quad2p (problem 3.2.1, positive)

Percentage Accurate: 59.6% → 83.3%

Time: 14.0s

Alternatives: 11

Speedup: 8.0×

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: 59.6% 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: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -8.2 \cdot 10^{+67}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.2e+67)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 3.2)
     (/ (- (sqrt (fma a (- c) (* b_2 b_2))) b_2) a)
     (/ (* -0.5 (/ c (/ b_2 a))) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.2e+67) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 3.2) {
		tmp = (sqrt(fma(a, -c, (b_2 * b_2))) - b_2) / a;
	} else {
		tmp = (-0.5 * (c / (b_2 / a))) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.2e+67)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 3.2)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-c), Float64(b_2 * b_2))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * Float64(c / Float64(b_2 / a))) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.2e+67], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.2], N[(N[(N[Sqrt[N[(a * (-c) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.19999999999999959e67

   1. Initial program 61.0%

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

      1. +-commutative 61.0%

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

      2. unsub-neg 61.0%

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

   3. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 97.8%

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

   if -8.19999999999999959e67 < b_2 < 3.2000000000000002

   1. Initial program 80.1%

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

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} - b_2}{a}} \]
   4. Add Preprocessing

   if 3.2000000000000002 < b_2

   1. Initial program 30.7%

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

      1. +-commutative 30.7%

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

      2. unsub-neg 30.7%

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

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 73.6%

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

      1. associate-/l* 78.8%

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

      2. associate-/r/ 79.2%

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

   7. Simplified 79.2%

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

      1. *-commutative 79.2%

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

      2. clear-num 79.9%

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

      3. un-div-inv 79.9%

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

   9. Applied egg-rr 79.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8.2 \cdot 10^{+67}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{if}\;b_2 \leq -4.6 \cdot 10^{-41}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.55 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{-63}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot c\right)}{b_2} \cdot \frac{1}{-a}\\ \mathbf{elif}\;b_2 \leq 0.0016:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* a (- c))) b_2) a)))
   (if (<= b_2 -4.6e-41)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
     (if (<= b_2 1.55e-113)
       t_0
       (if (<= b_2 8e-63)
         (* (/ (* 0.5 (* a c)) b_2) (/ 1.0 (- a)))
         (if (<= b_2 0.0016) t_0 (/ (* -0.5 (/ c (/ b_2 a))) a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = (sqrt((a * -c)) - b_2) / a;
	double tmp;
	if (b_2 <= -4.6e-41) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.55e-113) {
		tmp = t_0;
	} else if (b_2 <= 8e-63) {
		tmp = ((0.5 * (a * c)) / b_2) * (1.0 / -a);
	} else if (b_2 <= 0.0016) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * (c / (b_2 / a))) / 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((a * -c)) - b_2) / a
    if (b_2 <= (-4.6d-41)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 1.55d-113) then
        tmp = t_0
    else if (b_2 <= 8d-63) then
        tmp = ((0.5d0 * (a * c)) / b_2) * (1.0d0 / -a)
    else if (b_2 <= 0.0016d0) then
        tmp = t_0
    else
        tmp = ((-0.5d0) * (c / (b_2 / a))) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = (Math.sqrt((a * -c)) - b_2) / a;
	double tmp;
	if (b_2 <= -4.6e-41) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.55e-113) {
		tmp = t_0;
	} else if (b_2 <= 8e-63) {
		tmp = ((0.5 * (a * c)) / b_2) * (1.0 / -a);
	} else if (b_2 <= 0.0016) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * (c / (b_2 / a))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = (math.sqrt((a * -c)) - b_2) / a
	tmp = 0
	if b_2 <= -4.6e-41:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 1.55e-113:
		tmp = t_0
	elif b_2 <= 8e-63:
		tmp = ((0.5 * (a * c)) / b_2) * (1.0 / -a)
	elif b_2 <= 0.0016:
		tmp = t_0
	else:
		tmp = (-0.5 * (c / (b_2 / a))) / a
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a)
	tmp = 0.0
	if (b_2 <= -4.6e-41)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.55e-113)
		tmp = t_0;
	elseif (b_2 <= 8e-63)
		tmp = Float64(Float64(Float64(0.5 * Float64(a * c)) / b_2) * Float64(1.0 / Float64(-a)));
	elseif (b_2 <= 0.0016)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * Float64(c / Float64(b_2 / a))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = (sqrt((a * -c)) - b_2) / a;
	tmp = 0.0;
	if (b_2 <= -4.6e-41)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 1.55e-113)
		tmp = t_0;
	elseif (b_2 <= 8e-63)
		tmp = ((0.5 * (a * c)) / b_2) * (1.0 / -a);
	elseif (b_2 <= 0.0016)
		tmp = t_0;
	else
		tmp = (-0.5 * (c / (b_2 / a))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[b$95$2, -4.6e-41], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.55e-113], t$95$0, If[LessEqual[b$95$2, 8e-63], N[(N[(N[(0.5 * N[(a * c), $MachinePrecision]), $MachinePrecision] / b$95$2), $MachinePrecision] * N[(1.0 / (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 0.0016], t$95$0, N[(N[(-0.5 * N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -4.6000000000000002e-41

