quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.9% → 83.3%

Time: 16.0s

Alternatives: 10

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.9 \cdot 10^{-25}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 8 \cdot 10^{-56}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.9e-25)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 8e-56)
     (- (/ (hypot b_2 (sqrt (* a (- c)))) a) (/ b_2 a))
     (/ 1.0 (+ (* -2.0 (/ b_2 c)) (* 0.5 (/ a b_2)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.9e-25) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 8e-56) {
		tmp = (hypot(b_2, sqrt((a * -c))) / a) - (b_2 / a);
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.9e-25) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 8e-56) {
		tmp = (Math.hypot(b_2, Math.sqrt((a * -c))) / a) - (b_2 / a);
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.9e-25:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 8e-56:
		tmp = (math.hypot(b_2, math.sqrt((a * -c))) / a) - (b_2 / a)
	else:
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.9e-25)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 8e-56)
		tmp = Float64(Float64(hypot(b_2, sqrt(Float64(a * Float64(-c)))) / a) - Float64(b_2 / a));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b_2 / c)) + Float64(0.5 * Float64(a / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.9e-25)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 8e-56)
		tmp = (hypot(b_2, sqrt((a * -c))) / a) - (b_2 / a);
	else
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.9e-25], 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, 8e-56], N[(N[(N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.9e-25

   1. Initial program 62.0%

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

      1. +-commutative 62.0%

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

      2. unsub-neg 62.0%

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

   3. Simplified 62.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 94.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]

   if -3.9e-25 < b_2 < 8.0000000000000003e-56

   1. Initial program 79.6%

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

      1. +-commutative 79.6%

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

      2. unsub-neg 79.6%

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

   3. Simplified 79.6%

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

      1. div-sub 79.5%

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

      2. sub-neg 79.5%

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

      3. add-sqr-sqrt 78.3%

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

      4. hypot-define 82.3%

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

      5. *-commutative 82.3%

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

      6. distribute-rgt-neg-in 82.3%

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

   6. Applied egg-rr 82.3%

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

   if 8.0000000000000003e-56 < b_2

   1. Initial program 23.6%

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

      1. +-commutative 23.6%

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

      2. unsub-neg 23.6%

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

   3. Simplified 23.6%

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

      1. add-sqr-sqrt 18.8%

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

      2. pow2 18.8%

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

      3. pow1/2 18.8%

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

      4. sqrt-pow1 18.7%

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

      5. pow2 18.7%

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

      6. metadata-eval 18.7%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 18.7%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]
   7. Step-by-step derivation

      1. div-sub 18.3%

         \[\leadsto \color{blue}{\frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}{a} - \frac{b\_2}{a}} \]

      2. pow-pow 22.9%

         \[\leadsto \frac{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}{a} - \frac{b\_2}{a} \]

      3. metadata-eval 22.9%

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

      4. metadata-eval 22.9%

         \[\leadsto \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}{a} - \frac{b\_2}{a} \]

      5. pow-pow 8.7%

         \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}}}{a} - \frac{b\_2}{a} \]

      6. pow1/3 9.7%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}}}}{a} - \frac{b\_2}{a} \]

      7. div-sub 10.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}{a}} \]

      8. clear-num 10.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}}} \]

      9. inv-pow 10.0%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}\right)}^{-1}} \]

      10. pow1/3 8.9%

          \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}} - b\_2}\right)}^{-1} \]

      11. pow-pow 23.5%

          \[\leadsto {\left(\frac{a}{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b\_2}\right)}^{-1} \]

