quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.0% → 84.4%

Time: 15.2s

Alternatives: 11

Speedup: 15.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.0% 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: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -8 \cdot 10^{+114}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.25 \cdot 10^{-255}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{0.5} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.5 \cdot 10^{-292}:\\ \;\;\;\;\frac{{\left(e^{0.25 \cdot \left(\log a - \log \left(\frac{-1}{c}\right)\right)}\right)}^{2} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8e+114)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 -2.25e-255)
     (/ (- (pow (fma b_2 b_2 (* a (- c))) 0.5) b_2) a)
     (if (<= b_2 3.5e-292)
       (/ (- (pow (exp (* 0.25 (- (log a) (log (/ -1.0 c))))) 2.0) b_2) a)
       (if (<= b_2 2.2e-25)
         (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
         (/ (* c -0.5) b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8e+114) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= -2.25e-255) {
		tmp = (pow(fma(b_2, b_2, (a * -c)), 0.5) - b_2) / a;
	} else if (b_2 <= 3.5e-292) {
		tmp = (pow(exp((0.25 * (log(a) - log((-1.0 / c))))), 2.0) - b_2) / a;
	} else if (b_2 <= 2.2e-25) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= -8e+114)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= -2.25e-255)
		tmp = Float64(Float64((fma(b_2, b_2, Float64(a * Float64(-c))) ^ 0.5) - b_2) / a);
	elseif (b_2 <= 3.5e-292)
		tmp = Float64(Float64((exp(Float64(0.25 * Float64(log(a) - log(Float64(-1.0 / c))))) ^ 2.0) - b_2) / a);
	elseif (b_2 <= 2.2e-25)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8e+114], 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, -2.25e-255], N[(N[(N[Power[N[(b$95$2 * b$95$2 + N[(a * (-c)), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-292], N[(N[(N[Power[N[Exp[N[(0.25 * N[(N[Log[a], $MachinePrecision] - N[Log[N[(-1.0 / c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.2e-25], 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[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b_2 < -8e114

   1. Initial program 35.5%

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

      1. +-commutative 35.5%

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

      2. unsub-neg 35.5%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in b_2 around -inf 93.8%

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

   if -8e114 < b_2 < -2.2499999999999999e-255

   1. Initial program 86.6%

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

      1. +-commutative 86.6%

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

      2. unsub-neg 86.6%

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

   3. Simplified 86.6%

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

      1. pow1/2 86.6%

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

      2. fma-neg 86.6%

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

      3. *-commutative 86.6%

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

      4. distribute-rgt-neg-in 86.6%

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

   5. Applied egg-rr 86.6%

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

   if -2.2499999999999999e-255 < b_2 < 3.5e-292

   1. Initial program 39.3%

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

      1. +-commutative 39.3%

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

      2. unsub-neg 39.3%

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

   3. Simplified 39.3%

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

      1. add-sqr-sqrt 39.1%

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

      2. pow2 39.1%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}} - b_2}{a} \]

      3. pow1/2 39.1%

         \[\leadsto \frac{{\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b_2}{a} \]

      4. sqrt-pow1 39.1%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b_2}{a} \]

      5. fma-neg 39.1%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      6. *-commutative 39.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      7. distribute-rgt-neg-in 39.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      8. metadata-eval 39.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b_2}{a} \]

   5. Applied egg-rr 39.1%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}} - b_2}{a} \]

   6. Taylor expanded in c around -inf 89.7%

      \[\leadsto \frac{{\color{blue}{\left(e^{0.25 \cdot \left(\log a + -1 \cdot \log \left(\frac{-1}{c}\right)\right)}\right)}}^{2} - b_2}{a} \]

   if 3.5e-292 < b_2 < 2.2000000000000002e-25

   1. Initial program 73.8%

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

      1. +-commutative 73.8%

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

      2. unsub-neg 73.8%

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

   3. Simplified 73.8%

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

   if 2.2000000000000002e-25 < b_2

   1. Initial program 8.6%

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

      1. +-commutative 8.6%

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

      2. unsub-neg 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in b_2 around inf 73.2%

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

      1. associate-*r/ 73.2%

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

      2. associate-/l* 73.1%

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

   6. Simplified 73.1%

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

      1. *-un-lft-identity 73.1%

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

      2. times-frac 75.0%

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

   8. Applied egg-rr 75.0%

      \[\leadsto \frac{\frac{-0.5}{\color{blue}{\frac{1}{a} \cdot \frac{b_2}{c}}}}{a} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.0%