   1. Initial program 70.1%

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

      1. +-commutative 70.1%

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

      2. unsub-neg 70.1%

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

   3. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 89.4%

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

   if -4.6000000000000002e-41 < b_2 < 1.55000000000000006e-113 or 8.00000000000000053e-63 < b_2 < 0.00160000000000000008

   1. Initial program 81.9%

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

      1. +-commutative 81.9%

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

      2. unsub-neg 81.9%

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

   3. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around 0 67.4%

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

      1. associate-*r* 67.4%

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

      2. neg-mul-1 67.4%

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

   7. Simplified 67.4%

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

   if 1.55000000000000006e-113 < b_2 < 8.00000000000000053e-63

   1. Initial program 57.7%

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

      1. +-commutative 57.7%

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

      2. unsub-neg 57.7%

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

   3. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 75.9%

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

      1. associate-/l* 58.3%

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

      2. associate-/r/ 58.2%

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

   7. Simplified 58.2%

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

      1. frac-2neg 58.2%

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

      2. div-inv 58.2%

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

      3. *-commutative 58.2%

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

      4. distribute-rgt-neg-in 58.2%

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

      5. metadata-eval 58.2%

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

      6. associate-*l/ 76.1%

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

      7. associate-*l/ 76.1%

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

   9. Applied egg-rr 76.1%

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

   if 0.00160000000000000008 < b_2

   1. Initial program 30.7%

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

      1. +-commutative 30.7%

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

      2. unsub-neg 30.7%

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

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 73.6%

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

      1. associate-/l* 78.8%

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

      2. associate-/r/ 79.2%

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

   7. Simplified 79.2%

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

      1. *-commutative 79.2%

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

      2. clear-num 79.9%

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

      3. un-div-inv 79.9%

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

   9. Applied egg-rr 79.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.6 \cdot 10^{-41}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.55 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 8 \cdot 10^{-63}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot c\right)}{b_2} \cdot \frac{1}{-a}\\ \mathbf{elif}\;b_2 \leq 0.0016:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{+67}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 11:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+67)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 11.0)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* -0.5 (/ c (/ b_2 a))) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+67) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 11.0) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (-0.5 * (c / (b_2 / a))) / 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+67)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 11.0d0) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = ((-0.5d0) * (c / (b_2 / a))) / 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+67) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 11.0) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (-0.5 * (c / (b_2 / a))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e+67:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 11.0:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (-0.5 * (c / (b_2 / a))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+67)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 11.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * Float64(c / Float64(b_2 / a))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e+67)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 11.0)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = (-0.5 * (c / (b_2 / a))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+67], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 11.0], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.99999999999999976e67

