quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.1% → 85.4%

Time: 11.7s

Alternatives: 8

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: 52.1% 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: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -3.6 \cdot 10^{+108}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 5.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.6e+108)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 5.1e-54)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.6e+108) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 5.1e-54) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.6d+108)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 5.1d-54) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.6e+108) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 5.1e-54) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.6e+108:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 5.1e-54:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.6e+108)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 5.1e-54)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.6e+108)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 5.1e-54)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.6e+108], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 5.1e-54], 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[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.6e108

   1. Initial program 47.0%

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

      1. +-commutative 47.0%

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

      2. unsub-neg 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in b_2 around -inf 94.9%

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

      1. *-commutative 94.9%

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

   6. Simplified 94.9%

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

   if -3.6e108 < b_2 < 5.1000000000000001e-54

   1. Initial program 84.0%

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

      1. +-commutative 84.0%

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

      2. unsub-neg 84.0%

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

   3. Simplified 84.0%

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

   if 5.1000000000000001e-54 < b_2

   1. Initial program 13.5%

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

      1. +-commutative 13.5%

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

      2. unsub-neg 13.5%

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

   3. Simplified 13.5%

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

      1. add-cbrt-cube 7.3%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow3 7.3%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

      3. pow1/3 6.5%

         \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

      4. sqrt-pow2 6.5%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

      5. fma-neg 6.6%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      6. *-commutative 6.6%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      7. distribute-rgt-neg-in 6.6%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      8. metadata-eval 6.6%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

   5. Applied egg-rr 6.6%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow1/3 7.4%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

      2. distribute-rgt-neg-out 7.4%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

      3. *-commutative 7.4%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

      4. fma-neg 7.3%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

      5. *-commutative 7.3%

         \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

   7. Simplified 7.3%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]
   8. Step-by-step derivation

      1. clear-num 7.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}}} \]

      2. inv-pow 7.3%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}\right)}^{-1}} \]

      3. pow1/3 6.5%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}\right)}^{-1} \]

      4. pow-pow 13.4%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b_2}\right)}^{-1} \]

      5. *-commutative 13.4%

         \[\leadsto {\left(\frac{a}{{\left(b_2 \cdot b_2 - \color{blue}{a \cdot c}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} - b_2}\right)}^{-1} \]

      6. metadata-eval 13.4%

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

   9. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{{\left(\frac{a}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5} - b_2}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 13.4%

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

       2. unpow1/2 13.4%

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

       3. *-commutative 13.4%

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

   11. Simplified 13.4%

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

   12. Taylor expanded in a around 0 87.0%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.6 \cdot 10^{+108}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 5.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \end{array} \]

Alternative 2: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -3.6 \cdot 10^{-51}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.6e-51)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.1e-55)
     (/ (- (sqrt (* c (- a))) b_2) a)
     (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.6e-51) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.1e-55) {
		tmp = (sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.6d-51)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 1.1d-55) then
        tmp = (sqrt((c * -a)) - b_2) / a
    else
        tmp = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.6e-51) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.1e-55) {
		tmp = (Math.sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.6e-51:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 1.1e-55:
		tmp = (math.sqrt((c * -a)) - b_2) / a
	else:
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.6e-51)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.1e-55)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(-a))) - b_2) / a);
	else
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.6e-51)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 1.1e-55)
		tmp = (sqrt((c * -a)) - b_2) / a;
	else
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.6e-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, 1.1e-55], N[(N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.6e-51

   1. Initial program 65.2%

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

      1. +-commutative 65.2%

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

      2. unsub-neg 65.2%

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

   3. Simplified 65.2%

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

   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 -3.6e-51 < b_2 < 1.1e-55

   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. Taylor expanded in b_2 around 0 74.3%

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

      1. mul-1-neg 74.3%

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

      2. distribute-rgt-neg-out 74.3%

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

   6. Simplified 74.3%

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

   if 1.1e-55 < b_2

   1. Initial program 13.5%

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

      1. +-commutative 13.5%

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

      2. unsub-neg 13.5%

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

   3. Simplified 13.5%

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

      1. add-cbrt-cube 7.3%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow3 7.3%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

