quadp (p42, positive)

Percentage Accurate: 51.8% → 85.4%

Time: 17.0s

Alternatives: 6

Speedup: 19.1×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+93)
   (- (/ c b) (/ b a))
   (if (<= b 1.95e-57)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+93) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-57) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d+93)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.95d-57) then
        tmp = (sqrt(((b * b) - (4.0d0 * (c * a)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+93) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-57) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+93:
		tmp = (c / b) - (b / a)
	elif b <= 1.95e-57:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+93)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.95e-57)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+93)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.95e-57)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+93], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-57], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000001e93

   1. Initial program 52.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 93.5%

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

      1. +-commutative 93.5%

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

      2. mul-1-neg 93.5%

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

      3. unsub-neg 93.5%

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

   4. Simplified 93.5%

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

   if -5.0000000000000001e93 < b < 1.95000000000000003e-57

   1. Initial program 77.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   if 1.95000000000000003e-57 < b

   1. Initial program 14.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 86.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

   4. Simplified 86.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \frac{c \cdot \frac{a \cdot 2}{b} - \left(b + b\right)}{a}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.3e-69)
   (* 0.5 (/ (- (* c (/ (* a 2.0) b)) (+ b b)) a))
   (if (<= b 9.2e-57)
     (/ (- (sqrt (* (* c a) -4.0)) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-69) {
		tmp = 0.5 * (((c * ((a * 2.0) / b)) - (b + b)) / a);
	} else if (b <= 9.2e-57) {
		tmp = (sqrt(((c * a) * -4.0)) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.3d-69)) then
        tmp = 0.5d0 * (((c * ((a * 2.0d0) / b)) - (b + b)) / a)
    else if (b <= 9.2d-57) then
        tmp = (sqrt(((c * a) * (-4.0d0))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-69) {
		tmp = 0.5 * (((c * ((a * 2.0) / b)) - (b + b)) / a);
	} else if (b <= 9.2e-57) {
		tmp = (Math.sqrt(((c * a) * -4.0)) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.3e-69:
		tmp = 0.5 * (((c * ((a * 2.0) / b)) - (b + b)) / a)
	elif b <= 9.2e-57:
		tmp = (math.sqrt(((c * a) * -4.0)) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.3e-69)
		tmp = Float64(0.5 * Float64(Float64(Float64(c * Float64(Float64(a * 2.0) / b)) - Float64(b + b)) / a));
	elseif (b <= 9.2e-57)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -4.0)) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.3e-69)
		tmp = 0.5 * (((c * ((a * 2.0) / b)) - (b + b)) / a);
	elseif (b <= 9.2e-57)
		tmp = (sqrt(((c * a) * -4.0)) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.3e-69], N[(0.5 * N[(N[(N[(c * N[(N[(a * 2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(b + b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.2e-57], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.3e-69

   1. Initial program 67.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 67.2%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \]

      2. pow2 67.2%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}^{2}}}{2 \cdot a} \]

      3. pow1/2 67.2%

         \[\leadsto \frac{\left(-b\right) + {\left(\sqrt{\color{blue}{{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}\right)}^{2}}{2 \cdot a} \]

      4. sqrt-pow1 67.2%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{2 \cdot a} \]

      5. fma-neg 67.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      6. distribute-lft-neg-in 67.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      7. associate-*r* 67.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{\left(\left(-4\right) \cdot a\right) \cdot c}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      8. metadata-eval 67.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \left(\color{blue}{-4} \cdot a\right) \cdot c\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      9. metadata-eval 67.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{2 \cdot a} \]

   3. Applied egg-rr 67.2%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{0.25}\right)}^{2}}}{2 \cdot a} \]

   4. Taylor expanded in b around -inf 87.5%

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

      1. neg-mul-1 87.5%

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

      2. +-commutative 87.5%

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

      3. unsub-neg 87.5%

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

      4. associate-/l* 89.0%

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

   6. Simplified 89.0%

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

      1. expm1-log1p-u 52.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \left(2 \cdot \frac{a}{\frac{b}{c}} - b\right)}{2 \cdot a}\right)\right)} \]