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

      2. *-lft-identity 75.0%

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

   10. Simplified 75.0%

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

   11. Taylor expanded in b_2 around 0 95.2%

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

       1. associate-*r/ 95.3%

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

   13. Simplified 95.3%

       \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8 \cdot 10^{+114}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.25 \cdot 10^{-255}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{0.5} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.5 \cdot 10^{-292}:\\ \;\;\;\;\frac{{\left(e^{0.25 \cdot \left(\log a - \log \left(\frac{-1}{c}\right)\right)}\right)}^{2} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.55 \cdot 10^{+116}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.55 \cdot 10^{-24}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{0.5} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.55e+116)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.55e-24)
     (/ (- (pow (fma b_2 b_2 (* a (- c))) 0.5) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.55e+116) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.55e-24) {
		tmp = (pow(fma(b_2, b_2, (a * -c)), 0.5) - 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 <= -2.55e+116)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.55e-24)
		tmp = Float64(Float64((fma(b_2, b_2, Float64(a * Float64(-c))) ^ 0.5) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.55e+116], 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-24], N[(N[(N[Power[N[(b$95$2 * b$95$2 + N[(a * (-c)), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.55e116

   1. Initial program 35.5%

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

      1. +-commutative 35.5%

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

      2. unsub-neg 35.5%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in b_2 around -inf 93.8%

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

   if -2.55e116 < b_2 < 1.55e-24

   1. Initial program 78.8%

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

      1. +-commutative 78.8%

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

      2. unsub-neg 78.8%

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

   3. Simplified 78.8%

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

      1. pow1/2 78.8%

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

      2. fma-neg 78.8%

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

      3. *-commutative 78.8%

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

      4. distribute-rgt-neg-in 78.8%

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

   5. Applied egg-rr 78.8%

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

   if 1.55e-24 < b_2

   1. Initial program 8.6%

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

      1. +-commutative 8.6%

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

      2. unsub-neg 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in b_2 around inf 73.2%

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

      1. associate-*r/ 73.2%

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

      2. associate-/l* 73.1%

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

   6. Simplified 73.1%

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

      1. *-un-lft-identity 73.1%

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

      2. times-frac 75.0%

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

   8. Applied egg-rr 75.0%

      \[\leadsto \frac{\frac{-0.5}{\color{blue}{\frac{1}{a} \cdot \frac{b_2}{c}}}}{a} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.0%

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

      2. *-lft-identity 75.0%

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

   10. Simplified 75.0%

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

   11. Taylor expanded in b_2 around 0 95.2%

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

       1. associate-*r/ 95.3%

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

   13. Simplified 95.3%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.55 \cdot 10^{+116}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.55 \cdot 10^{-24}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{0.5} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.4 \cdot 10^{+116}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.4e+116)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 9.5e-26)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.4e+116) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 9.5e-26) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= (-2.4d+116)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 9.5d-26) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= -2.4e+116) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 9.5e-26) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.4e+116:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 9.5e-26:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - 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 <= -2.4e+116)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 9.5e-26)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / 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 <= -2.4e+116)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 9.5e-26)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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, -2.4e+116], 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, 9.5e-26], 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[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.4e116

   1. Initial program 35.5%

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

      1. +-commutative 35.5%

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

      2. unsub-neg 35.5%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in b_2 around -inf 93.8%

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

   if -2.4e116 < b_2 < 9.4999999999999995e-26

   1. Initial program 78.8%

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

      1. +-commutative 78.8%

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

      2. unsub-neg 78.8%

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

   3. Simplified 78.8%

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

   if 9.4999999999999995e-26 < b_2

   1. Initial program 8.6%

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

      1. +-commutative 8.6%

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

      2. unsub-neg 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in b_2 around inf 73.2%

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

      1. associate-*r/ 73.2%

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

      2. associate-/l* 73.1%

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

   6. Simplified 73.1%

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

      1. *-un-lft-identity 73.1%

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

      2. times-frac 75.0%

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

   8. Applied egg-rr 75.0%

      \[\leadsto \frac{\frac{-0.5}{\color{blue}{\frac{1}{a} \cdot \frac{b_2}{c}}}}{a} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.0%