      3. pow1/3 6.5%

         \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

      4. sqrt-pow2 6.5%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

      5. fma-neg 6.6%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      6. *-commutative 6.6%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      7. distribute-rgt-neg-in 6.6%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      8. metadata-eval 6.6%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

   5. Applied egg-rr 6.6%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow1/3 7.4%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

      2. distribute-rgt-neg-out 7.4%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

      3. *-commutative 7.4%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

      4. fma-neg 7.3%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

      5. *-commutative 7.3%

         \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

   7. Simplified 7.3%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]
   8. Step-by-step derivation

      1. clear-num 7.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}}} \]

      2. inv-pow 7.3%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}\right)}^{-1}} \]

      3. pow1/3 6.5%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}\right)}^{-1} \]

      4. pow-pow 13.4%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b_2}\right)}^{-1} \]

      5. *-commutative 13.4%

         \[\leadsto {\left(\frac{a}{{\left(b_2 \cdot b_2 - \color{blue}{a \cdot c}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} - b_2}\right)}^{-1} \]

      6. metadata-eval 13.4%

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

   9. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{{\left(\frac{a}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5} - b_2}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 13.4%

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

       2. unpow1/2 13.4%

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

       3. *-commutative 13.4%

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

   11. Simplified 13.4%

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

   12. Taylor expanded in a around 0 87.0%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.6 \cdot 10^{-51}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \end{array} \]

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.55 \cdot 10^{-61}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.55e-61)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 2.25e-57)
     (/ (sqrt (* c (- a))) a)
     (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.55e-61) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.25e-57) {
		tmp = sqrt((c * -a)) / a;
	} else {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.55d-61)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 2.25d-57) then
        tmp = sqrt((c * -a)) / a
    else
        tmp = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.55e-61) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.25e-57) {
		tmp = Math.sqrt((c * -a)) / a;
	} else {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.55e-61:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 2.25e-57:
		tmp = math.sqrt((c * -a)) / a
	else:
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.55e-61)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 2.25e-57)
		tmp = Float64(sqrt(Float64(c * Float64(-a))) / a);
	else
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.55e-61)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 2.25e-57)
		tmp = sqrt((c * -a)) / a;
	else
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.55e-61], 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-57], N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.54999999999999984e-61

   1. Initial program 66.8%

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

      1. +-commutative 66.8%

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

      2. unsub-neg 66.8%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in b_2 around -inf 86.4%

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

   if -2.54999999999999984e-61 < b_2 < 2.24999999999999986e-57

   1. Initial program 78.5%

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

      1. +-commutative 78.5%

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

      2. unsub-neg 78.5%

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

   3. Simplified 78.5%

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

      1. prod-diff 78.1%

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

      2. *-commutative 78.1%

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

      3. fma-def 78.1%

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

      4. associate-+l+ 78.1%

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

      5. distribute-rgt-neg-in 78.1%

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

      6. fma-def 78.0%

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

      7. *-commutative 78.0%

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

      8. fma-udef 78.1%

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

      9. distribute-lft-neg-in 78.1%

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

      10. *-commutative 78.1%

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

      11. distribute-rgt-neg-in 78.1%

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

      12. fma-def 78.0%

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

   5. Applied egg-rr 78.0%

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

   6. Taylor expanded in b_2 around 0 74.0%

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

      1. associate-*l/ 74.1%

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

      2. distribute-rgt1-in 74.5%

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

      3. metadata-eval 74.5%

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

      4. mul-1-neg 74.5%

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

      5. distribute-rgt-neg-out 74.5%

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

      6. *-lft-identity 74.5%

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

   8. Simplified 74.5%

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

   if 2.24999999999999986e-57 < b_2

   1. Initial program 13.5%

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

      1. +-commutative 13.5%

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

      2. unsub-neg 13.5%

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

   3. Simplified 13.5%

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

      1. add-cbrt-cube 7.3%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow3 7.3%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

      3. pow1/3 6.5%

         \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

      4. sqrt-pow2 6.5%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

      5. fma-neg 6.6%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      6. *-commutative 6.6%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      7. distribute-rgt-neg-in 6.6%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      8. metadata-eval 6.6%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