      12. metadata-eval 23.5%

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

      13. pow1/2 23.5%

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

   8. Applied egg-rr 23.5%

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

      1. unpow-1 23.5%

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

   10. Simplified 23.5%

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

   11. Taylor expanded in a around 0 84.9%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.9 \cdot 10^{-25}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 8 \cdot 10^{-56}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.6 \cdot 10^{-32}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right) - b\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.6e-32)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 3.5e-56)
     (* (/ 1.0 a) (- (hypot b_2 (sqrt (* a (- c)))) b_2))
     (/ 1.0 (+ (* -2.0 (/ b_2 c)) (* 0.5 (/ a b_2)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.6e-32) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 3.5e-56) {
		tmp = (1.0 / a) * (hypot(b_2, sqrt((a * -c))) - b_2);
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.6e-32) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 3.5e-56) {
		tmp = (1.0 / a) * (Math.hypot(b_2, Math.sqrt((a * -c))) - b_2);
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8.6e-32:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 3.5e-56:
		tmp = (1.0 / a) * (math.hypot(b_2, math.sqrt((a * -c))) - b_2)
	else:
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.6e-32)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 3.5e-56)
		tmp = Float64(Float64(1.0 / a) * Float64(hypot(b_2, sqrt(Float64(a * Float64(-c)))) - b_2));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b_2 / c)) + Float64(0.5 * Float64(a / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8.6e-32)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 3.5e-56)
		tmp = (1.0 / a) * (hypot(b_2, sqrt((a * -c))) - b_2);
	else
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.6e-32], 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, 3.5e-56], N[(N[(1.0 / a), $MachinePrecision] * N[(N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.5999999999999998e-32

   1. Initial program 62.4%

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

      1. +-commutative 62.4%

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

      2. unsub-neg 62.4%

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

   3. Simplified 62.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 95.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]

   if -8.5999999999999998e-32 < b_2 < 3.4999999999999998e-56

   1. Initial program 79.3%

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

      1. +-commutative 79.3%

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

      2. unsub-neg 79.3%

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

   3. Simplified 79.3%

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

      1. clear-num 79.2%

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

      2. associate-/r/ 79.2%

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

      3. sub-neg 79.2%

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

      4. add-sqr-sqrt 78.0%

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

      5. hypot-define 82.1%

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

      6. *-commutative 82.1%

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

      7. distribute-rgt-neg-in 82.1%

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

   6. Applied egg-rr 82.1%

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

   if 3.4999999999999998e-56 < b_2

   1. Initial program 23.6%

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

      1. +-commutative 23.6%

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

      2. unsub-neg 23.6%

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

   3. Simplified 23.6%

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

      1. add-sqr-sqrt 18.8%

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

      2. pow2 18.8%

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

      3. pow1/2 18.8%

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

      4. sqrt-pow1 18.7%

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

      5. pow2 18.7%

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

      6. metadata-eval 18.7%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 18.7%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]
   7. Step-by-step derivation

      1. div-sub 18.3%

         \[\leadsto \color{blue}{\frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}{a} - \frac{b\_2}{a}} \]

      2. pow-pow 22.9%

         \[\leadsto \frac{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}{a} - \frac{b\_2}{a} \]

      3. metadata-eval 22.9%

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

      4. metadata-eval 22.9%

         \[\leadsto \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}{a} - \frac{b\_2}{a} \]

      5. pow-pow 8.7%

         \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}}}{a} - \frac{b\_2}{a} \]

      6. pow1/3 9.7%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}}}}{a} - \frac{b\_2}{a} \]

      7. div-sub 10.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}{a}} \]

      8. clear-num 10.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}}} \]

      9. inv-pow 10.0%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}\right)}^{-1}} \]

      10. pow1/3 8.9%

          \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}} - b\_2}\right)}^{-1} \]

      11. pow-pow 23.5%

          \[\leadsto {\left(\frac{a}{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b\_2}\right)}^{-1} \]

      12. metadata-eval 23.5%

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

      13. pow1/2 23.5%

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

   8. Applied egg-rr 23.5%

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

      1. unpow-1 23.5%

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

   10. Simplified 23.5%

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

   11. Taylor expanded in a around 0 84.9%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.6 \cdot 10^{-32}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right) - b\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+115}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+115)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 3.5e-55)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ 1.0 (+ (* -2.0 (/ b_2 c)) (* 0.5 (/ a b_2)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+115) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 3.5e-55) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / 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 <= (-1d+115)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 3.5d-55) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = 1.0d0 / (((-2.0d0) * (b_2 / c)) + (0.5d0 * (a / 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 <= -1e+115) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 3.5e-55) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e+115:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 3.5e-55:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e+115)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 3.5e-55)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b_2 / c)) + Float64(0.5 * Float64(a / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e+115)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 3.5e-55)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+115], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-55], 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[(1.0 / N[(N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1e115