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

      2. *-lft-identity 75.0%

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

   10. Simplified 75.0%

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

   11. Taylor expanded in b_2 around 0 95.2%

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

       1. associate-*r/ 95.3%

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

   13. Simplified 95.3%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.4 \cdot 10^{+116}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.8e-11)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 9.5e-26) (/ (- (sqrt (* a (- c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.8e-11) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 9.5e-26) {
		tmp = (sqrt((a * -c)) - 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 <= (-1.8d-11)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 9.5d-26) then
        tmp = (sqrt((a * -c)) - 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 <= -1.8e-11) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 9.5e-26) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.8e-11:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 9.5e-26:
		tmp = (math.sqrt((a * -c)) - 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 <= -1.8e-11)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 9.5e-26)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / 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 <= -1.8e-11)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 9.5e-26)
		tmp = (sqrt((a * -c)) - 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, -1.8e-11], 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, 9.5e-26], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.79999999999999992e-11

   1. Initial program 56.5%

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

      1. +-commutative 56.5%

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

      2. unsub-neg 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in b_2 around -inf 87.9%

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

   if -1.79999999999999992e-11 < b_2 < 9.4999999999999995e-26

   1. Initial program 74.2%

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

      1. +-commutative 74.2%

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

      2. unsub-neg 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in b_2 around 0 68.4%

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

      1. associate-*r* 68.4%

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

      2. neg-mul-1 68.4%

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

      3. *-commutative 68.4%

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

   6. Simplified 68.4%

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

   if 9.4999999999999995e-26 < b_2

   1. Initial program 8.6%

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

      1. +-commutative 8.6%

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

      2. unsub-neg 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in b_2 around inf 73.2%

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

      1. associate-*r/ 73.2%

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

      2. associate-/l* 73.1%

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

   6. Simplified 73.1%

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

      1. *-un-lft-identity 73.1%

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

      2. times-frac 75.0%

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

   8. Applied egg-rr 75.0%

      \[\leadsto \frac{\frac{-0.5}{\color{blue}{\frac{1}{a} \cdot \frac{b_2}{c}}}}{a} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.0%

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

      2. *-lft-identity 75.0%

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

   10. Simplified 75.0%

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

   11. Taylor expanded in b_2 around 0 95.2%

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

       1. associate-*r/ 95.3%

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

   13. Simplified 95.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 5: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4.3 \cdot 10^{-51}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.3e-51)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 4.7e-26) (/ (sqrt (* a (- c))) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.3e-51) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4.7e-26) {
		tmp = sqrt((a * -c)) / 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 <= (-4.3d-51)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 4.7d-26) then
        tmp = sqrt((a * -c)) / 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 <= -4.3e-51) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4.7e-26) {
		tmp = Math.sqrt((a * -c)) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.3e-51:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 4.7e-26:
		tmp = math.sqrt((a * -c)) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.3e-51)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 4.7e-26)
		tmp = Float64(sqrt(Float64(a * Float64(-c))) / 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 <= -4.3e-51)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 4.7e-26)
		tmp = sqrt((a * -c)) / 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, -4.3e-51], 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, 4.7e-26], N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.2999999999999997e-51

   1. Initial program 61.3%

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

      1. +-commutative 61.3%

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

      2. unsub-neg 61.3%

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

   3. Simplified 61.3%

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

   4. Taylor expanded in b_2 around -inf 84.2%

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

   if -4.2999999999999997e-51 < b_2 < 4.69999999999999989e-26

   1. Initial program 71.5%

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

      1. +-commutative 71.5%

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

      2. unsub-neg 71.5%

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

   3. Simplified 71.5%

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

      1. prod-diff 71.1%

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

      2. *-commutative 71.1%

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

      3. fma-def 71.1%

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

      4. associate-+l+ 71.1%

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

      5. distribute-rgt-neg-in 71.1%

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

      6. fma-def 71.1%

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

      7. *-commutative 71.1%

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

      8. fma-udef 71.1%

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

      9. distribute-lft-neg-in 71.1%

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

      10. *-commutative 71.1%

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

      11. distribute-rgt-neg-in 71.1%

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

      12. fma-def 71.1%

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

   5. Applied egg-rr 71.1%

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

   6. Taylor expanded in b_2 around 0 68.4%

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

      1. associate-*l/ 68.5%

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

      2. *-lft-identity 68.5%

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

      3. distribute-lft1-in 68.8%

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

      4. metadata-eval 68.8%

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

      5. neg-mul-1 68.8%

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

      6. *-commutative 68.8%

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

      7. distribute-rgt-neg-out 68.8%

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

      8. distribute-rgt-neg-out 68.8%

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

      9. *-commutative 68.8%

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

   8. Simplified 68.8%

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

   if 4.69999999999999989e-26 < b_2

   1. Initial program 8.6%

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

      1. +-commutative 8.6%

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

      2. unsub-neg 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in b_2 around inf 73.2%

