   4. Simplified 55.7%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a}{\frac{b_2}{c}}}}{a} \]
   5. Step-by-step derivation

      1. associate-/l* 55.7%

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

      2. associate-/r/ 55.7%

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

      3. div-inv 55.7%

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

      4. clear-num 56.0%

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

   6. Applied egg-rr 56.0%

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

      1. associate-*r* 64.8%

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

      2. associate-*r/ 64.8%

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

      3. *-commutative 64.8%

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

      4. div-inv 64.8%

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

      5. associate-*l* 64.8%

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

      6. lft-mult-inverse 64.9%

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

      7. metadata-eval 64.9%

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

   8. Applied egg-rr 64.9%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 72.7%

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

   2. Taylor expanded in b_2 around inf 70.3%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 7: 15.5% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \frac{b_2}{a} \cdot -3 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* (/ b_2 a) -3.0))

C

double code(double a, double b_2, double c) {
	return (b_2 / a) * -3.0;
}

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) * (-3.0d0)
end function

Java

public static double code(double a, double b_2, double c) {
	return (b_2 / a) * -3.0;
}

Python

def code(a, b_2, c):
	return (b_2 / a) * -3.0

Julia

function code(a, b_2, c)
	return Float64(Float64(b_2 / a) * -3.0)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (b_2 / a) * -3.0;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(b$95$2 / a), $MachinePrecision] * -3.0), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

3. Applied egg-rr 45.7%

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

4. Taylor expanded in b_2 around inf 23.2%

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

5. Taylor expanded in b_2 around 0 14.3%

   \[\leadsto \color{blue}{-3 \cdot \frac{b_2}{a}} \]

6. Final simplification 14.3%

   \[\leadsto \frac{b_2}{a} \cdot -3 \]

Alternative 8: 34.7% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{b_2}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

2. Taylor expanded in b_2 around inf 32.3%

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

3. Final simplification 32.3%

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

Alternative 9: 15.2% 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(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 53.0%

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

2. Taylor expanded in b_2 around 0 34.8%

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

   1. mul-1-neg 34.8%

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

   2. distribute-rgt-neg-out 34.8%

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

4. Simplified 34.8%

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

5. Taylor expanded in b_2 around inf 14.0%

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

   1. associate-*r/ 14.0%

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

   2. neg-mul-1 14.0%

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

7. Simplified 14.0%

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

8. Final simplification 14.0%

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

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 2023297 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

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

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