quadp (p42, positive)

Percentage Accurate: 52.0% → 86.1%

Time: 17.1s

Alternatives: 7

Speedup: 12.9×

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: 52.0% 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: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+150)
   (/ (- b) a)
   (if (<= b 2.8e-68)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+150) {
		tmp = -b / a;
	} else if (b <= 2.8e-68) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - 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 <= (-2d+150)) then
        tmp = -b / a
    else if (b <= 2.8d-68) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - 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 <= -2e+150) {
		tmp = -b / a;
	} else if (b <= 2.8e-68) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+150:
		tmp = -b / a
	elif b <= 2.8e-68:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+150)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2.8e-68)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - 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 <= -2e+150)
		tmp = -b / a;
	elseif (b <= 2.8e-68)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+150], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2.8e-68], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $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 < -1.99999999999999996e150

   1. Initial program 36.3%

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

      1. *-commutative 36.3%

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

   3. Simplified 36.3%

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

   5. Taylor expanded in b around -inf 98.0%

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

      1. associate-*r/ 98.0%

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

      2. mul-1-neg 98.0%

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

   7. Simplified 98.0%

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

   if -1.99999999999999996e150 < b < 2.8000000000000001e-68

   1. Initial program 80.5%

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

   if 2.8000000000000001e-68 < b

   1. Initial program 11.3%

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

      1. *-commutative 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in b around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. neg-mul-1 91.3%

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

   7. Simplified 91.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-70}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.1e-52)
   (/ (- b) a)
   (if (<= b 6e-70) (* (/ -0.5 a) (- b (sqrt (* a (* c -4.0))))) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.1e-52) {
		tmp = -b / a;
	} else if (b <= 6e-70) {
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	} 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 <= (-6.1d-52)) then
        tmp = -b / a
    else if (b <= 6d-70) then
        tmp = ((-0.5d0) / a) * (b - sqrt((a * (c * (-4.0d0)))))
    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 <= -6.1e-52) {
		tmp = -b / a;
	} else if (b <= 6e-70) {
		tmp = (-0.5 / a) * (b - Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.1e-52:
		tmp = -b / a
	elif b <= 6e-70:
		tmp = (-0.5 / a) * (b - math.sqrt((a * (c * -4.0))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.1e-52)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 6e-70)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.1e-52)
		tmp = -b / a;
	elseif (b <= 6e-70)
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.1e-52], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 6e-70], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.0999999999999999e-52

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in b around -inf 94.5%

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

      1. associate-*r/ 94.5%

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

      2. mul-1-neg 94.5%

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

   7. Simplified 94.5%

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

   if -6.0999999999999999e-52 < b < 6.0000000000000003e-70

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in b around 0 64.6%

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

      1. associate-*r* 64.6%

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

   7. Simplified 64.6%

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

      1. add-sqr-sqrt 64.2%

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

      2. pow2 64.2%

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

      3. pow1/2 64.2%

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

      4. sqrt-pow1 64.2%

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

      5. associate-*r* 64.2%

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

      6. *-commutative 64.2%

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

      7. metadata-eval 64.2%

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

   9. Applied egg-rr 64.2%

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

       1. frac-2neg 64.2%

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

       2. div-inv 64.2%

          \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + {\left({\left(-4 \cdot \left(c \cdot a\right)\right)}^{0.25}\right)}^{2}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   11. Applied egg-rr 64.6%

       \[\leadsto \color{blue}{\left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{1}{a \cdot -2}} \]
   12. Step-by-step derivation

       1. *-commutative 64.6%

          \[\leadsto \color{blue}{\frac{1}{a \cdot -2} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

       2. *-commutative 64.6%

          \[\leadsto \frac{1}{\color{blue}{-2 \cdot a}} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \]

       3. associate-/r* 64.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{-2}}{a}} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \]

       4. metadata-eval 64.6%

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

       5. associate-*r* 64.6%

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

       6. *-commutative 64.6%

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

       7. associate-*l* 64.6%

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

   13. Simplified 64.6%

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

   if 6.0000000000000003e-70 < b

   1. Initial program 11.3%

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

      1. *-commutative 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in b around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. neg-mul-1 91.3%

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

   7. Simplified 91.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-70}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.3e-52)
   (/ (- b) a)
   (if (<= b 5.8e-65)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.3e-52) {
		tmp = -b / a;
	} else if (b <= 5.8e-65) {
		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 <= (-5.3d-52)) then
        tmp = -b / a
    else if (b <= 5.8d-65) 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 <= -5.3e-52) {
		tmp = -b / a;
	} else if (b <= 5.8e-65) {
		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 <= -5.3e-52:
		tmp = -b / a
	elif b <= 5.8e-65:
		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 <= -5.3e-52)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 5.8e-65)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(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 <= -5.3e-52)
		tmp = -b / a;
	elseif (b <= 5.8e-65)
		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, -5.3e-52], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 5.8e-65], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $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 < -5.3000000000000003e-52