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. unsub-neg 48.5%

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

   3. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 96.9%

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

      1. *-commutative 96.9%

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

   7. Simplified 96.9%

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

   if -1e115 < b_2 < 3.50000000000000025e-55

   1. Initial program 82.7%

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

      1. +-commutative 82.7%

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

      2. unsub-neg 82.7%

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

   3. Simplified 82.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   if 3.50000000000000025e-55 < b_2

   1. Initial program 23.6%

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

      1. +-commutative 23.6%

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

      2. unsub-neg 23.6%

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

   3. Simplified 23.6%

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

      1. add-sqr-sqrt 18.8%

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

      2. pow2 18.8%

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

      3. pow1/2 18.8%

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

      4. sqrt-pow1 18.7%

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

      5. pow2 18.7%

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

      6. metadata-eval 18.7%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 18.7%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]
   7. Step-by-step derivation

      1. div-sub 18.3%

         \[\leadsto \color{blue}{\frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}{a} - \frac{b\_2}{a}} \]

      2. pow-pow 22.9%

         \[\leadsto \frac{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}{a} - \frac{b\_2}{a} \]

      3. metadata-eval 22.9%

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

      4. metadata-eval 22.9%

         \[\leadsto \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}{a} - \frac{b\_2}{a} \]

      5. pow-pow 8.7%

         \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}}}{a} - \frac{b\_2}{a} \]

      6. pow1/3 9.7%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}}}}{a} - \frac{b\_2}{a} \]

      7. div-sub 10.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}{a}} \]

      8. clear-num 10.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}}} \]

      9. inv-pow 10.0%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}\right)}^{-1}} \]

      10. pow1/3 8.9%

          \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}} - b\_2}\right)}^{-1} \]

      11. pow-pow 23.5%

          \[\leadsto {\left(\frac{a}{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b\_2}\right)}^{-1} \]

      12. metadata-eval 23.5%

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

      13. pow1/2 23.5%

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

   8. Applied egg-rr 23.5%

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

      1. unpow-1 23.5%

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

   10. Simplified 23.5%

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

   11. Taylor expanded in a around 0 84.9%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+115}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-126}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.8e-126)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 4.2e-56)
     (/ (- (sqrt (* a (- c))) b_2) a)
     (/ 1.0 (+ (* -2.0 (/ b_2 c)) (* 0.5 (/ a b_2)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e-126) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4.2e-56) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / 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 <= (-3.8d-126)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 4.2d-56) then
        tmp = (sqrt((a * -c)) - b_2) / a
    else
        tmp = 1.0d0 / (((-2.0d0) * (b_2 / c)) + (0.5d0 * (a / 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 <= -3.8e-126) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4.2e-56) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.8e-126:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 4.2e-56:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.8e-126)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 4.2e-56)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b_2 / c)) + Float64(0.5 * Float64(a / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.8e-126)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 4.2e-56)
		tmp = (sqrt((a * -c)) - b_2) / a;
	else
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.8e-126], 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, 4.2e-56], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.7999999999999999e-126

   1. Initial program 65.1%

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

      1. +-commutative 65.1%

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

      2. unsub-neg 65.1%

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

   3. Simplified 65.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 88.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]

   if -3.7999999999999999e-126 < b_2 < 4.20000000000000012e-56

   1. Initial program 79.1%

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

      1. +-commutative 79.1%

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

      2. unsub-neg 79.1%

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

   3. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around 0 75.2%

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

      1. associate-*r* 75.2%

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

      2. neg-mul-1 75.2%

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

      3. *-commutative 75.2%

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

   7. Simplified 75.2%

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

   if 4.20000000000000012e-56 < b_2

   1. Initial program 23.6%

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

      1. +-commutative 23.6%

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

      2. unsub-neg 23.6%

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

   3. Simplified 23.6%

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

      1. add-sqr-sqrt 18.8%

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

      2. pow2 18.8%

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

      3. pow1/2 18.8%

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

      4. sqrt-pow1 18.7%

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

      5. pow2 18.7%

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

      6. metadata-eval 18.7%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 18.7%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]
   7. Step-by-step derivation