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

      1. associate-*r/ 73.2%

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

      2. associate-/l* 73.1%

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

   6. Simplified 73.1%

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

      1. *-un-lft-identity 73.1%

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

      2. times-frac 75.0%

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

   8. Applied egg-rr 75.0%

      \[\leadsto \frac{\frac{-0.5}{\color{blue}{\frac{1}{a} \cdot \frac{b_2}{c}}}}{a} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.0%

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

      2. *-lft-identity 75.0%

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

   10. Simplified 75.0%

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

   11. Taylor expanded in b_2 around 0 95.2%

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

       1. associate-*r/ 95.3%

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

   13. Simplified 95.3%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.3 \cdot 10^{-51}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 6: 68.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311)
   (+ (* -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-311) {
		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-311)) 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-311) {
		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-311:
		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-311)
		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-311)
		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-311], 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 < -5.00000000000023e-311

   1. Initial program 64.6%

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

      1. +-commutative 64.6%

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

      2. unsub-neg 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in b_2 around -inf 62.3%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 36.0%

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

      1. +-commutative 36.0%

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

      2. unsub-neg 36.0%

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

   3. Simplified 36.0%

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

   4. Taylor expanded in b_2 around inf 46.9%

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

      1. associate-*r/ 46.9%

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

      2. associate-/l* 46.9%

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

   6. Simplified 46.9%

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

      1. *-un-lft-identity 46.9%

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

      2. times-frac 52.5%

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

   8. Applied egg-rr 52.5%

      \[\leadsto \frac{\frac{-0.5}{\color{blue}{\frac{1}{a} \cdot \frac{b_2}{c}}}}{a} \]
   9. Step-by-step derivation

      1. associate-*l/ 52.5%

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

      2. *-lft-identity 52.5%

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

   10. Simplified 52.5%

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

   11. Taylor expanded in b_2 around 0 64.7%

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

       1. associate-*r/ 64.8%

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

   13. Simplified 64.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 7: 44.5% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-308}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-308) (* -2.0 (/ b_2 a)) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-308) {
		tmp = -2.0 * (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 <= (-9d-308)) then
        tmp = (-2.0d0) * (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 <= -9e-308) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-308:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-308)
		tmp = Float64(-2.0 * 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 <= -9e-308)
		tmp = -2.0 * (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, -9e-308], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.00000000000000017e-308

   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. Taylor expanded in b_2 around -inf 62.5%

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

   if -9.00000000000000017e-308 < b_2

   1. Initial program 35.8%

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

      1. +-commutative 35.8%

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

      2. unsub-neg 35.8%

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

   3. Simplified 35.8%

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

      1. add-sqr-sqrt 35.5%

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

      2. pow2 35.5%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}} - b_2}{a} \]

      3. pow1/2 35.5%

         \[\leadsto \frac{{\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b_2}{a} \]

      4. sqrt-pow1 35.4%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b_2}{a} \]

      5. fma-neg 35.4%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      6. *-commutative 35.4%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      7. distribute-rgt-neg-in 35.4%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      8. metadata-eval 35.4%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b_2}{a} \]

   5. Applied egg-rr 35.4%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}} - b_2}{a} \]

   6. Taylor expanded in b_2 around inf 20.6%

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

      1. distribute-rgt1-in 20.6%

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

      2. metadata-eval 20.6%

         \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

      3. mul0-lft 20.6%

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

   8. Simplified 20.6%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-308}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 8: 68.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311) (* -2.0 (/ b_2 a)) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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-311)) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-311:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-311)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-311)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-311], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.00000000000023e-311

   1. Initial program 64.6%

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

      1. +-commutative 64.6%

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

      2. unsub-neg 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in b_2 around -inf 62.0%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 36.0%