   1. Initial program 61.0%

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

      1. +-commutative 61.0%

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

      2. unsub-neg 61.0%

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

   3. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 97.8%

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

   if -4.99999999999999976e67 < b_2 < 11

   1. Initial program 80.1%

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

      1. +-commutative 80.1%

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

      2. unsub-neg 80.1%

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

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   if 11 < b_2

   1. Initial program 30.7%

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

      1. +-commutative 30.7%

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

      2. unsub-neg 30.7%

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

   3. Simplified 30.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 73.6%

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

      1. associate-/l* 78.8%

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

      2. associate-/r/ 79.2%

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

   7. Simplified 79.2%

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

      1. *-commutative 79.2%

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

      2. clear-num 79.9%

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

      3. un-div-inv 79.9%

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

   9. Applied egg-rr 79.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{+67}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 11:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 7.6 \cdot 10^{+261}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 1.6 \cdot 10^{+294}:\\ \;\;\;\;\frac{b_2 - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 7.6e+261)
     (/ -0.5 (/ b_2 c))
     (if (<= b_2 1.6e+294) (/ (- b_2 b_2) a) (* c (/ -0.5 b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 7.6e+261) {
		tmp = -0.5 / (b_2 / c);
	} else if (b_2 <= 1.6e+294) {
		tmp = (b_2 - b_2) / a;
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d-310)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 7.6d+261) then
        tmp = (-0.5d0) / (b_2 / c)
    else if (b_2 <= 1.6d+294) then
        tmp = (b_2 - b_2) / a
    else
        tmp = c * ((-0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 7.6e+261) {
		tmp = -0.5 / (b_2 / c);
	} else if (b_2 <= 1.6e+294) {
		tmp = (b_2 - b_2) / a;
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 7.6e+261:
		tmp = -0.5 / (b_2 / c)
	elif b_2 <= 1.6e+294:
		tmp = (b_2 - b_2) / a
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 7.6e+261)
		tmp = Float64(-0.5 / Float64(b_2 / c));
	elseif (b_2 <= 1.6e+294)
		tmp = Float64(Float64(b_2 - b_2) / a);
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 7.6e+261)
		tmp = -0.5 / (b_2 / c);
	elseif (b_2 <= 1.6e+294)
		tmp = (b_2 - b_2) / a;
	else
		tmp = c * (-0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 7.6e+261], N[(-0.5 / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.6e+294], N[(N[(b$95$2 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 75.5%

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

      1. +-commutative 75.5%

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

      2. unsub-neg 75.5%

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

   3. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 64.2%

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

      1. *-commutative 64.2%

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

   7. Simplified 64.2%

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

   if -3.999999999999988e-310 < b_2 < 7.6000000000000003e261

   1. Initial program 52.9%

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

      1. +-commutative 52.9%

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

      2. unsub-neg 52.9%

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

   3. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 54.7%

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

      1. associate-/l* 56.4%

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

      2. associate-/r/ 55.8%

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

   7. Simplified 55.8%

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

   8. Taylor expanded in a around 0 47.2%

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

      1. associate-*r/ 47.2%

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

      2. associate-/l* 47.8%

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

   10. Simplified 47.8%

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

   if 7.6000000000000003e261 < b_2 < 1.5999999999999999e294

   1. Initial program 1.6%

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

      1. +-commutative 1.6%

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

      2. unsub-neg 1.6%

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

   3. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 98.4%

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

   if 1.5999999999999999e294 < b_2

   1. Initial program 1.6%

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

      1. +-commutative 1.6%

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

      2. unsub-neg 1.6%

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

   3. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 75.7%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in a around 0 88.2%

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

      1. associate-*r/ 88.2%

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

      2. associate-/l* 88.0%

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

   10. Simplified 88.0%

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

       1. associate-/r/ 88.2%

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

   12. Applied egg-rr 88.2%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 7.6 \cdot 10^{+261}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 1.6 \cdot 10^{+294}:\\ \;\;\;\;\frac{b_2 - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (/ (* -0.5 (/ c (/ b_2 a))) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (-0.5 * (c / (b_2 / a))) / 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 <= (-4d-310)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else
        tmp = ((-0.5d0) * (c / (b_2 / a))) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (-0.5 * (c / (b_2 / a))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	else:
		tmp = (-0.5 * (c / (b_2 / a))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / Float64(b_2 / a))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	else
		tmp = (-0.5 * (c / (b_2 / a))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 75.5%