   5. Applied egg-rr 6.6%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow1/3 7.4%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

      2. distribute-rgt-neg-out 7.4%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

      3. *-commutative 7.4%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

      4. fma-neg 7.3%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

      5. *-commutative 7.3%

         \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

   7. Simplified 7.3%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]
   8. Step-by-step derivation

      1. clear-num 7.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}}} \]

      2. inv-pow 7.3%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}\right)}^{-1}} \]

      3. pow1/3 6.5%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}\right)}^{-1} \]

      4. pow-pow 13.4%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b_2}\right)}^{-1} \]

      5. *-commutative 13.4%

         \[\leadsto {\left(\frac{a}{{\left(b_2 \cdot b_2 - \color{blue}{a \cdot c}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} - b_2}\right)}^{-1} \]

      6. metadata-eval 13.4%

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

   9. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{{\left(\frac{a}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5} - b_2}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 13.4%

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

       2. unpow1/2 13.4%

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

       3. *-commutative 13.4%

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

   11. Simplified 13.4%

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

   12. Taylor expanded in a around 0 87.0%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.55 \cdot 10^{-61}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \end{array} \]

Alternative 4: 67.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4.1 \cdot 10^{-284}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.1e-284)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.1e-284) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.1d-284)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else
        tmp = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.1e-284) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.1e-284:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	else:
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.1e-284)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	else
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.1e-284)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	else
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.1e-284], 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[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.09999999999999999e-284

   1. Initial program 71.0%

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

      1. +-commutative 71.0%

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

      2. unsub-neg 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in b_2 around -inf 65.7%

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

   if -4.09999999999999999e-284 < b_2

   1. Initial program 31.5%

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

      1. +-commutative 31.5%

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

      2. unsub-neg 31.5%

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

   3. Simplified 31.5%

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

      1. add-cbrt-cube 22.9%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow3 22.9%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

      3. pow1/3 21.1%

         \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

      4. sqrt-pow2 21.1%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

      5. fma-neg 21.1%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      6. *-commutative 21.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      7. distribute-rgt-neg-in 21.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      8. metadata-eval 21.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

   5. Applied egg-rr 21.1%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow1/3 23.0%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

      2. distribute-rgt-neg-out 23.0%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

      3. *-commutative 23.0%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

      4. fma-neg 22.9%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

      5. *-commutative 22.9%

         \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

   7. Simplified 22.9%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]
   8. Step-by-step derivation

      1. clear-num 22.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}}} \]

      2. inv-pow 22.9%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}\right)}^{-1}} \]

      3. pow1/3 21.1%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}\right)}^{-1} \]

      4. pow-pow 31.4%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b_2}\right)}^{-1} \]

      5. *-commutative 31.4%

         \[\leadsto {\left(\frac{a}{{\left(b_2 \cdot b_2 - \color{blue}{a \cdot c}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} - b_2}\right)}^{-1} \]

      6. metadata-eval 31.4%

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

   9. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{{\left(\frac{a}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5} - b_2}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 31.4%

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

       2. unpow1/2 31.4%

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

       3. *-commutative 31.4%

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

   11. Simplified 31.4%

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

   12. Taylor expanded in a around 0 63.8%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.1 \cdot 10^{-284}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \end{array} \]

Alternative 5: 67.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} 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 <= (-1d-309)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    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 <= -1e-309) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	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 <= -1e-309)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], 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 / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 71.6%

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

      1. +-commutative 71.6%

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

      2. unsub-neg 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in b_2 around -inf 64.3%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 29.8%

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

      1. +-commutative 29.8%

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

      2. unsub-neg 29.8%

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

   3. Simplified 29.8%

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

   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 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 6: 47.9% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309) (/ (- b_2) a) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = -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 <= (-1d-309)) then
        tmp = -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 <= -1e-309) {
		tmp = -b_2 / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = -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 <= -1e-309)
		tmp = Float64(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 <= -1e-309)
		tmp = -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, -1e-309], N[((-b$95$2) / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 71.6%