      2. expm1-udef 41.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \left(2 \cdot \frac{a}{\frac{b}{c}} - b\right)}{2 \cdot a}\right)} - 1} \]

      3. associate-+r- 41.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\left(-b\right) + 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}}{2 \cdot a}\right)} - 1 \]

      4. neg-mul-1 41.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\left(\color{blue}{-1 \cdot b} + 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}{2 \cdot a}\right)} - 1 \]

      5. fma-def 41.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)} - b}{2 \cdot a}\right)} - 1 \]

      6. associate-/r/ 41.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, 2 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)}\right) - b}{2 \cdot a}\right)} - 1 \]

      7. associate-*l/ 41.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, 2 \cdot \color{blue}{\frac{a \cdot c}{b}}\right) - b}{2 \cdot a}\right)} - 1 \]

      8. *-commutative 41.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a \cdot c}{b}\right) - b}{\color{blue}{a \cdot 2}}\right)} - 1 \]

   8. Applied egg-rr 41.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a \cdot c}{b}\right) - b}{a \cdot 2}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 51.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a \cdot c}{b}\right) - b}{a \cdot 2}\right)\right)} \]

      2. expm1-log1p 87.5%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a \cdot c}{b}\right) - b}{a \cdot 2}} \]

      3. *-lft-identity 87.5%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(\mathsf{fma}\left(-1, b, 2 \cdot \frac{a \cdot c}{b}\right) - b\right)}}{a \cdot 2} \]

      4. *-commutative 87.5%

         \[\leadsto \frac{1 \cdot \left(\mathsf{fma}\left(-1, b, 2 \cdot \frac{a \cdot c}{b}\right) - b\right)}{\color{blue}{2 \cdot a}} \]

      5. times-frac 87.5%

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a \cdot c}{b}\right) - b}{a}} \]

      6. metadata-eval 87.5%

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

      7. fma-def 87.5%

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

      8. neg-mul-1 87.5%

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

      9. +-commutative 87.5%

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

      10. sub-neg 87.5%

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

      11. associate--l- 87.5%

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

      12. associate-/l* 89.0%

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

      13. associate-*r/ 89.0%

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

      14. *-commutative 89.0%

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

      15. associate-/r/ 89.0%

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

   10. Simplified 89.0%

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

   if -4.3e-69 < b < 9.2000000000000001e-57

   1. Initial program 70.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 70.3%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \]

      2. pow2 70.3%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}^{2}}}{2 \cdot a} \]

      3. pow1/2 70.3%

         \[\leadsto \frac{\left(-b\right) + {\left(\sqrt{\color{blue}{{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}\right)}^{2}}{2 \cdot a} \]

      4. sqrt-pow1 70.4%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{2 \cdot a} \]

      5. fma-neg 70.4%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      6. distribute-lft-neg-in 70.4%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      7. associate-*r* 70.4%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{\left(\left(-4\right) \cdot a\right) \cdot c}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      8. metadata-eval 70.4%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \left(\color{blue}{-4} \cdot a\right) \cdot c\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      9. metadata-eval 70.4%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{2 \cdot a} \]

   3. Applied egg-rr 70.4%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{0.25}\right)}^{2}}}{2 \cdot a} \]

   4. Taylor expanded in c around inf 38.4%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25 \cdot \left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)}\right)}^{2} - b}}{2 \cdot a} \]
   5. Step-by-step derivation

   6. Simplified 65.5%

      \[\leadsto \frac{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a} \]

   if 9.2000000000000001e-57 < b

   1. Initial program 14.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 86.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

   4. Simplified 86.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \frac{c \cdot \frac{a \cdot 2}{b} - \left(b + b\right)}{a}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 67.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 68.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 64.8%

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

      1. +-commutative 64.8%

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

      2. mul-1-neg 64.8%

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

      3. unsub-neg 64.8%

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

   4. Simplified 64.8%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 67.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 67.9%