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

      1. +-commutative 75.5%

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

      2. unsub-neg 75.5%

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

   3. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 64.5%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 47.0%

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

      1. +-commutative 47.0%

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

      2. unsub-neg 47.0%

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

   3. Simplified 47.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 56.7%

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

      1. associate-/l* 59.4%

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

      2. associate-/r/ 59.6%

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

   7. Simplified 59.6%

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

      1. *-commutative 59.6%

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

      2. clear-num 60.0%

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

      3. un-div-inv 60.0%

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

   9. Applied egg-rr 60.0%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{a}{\frac{b_2}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310) (/ (* b_2 -2.0) a) (* (/ -0.5 a) (/ a (/ b_2 c)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 / a) * (a / (b_2 / c));
	}
	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 <= (-4d-310)) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = ((-0.5d0) / a) * (a / (b_2 / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 / a) * (a / (b_2 / c));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (-0.5 / a) * (a / (b_2 / c))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(-0.5 / a) * Float64(a / Float64(b_2 / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (-0.5 / a) * (a / (b_2 / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(a / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 75.5%

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

      1. +-commutative 75.5%

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

      2. unsub-neg 75.5%

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

   3. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 64.2%

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

      1. *-commutative 64.2%

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

   7. Simplified 64.2%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 47.0%

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

      1. +-commutative 47.0%

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

      2. unsub-neg 47.0%

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

   3. Simplified 47.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 56.7%

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

      1. associate-/l* 59.4%

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

      2. associate-/r/ 59.6%

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

   7. Simplified 59.6%

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

      1. associate-/l* 59.4%

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

      2. associate-/r/ 59.5%

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

      3. associate-/r/ 59.4%

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

   9. Applied egg-rr 59.4%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{a}{\frac{b_2}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.3% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 4.6 \cdot 10^{-263}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \left(c \cdot \frac{a}{b_2}\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 4.6e-263) (/ (* b_2 -2.0) a) (/ (* -0.5 (* c (/ a b_2))) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 4.6e-263) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 * (c * (a / 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 <= 4.6d-263) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = ((-0.5d0) * (c * (a / 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 <= 4.6e-263) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 * (c * (a / b_2))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 4.6e-263:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (-0.5 * (c * (a / b_2))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 4.6e-263)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(-0.5 * Float64(c * Float64(a / b_2))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 4.6e-263)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (-0.5 * (c * (a / b_2))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 4.6e-263], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * N[(c * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 4.60000000000000006e-263

   1. Initial program 76.2%

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

      1. +-commutative 76.2%

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

      2. unsub-neg 76.2%

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

   3. Simplified 76.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 62.6%

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

      1. *-commutative 62.6%

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

   7. Simplified 62.6%

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

   if 4.60000000000000006e-263 < b_2

   1. Initial program 45.9%

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

      1. +-commutative 45.9%

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

      2. unsub-neg 45.9%

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

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 57.8%

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

      1. associate-/l* 60.6%

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

      2. associate-/r/ 60.8%

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

   7. Simplified 60.8%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 4.6 \cdot 10^{-263}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \left(c \cdot \frac{a}{b_2}\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 4.6 \cdot 10^{-263}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 4.6e-263) (/ (* b_2 -2.0) a) (/ (* -0.5 (/ c (/ b_2 a))) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 4.6e-263) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 * (c / (b_2 / a))) / 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 <= 4.6d-263) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = ((-0.5d0) * (c / (b_2 / a))) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 4.6e-263) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 * (c / (b_2 / a))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 4.6e-263:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (-0.5 * (c / (b_2 / a))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 4.6e-263)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(-0.5 * Float64(c / Float64(b_2 / a))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 4.6e-263)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (-0.5 * (c / (b_2 / a))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 4.6e-263], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 4.60000000000000006e-263