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

      1. +-commutative 36.0%

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

      2. unsub-neg 36.0%

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

   3. Simplified 36.0%

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

   4. Taylor expanded in b_2 around inf 64.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 9: 68.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311) (* -2.0 (/ b_2 a)) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = -2.0 * (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 <= (-5d-311)) then
        tmp = (-2.0d0) * (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 <= -5e-311) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-311:
		tmp = -2.0 * (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 <= -5e-311)
		tmp = Float64(-2.0 * Float64(b_2 / 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-311)
		tmp = -2.0 * (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, -5e-311], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.00000000000023e-311

   1. Initial program 64.6%

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

      1. +-commutative 64.6%

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

      2. unsub-neg 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in b_2 around -inf 62.0%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 36.0%

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

      1. +-commutative 36.0%

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

      2. unsub-neg 36.0%

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

   3. Simplified 36.0%

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

   4. Taylor expanded in b_2 around inf 46.9%

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

      1. associate-*r/ 46.9%

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

      2. associate-/l* 46.9%

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

   6. Simplified 46.9%

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

      1. *-un-lft-identity 46.9%

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

      2. times-frac 52.5%

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

   8. Applied egg-rr 52.5%

      \[\leadsto \frac{\frac{-0.5}{\color{blue}{\frac{1}{a} \cdot \frac{b_2}{c}}}}{a} \]
   9. Step-by-step derivation

      1. associate-*l/ 52.5%

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

      2. *-lft-identity 52.5%

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

   10. Simplified 52.5%

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

   11. Taylor expanded in b_2 around 0 64.7%

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

       1. associate-*r/ 64.8%

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

   13. Simplified 64.8%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 10: 23.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-308) (/ (- b_2) a) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-308) {
		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 <= (-9d-308)) 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 <= -9e-308) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-308:
		tmp = -b_2 / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-308)
		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 <= -9e-308)
		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, -9e-308], N[((-b$95$2) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.00000000000000017e-308

   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. Taylor expanded in b_2 around 0 41.5%

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

      1. associate-*r* 41.5%

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

      2. neg-mul-1 41.5%

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

      3. *-commutative 41.5%

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

   6. Simplified 41.5%

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

   7. Taylor expanded in c around 0 21.7%

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

      1. associate-*r/ 21.7%

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

      2. neg-mul-1 21.7%

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

   9. Simplified 21.7%

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

   if -9.00000000000000017e-308 < b_2

   1. Initial program 35.8%

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

      1. +-commutative 35.8%

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

      2. unsub-neg 35.8%

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

   3. Simplified 35.8%

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

      1. add-sqr-sqrt 35.5%

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

      2. pow2 35.5%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}} - b_2}{a} \]

      3. pow1/2 35.5%

         \[\leadsto \frac{{\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b_2}{a} \]

      4. sqrt-pow1 35.4%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b_2}{a} \]

      5. fma-neg 35.4%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      6. *-commutative 35.4%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      7. distribute-rgt-neg-in 35.4%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

      8. metadata-eval 35.4%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b_2}{a} \]

   5. Applied egg-rr 35.4%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}} - b_2}{a} \]

   6. Taylor expanded in b_2 around inf 20.6%

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

      1. distribute-rgt1-in 20.6%

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

      2. metadata-eval 20.6%

         \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

      3. mul0-lft 20.6%

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

   8. Simplified 20.6%

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

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 11: 11.1% 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 50.9%

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

   1. +-commutative 50.9%

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

   2. unsub-neg 50.9%

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

3. Simplified 50.9%

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

   1. add-sqr-sqrt 50.6%

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

   2. pow2 50.6%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}} - b_2}{a} \]

   3. pow1/2 50.6%

      \[\leadsto \frac{{\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b_2}{a} \]

   4. sqrt-pow1 50.6%

      \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b_2}{a} \]

   5. fma-neg 50.6%

      \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

   6. *-commutative 50.6%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

   7. distribute-rgt-neg-in 50.6%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b_2}{a} \]

   8. metadata-eval 50.6%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2} - b_2}{a} \]

5. Applied egg-rr 50.6%

   \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{0.25}\right)}^{2}} - b_2}{a} \]

6. Taylor expanded in b_2 around inf 11.5%

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

   1. distribute-rgt1-in 11.5%

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

   2. metadata-eval 11.5%

      \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

   3. mul0-lft 11.5%

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

8. Simplified 11.5%

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

9. Final simplification 11.5%

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

Reproduce

herbie shell --seed 2023285 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))