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in b around -inf 94.5%

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

      1. associate-*r/ 94.5%

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

      2. mul-1-neg 94.5%

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

   7. Simplified 94.5%

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

   if -5.3000000000000003e-52 < b < 5.7999999999999996e-65

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in b around 0 64.6%

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

      1. associate-*r* 64.6%

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

   7. Simplified 64.6%

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

      1. +-commutative 64.6%

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

      2. unsub-neg 64.6%

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

      3. associate-*r* 64.6%

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

      4. *-commutative 64.6%

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

   9. Applied egg-rr 64.6%

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

       1. *-commutative 64.6%

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

       2. associate-*r* 64.6%

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

       3. *-commutative 64.6%

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

   11. Simplified 64.6%

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

   if 5.7999999999999996e-65 < b

   1. Initial program 11.3%

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

      1. *-commutative 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in b around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. neg-mul-1 91.3%

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

   7. Simplified 91.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.25 \cdot 10^{+22}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 3.25e+22) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.25e+22) {
		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 <= 3.25d+22) 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 <= 3.25e+22) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.25e+22:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.25e+22)
		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 <= 3.25e+22)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.25e+22], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.2499999999999999e22

   1. Initial program 64.4%

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

      1. *-commutative 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in b around -inf 50.4%

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

      1. associate-*r/ 50.4%

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

      2. mul-1-neg 50.4%

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

   7. Simplified 50.4%

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

   if 3.2499999999999999e22 < b

   1. Initial program 11.4%

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

      1. *-commutative 11.4%

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

   3. Simplified 11.4%

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

   5. Taylor expanded in b around -inf 2.3%

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

      1. +-commutative 2.3%

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

      2. mul-1-neg 2.3%

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

      3. unsub-neg 2.3%

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

      4. associate-/l* 2.4%

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

      5. associate-*r/ 2.4%

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

      6. *-commutative 2.4%

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

   7. Simplified 2.4%

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

   8. Taylor expanded in b around 0 23.3%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.25 \cdot 10^{+22}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.4e-291) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.4e-291) {
		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 <= 6.4d-291) 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 <= 6.4e-291) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.4e-291:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.4e-291)
		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 <= 6.4e-291)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.4e-291], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.4000000000000003e-291

   1. Initial program 70.2%

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

      1. *-commutative 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in b around -inf 65.8%

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

      1. associate-*r/ 65.8%

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

      2. mul-1-neg 65.8%

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

   7. Simplified 65.8%

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

   if 6.4000000000000003e-291 < b

   1. Initial program 24.9%

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

      1. *-commutative 24.9%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in b around inf 73.4%

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

      1. associate-*r/ 73.4%

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

      2. neg-mul-1 73.4%

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

   7. Simplified 73.4%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 2.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return b / 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 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.5%

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

   1. *-commutative 49.5%

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

3. Simplified 49.5%

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

   1. add-sqr-sqrt 49.0%

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

   2. pow2 49.0%

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

   3. pow1/2 49.0%

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

   4. sqrt-pow1 49.1%

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

   5. fma-neg 49.1%

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

   6. distribute-lft-neg-in 49.1%

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

   7. *-commutative 49.1%

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

   8. associate-*r* 49.1%

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

   9. metadata-eval 49.1%

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

   10. metadata-eval 49.1%

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

6. Applied egg-rr 49.1%

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

   1. pow-pow 49.5%

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

   2. metadata-eval 49.5%

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

   3. metadata-eval 49.5%

      \[\leadsto \frac{\left(-b\right) + {\left(\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}{a \cdot 2} \]

   4. pow-pow 35.1%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{a \cdot 2} \]

   5. pow1/3 36.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)\right)}^{1.5}}}}{a \cdot 2} \]

   6. clear-num 36.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)\right)}^{1.5}}}}} \]

   7. inv-pow 36.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)\right)}^{1.5}}}\right)}^{-1}} \]

8. Applied egg-rr 24.4%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{b + \mathsf{hypot}\left(b, \sqrt{-4 \cdot \left(c \cdot a\right)}\right)}\right)}^{-1}} \]
9. Step-by-step derivation

   1. unpow-1 24.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{b + \mathsf{hypot}\left(b, \sqrt{-4 \cdot \left(c \cdot a\right)}\right)}}} \]

   2. associate-/l* 24.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\frac{b + \mathsf{hypot}\left(b, \sqrt{-4 \cdot \left(c \cdot a\right)}\right)}{2}}}} \]

   3. *-commutative 24.4%

      \[\leadsto \frac{1}{\frac{a}{\frac{b + \mathsf{hypot}\left(b, \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}\right)}{2}}} \]

10. Simplified 24.4%

    \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{b + \mathsf{hypot}\left(b, \sqrt{-4 \cdot \left(a \cdot c\right)}\right)}{2}}}} \]

11. Taylor expanded in a around 0 2.7%

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

12. Final simplification 2.7%

    \[\leadsto \frac{b}{a} \]
13. Add Preprocessing

Alternative 7: 11.0% 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 49.5%

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

   1. *-commutative 49.5%

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

3. Simplified 49.5%

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

5. Taylor expanded in b around -inf 34.3%

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

   1. +-commutative 34.3%

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

   2. mul-1-neg 34.3%

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

   3. unsub-neg 34.3%

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

   4. associate-/l* 36.7%

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

   5. associate-*r/ 36.7%

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

   6. *-commutative 36.7%

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

7. Simplified 36.7%

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

8. Taylor expanded in b around 0 8.7%

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

9. Final simplification 8.7%

   \[\leadsto \frac{c}{b} \]
10. Add Preprocessing

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t_0 - t_1} \cdot \sqrt{t_0 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

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

Reproduce

herbie shell --seed 2024021 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

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

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