      1. div-sub 18.3%

         \[\leadsto \color{blue}{\frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}{a} - \frac{b\_2}{a}} \]

      2. pow-pow 22.9%

         \[\leadsto \frac{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}{a} - \frac{b\_2}{a} \]

      3. metadata-eval 22.9%

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

      4. metadata-eval 22.9%

         \[\leadsto \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}{a} - \frac{b\_2}{a} \]

      5. pow-pow 8.7%

         \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}}}{a} - \frac{b\_2}{a} \]

      6. pow1/3 9.7%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}}}}{a} - \frac{b\_2}{a} \]

      7. div-sub 10.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}{a}} \]

      8. clear-num 10.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}}} \]

      9. inv-pow 10.0%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}\right)}^{-1}} \]

      10. pow1/3 8.9%

          \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}} - b\_2}\right)}^{-1} \]

      11. pow-pow 23.5%

          \[\leadsto {\left(\frac{a}{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b\_2}\right)}^{-1} \]

      12. metadata-eval 23.5%

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

      13. pow1/2 23.5%

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

   8. Applied egg-rr 23.5%

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

      1. unpow-1 23.5%

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

   10. Simplified 23.5%

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

   11. Taylor expanded in a around 0 84.9%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-126}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-253}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-253)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (/ 1.0 (+ (* -2.0 (/ b_2 c)) (* 0.5 (/ a b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-253) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.8d-253)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else
        tmp = 1.0d0 / (((-2.0d0) * (b_2 / c)) + (0.5d0 * (a / b_2)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-253) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-253:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	else:
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-253)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b_2 / c)) + Float64(0.5 * Float64(a / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-253)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	else
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-253], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.80000000000000006e-253

   1. Initial program 66.4%

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

      1. +-commutative 66.4%

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

      2. unsub-neg 66.4%

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

   3. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 78.7%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]

   if -2.80000000000000006e-253 < b_2

   1. Initial program 43.5%

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

      1. +-commutative 43.5%

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

      2. unsub-neg 43.5%

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

   3. Simplified 43.5%

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

      1. add-sqr-sqrt 40.2%

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

      2. pow2 40.2%

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

      3. pow1/2 40.2%

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

      4. sqrt-pow1 40.1%

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

      5. pow2 40.1%

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

      6. metadata-eval 40.1%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 40.1%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]
   7. Step-by-step derivation

      1. div-sub 39.8%

         \[\leadsto \color{blue}{\frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}{a} - \frac{b\_2}{a}} \]

      2. pow-pow 43.0%

         \[\leadsto \frac{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}{a} - \frac{b\_2}{a} \]

      3. metadata-eval 43.0%

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

      4. metadata-eval 43.0%

         \[\leadsto \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}{a} - \frac{b\_2}{a} \]

      5. pow-pow 25.7%

         \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}}}{a} - \frac{b\_2}{a} \]

      6. pow1/3 27.6%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}}}}{a} - \frac{b\_2}{a} \]

      7. div-sub 27.9%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}{a}} \]

      8. clear-num 27.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}}} \]

      9. inv-pow 27.9%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}} - b\_2}\right)}^{-1}} \]

      10. pow1/3 25.8%

          \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}} - b\_2}\right)}^{-1} \]

      11. pow-pow 43.4%

          \[\leadsto {\left(\frac{a}{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b\_2}\right)}^{-1} \]

      12. metadata-eval 43.4%

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

      13. pow1/2 43.4%

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

   8. Applied egg-rr 43.4%

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

      1. unpow-1 43.4%

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

   10. Simplified 43.4%

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

   11. Taylor expanded in a around 0 60.3%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-253}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 67.4%

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

      1. +-commutative 67.4%

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

      2. unsub-neg 67.4%

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

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 76.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. unsub-neg 41.7%

