quadp (p42, positive)

Percentage Accurate: 51.9% → 84.4%

Time: 17.7s

Alternatives: 10

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: 51.9% 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: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-70}:\\ \;\;\;\;{c}^{2} \cdot \frac{-a}{{b}^{3}} - \frac{c}{b}\\ \mathbf{elif}\;b \leq 7.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+146)
   (- (/ c b) (/ b a))
   (if (<= b 1.35e-88)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (if (<= b 1.05e-70)
       (- (* (pow c 2.0) (/ (- a) (pow b 3.0))) (/ c b))
       (if (<= b 7.7e+27)
         (/ (- (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 <= -2e+146) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.35e-88) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else if (b <= 1.05e-70) {
		tmp = (pow(c, 2.0) * (-a / pow(b, 3.0))) - (c / b);
	} else if (b <= 7.7e+27) {
		tmp = (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 <= -2e+146)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.35e-88)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	elseif (b <= 1.05e-70)
		tmp = Float64(Float64((c ^ 2.0) * Float64(Float64(-a) / (b ^ 3.0))) - Float64(c / b));
	elseif (b <= 7.7e+27)
		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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+146], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-88], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-70], N[(N[(N[Power[c, 2.0], $MachinePrecision] * N[((-a) / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.7e+27], 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 5 regimes

2. if b < -1.99999999999999987e146

   1. Initial program 39.5%

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

      1. *-commutative 39.5%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in b around -inf 96.2%

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

      1. +-commutative 96.2%

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

      2. mul-1-neg 96.2%

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

      3. unsub-neg 96.2%

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

   7. Simplified 96.2%

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

   if -1.99999999999999987e146 < b < 1.34999999999999997e-88

   1. Initial program 89.0%

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

      1. +-commutative 89.0%

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

      2. unsub-neg 89.0%

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

      3. fma-neg 89.0%

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

      4. distribute-lft-neg-in 89.0%

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

      5. *-commutative 89.0%

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

      6. *-commutative 89.0%

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

      7. associate-*l* 89.1%

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

      8. metadata-eval 89.1%

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

      9. *-commutative 89.1%

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

   3. Simplified 89.1%

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

   if 1.34999999999999997e-88 < b < 1.0500000000000001e-70

   1. Initial program 8.5%

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

      1. *-commutative 8.5%

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

   3. Simplified 8.5%

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

   5. Taylor expanded in b around inf 98.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. associate-*r/ 98.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      4. neg-mul-1 98.7%

         \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. associate-/l* 80.1%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]

      6. associate-/r/ 100.0%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{a}{{b}^{3}} \cdot {c}^{2}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot {c}^{2}} \]

   if 1.0500000000000001e-70 < b < 7.69999999999999953e27

   1. Initial program 71.5%

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

   if 7.69999999999999953e27 < b

   1. Initial program 11.5%

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

      1. *-commutative 11.5%

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

   3. Simplified 11.5%

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

   5. Taylor expanded in b around inf 97.1%

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

      1. associate-*r/ 97.1%

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

      2. neg-mul-1 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-70}:\\ \;\;\;\;{c}^{2} \cdot \frac{-a}{{b}^{3}} - \frac{c}{b}\\ \mathbf{elif}\;b \leq 7.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-67}:\\ \;\;\;\;{c}^{2} \cdot \frac{-a}{{b}^{3}} - \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))))
   (if (<= b -2e+146)
     (- (/ c b) (/ b a))
     (if (<= b 2.25e-87)
       t_0
       (if (<= b 3.1e-67)
         (- (* (pow c 2.0) (/ (- a) (pow b 3.0))) (/ c b))
         (if (<= b 6.5e+27) t_0 (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	double tmp;
	if (b <= -2e+146) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.25e-87) {
		tmp = t_0;
	} else if (b <= 3.1e-67) {
		tmp = (pow(c, 2.0) * (-a / pow(b, 3.0))) - (c / b);
	} else if (b <= 6.5e+27) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (4.0d0 * (c * a)))) - b) / (a * 2.0d0)
    if (b <= (-2d+146)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.25d-87) then
        tmp = t_0
    else if (b <= 3.1d-67) then
        tmp = ((c ** 2.0d0) * (-a / (b ** 3.0d0))) - (c / b)
    else if (b <= 6.5d+27) then
        tmp = t_0
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	double tmp;
	if (b <= -2e+146) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.25e-87) {
		tmp = t_0;
	} else if (b <= 3.1e-67) {
		tmp = (Math.pow(c, 2.0) * (-a / Math.pow(b, 3.0))) - (c / b);
	} else if (b <= 6.5e+27) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	tmp = 0
	if b <= -2e+146:
		tmp = (c / b) - (b / a)
	elif b <= 2.25e-87:
		tmp = t_0
	elif b <= 3.1e-67:
		tmp = (math.pow(c, 2.0) * (-a / math.pow(b, 3.0))) - (c / b)
	elif b <= 6.5e+27:
		tmp = t_0
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -2e+146)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.25e-87)
		tmp = t_0;
	elseif (b <= 3.1e-67)
		tmp = Float64(Float64((c ^ 2.0) * Float64(Float64(-a) / (b ^ 3.0))) - Float64(c / b));
	elseif (b <= 6.5e+27)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (b <= -2e+146)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.25e-87)
		tmp = t_0;
	elseif (b <= 3.1e-67)
		tmp = ((c ^ 2.0) * (-a / (b ^ 3.0))) - (c / b);
	elseif (b <= 6.5e+27)
		tmp = t_0;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, If[LessEqual[b, -2e+146], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.25e-87], t$95$0, If[LessEqual[b, 3.1e-67], N[(N[(N[Power[c, 2.0], $MachinePrecision] * N[((-a) / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e+27], t$95$0, N[((-c) / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.99999999999999987e146