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

      2. neg-mul-1 67.9%

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

   4. Simplified 67.9%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 42.9% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 7.5e+33) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.5e+33) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 7.5d+33) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.5e+33) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.5e+33:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.5e+33)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.5e+33)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.5e+33], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.50000000000000046e33

   1. Initial program 65.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 46.0%

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

      1. associate-*r/ 46.0%

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

      2. mul-1-neg 46.0%

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

   4. Simplified 46.0%

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

   if 7.50000000000000046e33 < b

   1. Initial program 9.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 8.9%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \]

      2. pow2 8.9%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}^{2}}}{2 \cdot a} \]

      3. pow1/2 8.9%

         \[\leadsto \frac{\left(-b\right) + {\left(\sqrt{\color{blue}{{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}\right)}^{2}}{2 \cdot a} \]

      4. sqrt-pow1 8.8%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{2 \cdot a} \]

      5. fma-neg 8.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      6. distribute-lft-neg-in 8.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      7. associate-*r* 8.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{\left(\left(-4\right) \cdot a\right) \cdot c}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      8. metadata-eval 8.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \left(\color{blue}{-4} \cdot a\right) \cdot c\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

      9. metadata-eval 8.9%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{2 \cdot a} \]

   3. Applied egg-rr 8.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{0.25}\right)}^{2}}}{2 \cdot a} \]

   4. Taylor expanded in b around -inf 2.2%

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

      1. neg-mul-1 2.2%

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

      2. +-commutative 2.2%

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

      3. unsub-neg 2.2%

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

      4. associate-/l* 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in b around 0 31.6%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 5: 67.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-261}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1e-261) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1e-261) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1d-261) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1e-261) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1e-261:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1e-261)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1e-261)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1e-261], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999984e-262

   1. Initial program 68.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 61.8%

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

      1. associate-*r/ 61.8%

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

      2. mul-1-neg 61.8%

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

   4. Simplified 61.8%

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

   if 9.99999999999999984e-262 < b

   1. Initial program 28.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 70.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

   4. Simplified 70.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-261}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 10.9% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

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

Fortran

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

Java

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

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 48.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Step-by-step derivation

   1. add-sqr-sqrt 48.6%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \]

   2. pow2 48.6%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}^{2}}}{2 \cdot a} \]

   3. pow1/2 48.6%

      \[\leadsto \frac{\left(-b\right) + {\left(\sqrt{\color{blue}{{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}\right)}^{2}}{2 \cdot a} \]

   4. sqrt-pow1 48.6%

      \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{2 \cdot a} \]

   5. fma-neg 48.6%

      \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

   6. distribute-lft-neg-in 48.6%

      \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

   7. associate-*r* 48.6%

      \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{\left(\left(-4\right) \cdot a\right) \cdot c}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

   8. metadata-eval 48.6%

      \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \left(\color{blue}{-4} \cdot a\right) \cdot c\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{2 \cdot a} \]

   9. metadata-eval 48.6%

      \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{2 \cdot a} \]

3. Applied egg-rr 48.6%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)\right)}^{0.25}\right)}^{2}}}{2 \cdot a} \]

4. Taylor expanded in b around -inf 32.6%

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

   1. neg-mul-1 32.6%

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

   2. +-commutative 32.6%

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

   3. unsub-neg 32.6%

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

   4. associate-/l* 33.3%

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

6. Simplified 33.3%

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

7. Taylor expanded in b around 0 11.7%

   \[\leadsto \color{blue}{\frac{c}{b}} \]

8. Final simplification 11.7%

   \[\leadsto \frac{c}{b} \]