   1. Initial program 76.2%

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

      1. +-commutative 76.2%

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

      2. unsub-neg 76.2%

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

   3. Simplified 76.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 62.6%

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

      1. *-commutative 62.6%

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

   7. Simplified 62.6%

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

   if 4.60000000000000006e-263 < b_2

   1. Initial program 45.9%

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

      1. +-commutative 45.9%

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

      2. unsub-neg 45.9%

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

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 57.8%

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

      1. associate-/l* 60.6%

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

      2. associate-/r/ 60.8%

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

   7. Simplified 60.8%

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

      1. *-commutative 60.8%

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

      2. clear-num 61.2%

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

      3. un-div-inv 61.2%

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

   9. Applied egg-rr 61.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 4.6 \cdot 10^{-263}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310) (/ (* b_2 -2.0) a) (/ -0.5 (/ b_2 c))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = -0.5 / (b_2 / c);
	}
	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 <= (-4d-310)) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (-0.5d0) / (b_2 / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = -0.5 / (b_2 / c);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = -0.5 / (b_2 / c)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(-0.5 / Float64(b_2 / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = -0.5 / (b_2 / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 75.5%

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

      1. +-commutative 75.5%

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

      2. unsub-neg 75.5%

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

   3. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 64.2%

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

      1. *-commutative 64.2%

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

   7. Simplified 64.2%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 47.0%

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

      1. +-commutative 47.0%

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

      2. unsub-neg 47.0%

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

   3. Simplified 47.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 56.7%

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

      1. associate-/l* 59.4%

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

      2. associate-/r/ 59.6%

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

   7. Simplified 59.6%

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

   8. Taylor expanded in a around 0 49.4%

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

      1. associate-*r/ 49.4%

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

      2. associate-/l* 49.9%

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

   10. Simplified 49.9%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 23.2% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b_2} \end{array} \]

FPCore

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

C

double code(double a, double b_2, double c) {
	return c * (-0.5 / 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 = c * ((-0.5d0) / b_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.2%

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

   1. +-commutative 59.2%

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

   2. unsub-neg 59.2%

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

3. Simplified 59.2%

   \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
4. Add Preprocessing

5. Taylor expanded in b_2 around inf 33.3%

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

   1. associate-/l* 34.8%

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

   2. associate-/r/ 34.9%

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

7. Simplified 34.9%

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

8. Taylor expanded in a around 0 29.1%

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

   1. associate-*r/ 29.1%

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

   2. associate-/l* 29.4%

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

10. Simplified 29.4%

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

    1. associate-/r/ 29.1%

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

12. Applied egg-rr 29.1%

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

13. Final simplification 29.1%

    \[\leadsto c \cdot \frac{-0.5}{b_2} \]
14. Add Preprocessing

Alternative 11: 23.4% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{\frac{b_2}{c}} \end{array} \]

FPCore

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

C

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

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) / (b_2 / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.2%

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

   1. +-commutative 59.2%

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

   2. unsub-neg 59.2%

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

3. Simplified 59.2%

   \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
4. Add Preprocessing

5. Taylor expanded in b_2 around inf 33.3%

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

   1. associate-/l* 34.8%

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

   2. associate-/r/ 34.9%

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

7. Simplified 34.9%

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

8. Taylor expanded in a around 0 29.1%

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

   1. associate-*r/ 29.1%

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

   2. associate-/l* 29.4%

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

10. Simplified 29.4%

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

11. Final simplification 29.4%

    \[\leadsto \frac{-0.5}{\frac{b_2}{c}} \]
12. Add Preprocessing

Developer target: 87.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b_2\right| - t_0} \cdot \sqrt{\left|b_2\right| + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b_2, t_0\right)\\ \end{array}\\ \mathbf{if}\;b_2 < 0:\\ \;\;\;\;\frac{t_1 - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + t_1}\\ \end{array} \end{array} \]

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) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	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(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	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 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	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[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024024 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b_2 0.0) (/ (- (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) a) (/ (- c) (+ 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))))))))

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