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

      1. +-commutative 71.6%

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

      2. unsub-neg 71.6%

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

   3. Simplified 71.6%

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

      1. add-cbrt-cube 60.5%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow3 60.5%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

      3. pow1/3 57.9%

         \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

      4. sqrt-pow2 57.9%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

      5. fma-neg 57.9%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      6. *-commutative 57.9%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      7. distribute-rgt-neg-in 57.9%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      8. metadata-eval 57.9%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

   5. Applied egg-rr 57.9%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow1/3 60.5%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

      2. distribute-rgt-neg-out 60.5%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

      3. *-commutative 60.5%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

      4. fma-neg 60.4%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

      5. *-commutative 60.4%

         \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

   7. Simplified 60.4%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]

   8. Taylor expanded in b_2 around 0 36.2%

      \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(-1 \cdot \left(c \cdot a\right)\right)}}^{1.5}} - b_2}{a} \]
   9. Step-by-step derivation

      1. mul-1-neg 36.2%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(-c \cdot a\right)}}^{1.5}} - b_2}{a} \]

      2. distribute-rgt-neg-in 36.2%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(c \cdot \left(-a\right)\right)}}^{1.5}} - b_2}{a} \]

   10. Simplified 36.2%

       \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(c \cdot \left(-a\right)\right)}}^{1.5}} - b_2}{a} \]

   11. Taylor expanded in c around 0 27.0%

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

       1. associate-*r/ 27.0%

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

       2. neg-mul-1 27.0%

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

   13. Simplified 27.0%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 29.8%

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

      1. +-commutative 29.8%

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

      2. unsub-neg 29.8%

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

   3. Simplified 29.8%

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

   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 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 7: 67.4% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309) (/ (* b_2 -2.0) a) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (b_2 * -2.0) / 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 <= (-1d-309)) then
        tmp = (b_2 * (-2.0d0)) / 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 <= -1e-309) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(b_2 * -2.0) / 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 <= -1e-309)
		tmp = (b_2 * -2.0) / 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, -1e-309], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 71.6%

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

      1. +-commutative 71.6%

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

      2. unsub-neg 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in b_2 around -inf 64.0%

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

      1. *-commutative 64.0%

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

   6. Simplified 64.0%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 29.8%

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

      1. +-commutative 29.8%

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

      2. unsub-neg 29.8%

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

   3. Simplified 29.8%

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

   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 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 8: 15.4% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. +-commutative 51.7%

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

   2. unsub-neg 51.7%

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

3. Simplified 51.7%

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

   1. add-cbrt-cube 42.0%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

   2. pow3 42.0%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

   3. pow1/3 39.8%

      \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

   4. sqrt-pow2 39.8%

      \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

   5. fma-neg 39.8%

      \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

   6. *-commutative 39.8%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

   7. distribute-rgt-neg-in 39.8%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

   8. metadata-eval 39.8%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

5. Applied egg-rr 39.8%

   \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
6. Step-by-step derivation

   1. unpow1/3 42.0%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

   2. distribute-rgt-neg-out 42.0%

      \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

   3. *-commutative 42.0%

      \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

   4. fma-neg 41.9%

      \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

   5. *-commutative 41.9%

      \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

7. Simplified 41.9%

   \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]

8. Taylor expanded in b_2 around 0 27.9%

   \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(-1 \cdot \left(c \cdot a\right)\right)}}^{1.5}} - b_2}{a} \]
9. Step-by-step derivation

   1. mul-1-neg 27.9%

      \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(-c \cdot a\right)}}^{1.5}} - b_2}{a} \]

   2. distribute-rgt-neg-in 27.9%

      \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(c \cdot \left(-a\right)\right)}}^{1.5}} - b_2}{a} \]

10. Simplified 27.9%

    \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(c \cdot \left(-a\right)\right)}}^{1.5}} - b_2}{a} \]

11. Taylor expanded in c around 0 15.4%

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

    1. associate-*r/ 15.4%

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

    2. neg-mul-1 15.4%

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

13. Simplified 15.4%

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

14. Final simplification 15.4%

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

Reproduce

herbie shell --seed 2023256 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))