Developer target: 59.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\left|b\right| > t_0:\\ \;\;\;\;\sqrt{\left|b\right| - t_0} \cdot \sqrt{\left|b\right| + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{NaN}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b, t_0\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t_1 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (if (> (fabs b) t_0)
             (* (sqrt (- (fabs b) t_0)) (sqrt (+ (fabs b) t_0)))
             NAN)
           (hypot b t_0))))
   (if (< b 0.0) (/ (- t_1 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp_1;
	if (copysign(a, c) == a) {
		double tmp_2;
		if (fabs(b) > t_0) {
			tmp_2 = sqrt((fabs(b) - t_0)) * sqrt((fabs(b) + t_0));
		} else {
			tmp_2 = (double) NAN;
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = hypot(b, t_0);
	}
	double t_1 = tmp_1;
	double tmp_3;
	if (b < 0.0) {
		tmp_3 = (t_1 - (b / 2.0)) / a;
	} else {
		tmp_3 = -c / ((b / 2.0) + t_1);
	}
	return tmp_3;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp_1;
	if (Math.copySign(a, c) == a) {
		double tmp_2;
		if (Math.abs(b) > t_0) {
			tmp_2 = Math.sqrt((Math.abs(b) - t_0)) * Math.sqrt((Math.abs(b) + t_0));
		} else {
			tmp_2 = Double.NaN;
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = Math.hypot(b, t_0);
	}
	double t_1 = tmp_1;
	double tmp_3;
	if (b < 0.0) {
		tmp_3 = (t_1 - (b / 2.0)) / a;
	} else {
		tmp_3 = -c / ((b / 2.0) + t_1);
	}
	return tmp_3;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp_1 = 0
	if math.copysign(a, c) == a:
		tmp_2 = 0
		if math.fabs(b) > t_0:
			tmp_2 = math.sqrt((math.fabs(b) - t_0)) * math.sqrt((math.fabs(b) + t_0))
		else:
			tmp_2 = math.nan
		tmp_1 = tmp_2
	else:
		tmp_1 = math.hypot(b, t_0)
	t_1 = tmp_1
	tmp_3 = 0
	if b < 0.0:
		tmp_3 = (t_1 - (b / 2.0)) / a
	else:
		tmp_3 = -c / ((b / 2.0) + t_1)
	return tmp_3

Julia

function code(a, b, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp_1 = 0.0
	if (copysign(a, c) == a)
		tmp_2 = 0.0
		if (abs(b) > t_0)
			tmp_2 = Float64(sqrt(Float64(abs(b) - t_0)) * sqrt(Float64(abs(b) + t_0)));
		else
			tmp_2 = NaN;
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = hypot(b, t_0);
	end
	t_1 = tmp_1
	tmp_3 = 0.0
	if (b < 0.0)
		tmp_3 = Float64(Float64(t_1 - Float64(b / 2.0)) / a);
	else
		tmp_3 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_1));
	end
	return tmp_3
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp_2 = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp_3 = 0.0;
		if (abs(b) > t_0)
			tmp_3 = sqrt((abs(b) - t_0)) * sqrt((abs(b) + t_0));
		else
			tmp_3 = NaN;
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = hypot(b, t_0);
	end
	t_1 = tmp_2;
	tmp_4 = 0.0;
	if (b < 0.0)
		tmp_4 = (t_1 - (b / 2.0)) / a;
	else
		tmp_4 = -c / ((b / 2.0) + t_1);
	end
	tmp_5 = tmp_4;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], If[Greater[N[Abs[b], $MachinePrecision], t$95$0], N[(N[Sqrt[N[(N[Abs[b], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], Indeterminate], N[Sqrt[b ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$1 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2023287 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (if (> (fabs b) (* (sqrt (fabs a)) (sqrt (fabs c)))) (* (sqrt (- (fabs b) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b) (* (sqrt (fabs a)) (sqrt (fabs c)))))) NAN) (hypot b (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (if (> (fabs b) (* (sqrt (fabs a)) (sqrt (fabs c)))) (* (sqrt (- (fabs b) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b) (* (sqrt (fabs a)) (sqrt (fabs c)))))) NAN) (hypot b (* (sqrt (fabs a)) (sqrt (fabs c))))))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))