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

   3. Simplified 41.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*l/ 61.4%

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

   7. Simplified 61.4%

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

4. Final simplification 68.8%

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

Alternative 7: 42.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (* b_2 -2.0) a) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 67.4%

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

      1. +-commutative 67.4%

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

      2. unsub-neg 67.4%

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

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 75.4%

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

      1. *-commutative 75.4%

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

   7. Simplified 75.4%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. unsub-neg 41.7%

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

   3. Simplified 41.7%

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

      1. add-sqr-sqrt 38.3%

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

      2. pow2 38.3%

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

      3. pow1/2 38.3%

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

      4. sqrt-pow1 38.3%

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

      5. pow2 38.3%

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

      6. metadata-eval 38.3%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 38.3%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around inf 18.6%

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

      1. distribute-rgt1-in 18.6%

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

      2. metadata-eval 18.6%

         \[\leadsto \frac{\color{blue}{0} \cdot b\_2}{a} \]

      3. mul0-lft 18.6%

         \[\leadsto \frac{\color{blue}{0}}{a} \]

   9. Simplified 18.6%

      \[\leadsto \frac{\color{blue}{0}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (b_2 * -2.0) / 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, -5e-310], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 67.4%

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

      1. +-commutative 67.4%

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

      2. unsub-neg 67.4%

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

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 75.4%

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

      1. *-commutative 75.4%

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

   7. Simplified 75.4%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. unsub-neg 41.7%

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

   3. Simplified 41.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*l/ 61.4%

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

   7. Simplified 61.4%

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

4. Final simplification 68.3%

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

Alternative 9: 23.4% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (- b_2) a) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = -b_2 / a
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -b_2 / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[((-b$95$2) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 67.4%

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

      1. +-commutative 67.4%

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

      2. unsub-neg 67.4%

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

   3. Simplified 67.4%

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

      1. add-sqr-sqrt 67.2%

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

      2. pow2 67.2%

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

      3. pow1/2 67.2%

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

      4. sqrt-pow1 67.3%

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

      5. pow2 67.3%

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

      6. metadata-eval 67.3%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 67.3%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around inf 31.6%

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

      1. neg-mul-1 31.6%

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

   9. Simplified 31.6%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. unsub-neg 41.7%

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

   3. Simplified 41.7%

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

      1. add-sqr-sqrt 38.3%

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

      2. pow2 38.3%

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

      3. pow1/2 38.3%

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

      4. sqrt-pow1 38.3%

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

      5. pow2 38.3%

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

      6. metadata-eval 38.3%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 38.3%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around inf 18.6%

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

      1. distribute-rgt1-in 18.6%

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

      2. metadata-eval 18.6%

         \[\leadsto \frac{\color{blue}{0} \cdot b\_2}{a} \]

      3. mul0-lft 18.6%

         \[\leadsto \frac{\color{blue}{0}}{a} \]

   9. Simplified 18.6%

      \[\leadsto \frac{\color{blue}{0}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 11.0% accurate, 37.3× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b_2, double c) {
	return 0.0 / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0 / a;
}

Python

def code(a, b_2, c):
	return 0.0 / a

Julia

function code(a, b_2, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

   1. +-commutative 54.5%

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

   2. unsub-neg 54.5%

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

3. Simplified 54.5%

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

   1. add-sqr-sqrt 52.7%

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

   2. pow2 52.7%

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

   3. pow1/2 52.7%

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

   4. sqrt-pow1 52.7%

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

   5. pow2 52.7%

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

   6. metadata-eval 52.7%

      \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

6. Applied egg-rr 52.7%

   \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]

7. Taylor expanded in b_2 around inf 10.7%

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

   1. distribute-rgt1-in 10.7%

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

   2. metadata-eval 10.7%

      \[\leadsto \frac{\color{blue}{0} \cdot b\_2}{a} \]

   3. mul0-lft 10.7%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

9. Simplified 10.7%

   \[\leadsto \frac{\color{blue}{0}}{a} \]

10. Final simplification 10.7%

    \[\leadsto \frac{0}{a} \]
11. Add Preprocessing

Developer target: 99.7% accurate, 0.1× speedup

Math

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

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 2024040 
(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))