   1. Initial program 39.5%

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

      1. *-commutative 39.5%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in b around -inf 96.2%

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

      1. +-commutative 96.2%

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

      2. mul-1-neg 96.2%

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

      3. unsub-neg 96.2%

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

   7. Simplified 96.2%

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

   if -1.99999999999999987e146 < b < 2.24999999999999979e-87 or 3.1000000000000003e-67 < b < 6.5000000000000005e27

   1. Initial program 87.1%

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

   if 2.24999999999999979e-87 < b < 3.1000000000000003e-67

   1. Initial program 8.5%

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

      1. *-commutative 8.5%

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

   3. Simplified 8.5%

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

   5. Taylor expanded in b around inf 98.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. associate-*r/ 98.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      4. neg-mul-1 98.7%

         \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. associate-/l* 80.1%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]

      6. associate-/r/ 100.0%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{a}{{b}^{3}} \cdot {c}^{2}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot {c}^{2}} \]

   if 6.5000000000000005e27 < b

   1. Initial program 11.5%

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

      1. *-commutative 11.5%

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

   3. Simplified 11.5%

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

   5. Taylor expanded in b around inf 97.1%

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

      1. associate-*r/ 97.1%

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

      2. neg-mul-1 97.1%

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

   7. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-67}:\\ \;\;\;\;{c}^{2} \cdot \frac{-a}{{b}^{3}} - \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-87} \lor \neg \left(b \leq 1.45 \cdot 10^{-67}\right) \land b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+146)
   (- (/ c b) (/ b a))
   (if (or (<= b 2.2e-87) (and (not (<= b 1.45e-67)) (<= b 6.5e+27)))
     (/ (- (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 <= -2e+146) {
		tmp = (c / b) - (b / a);
	} else if ((b <= 2.2e-87) || (!(b <= 1.45e-67) && (b <= 6.5e+27))) {
		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 <= (-2d+146)) then
        tmp = (c / b) - (b / a)
    else if ((b <= 2.2d-87) .or. (.not. (b <= 1.45d-67)) .and. (b <= 6.5d+27)) 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 <= -2e+146) {
		tmp = (c / b) - (b / a);
	} else if ((b <= 2.2e-87) || (!(b <= 1.45e-67) && (b <= 6.5e+27))) {
		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 <= -2e+146:
		tmp = (c / b) - (b / a)
	elif (b <= 2.2e-87) or (not (b <= 1.45e-67) and (b <= 6.5e+27)):
		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 <= -2e+146)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif ((b <= 2.2e-87) || (!(b <= 1.45e-67) && (b <= 6.5e+27)))
		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 <= -2e+146)
		tmp = (c / b) - (b / a);
	elseif ((b <= 2.2e-87) || (~((b <= 1.45e-67)) && (b <= 6.5e+27)))
		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, -2e+146], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 2.2e-87], And[N[Not[LessEqual[b, 1.45e-67]], $MachinePrecision], LessEqual[b, 6.5e+27]]], 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 < -1.99999999999999987e146

   1. Initial program 39.5%

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

      1. *-commutative 39.5%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in b around -inf 96.2%

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

      1. +-commutative 96.2%

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

      2. mul-1-neg 96.2%

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

      3. unsub-neg 96.2%

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

   7. Simplified 96.2%

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

   if -1.99999999999999987e146 < b < 2.19999999999999988e-87 or 1.45000000000000002e-67 < b < 6.5000000000000005e27

   1. Initial program 87.1%

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

   if 2.19999999999999988e-87 < b < 1.45000000000000002e-67 or 6.5000000000000005e27 < 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 97.2%

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

      1. associate-*r/ 97.2%

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

      2. neg-mul-1 97.2%

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

   7. Simplified 97.2%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-87} \lor \neg \left(b \leq 1.45 \cdot 10^{-67}\right) \land b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-88} \lor \neg \left(b \leq 1.42 \cdot 10^{-71}\right) \land b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-74)
   (/ (- b) a)
   (if (or (<= b 9.2e-88) (and (not (<= b 1.42e-71)) (<= b 6.5e+27)))
     (/ (+ b (sqrt (* c (* a -4.0)))) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-74) {
		tmp = -b / a;
	} else if ((b <= 9.2e-88) || (!(b <= 1.42e-71) && (b <= 6.5e+27))) {
		tmp = (b + sqrt((c * (a * -4.0)))) / (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 <= (-3.6d-74)) then
        tmp = -b / a
    else if ((b <= 9.2d-88) .or. (.not. (b <= 1.42d-71)) .and. (b <= 6.5d+27)) then
        tmp = (b + sqrt((c * (a * (-4.0d0))))) / (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 <= -3.6e-74) {
		tmp = -b / a;
	} else if ((b <= 9.2e-88) || (!(b <= 1.42e-71) && (b <= 6.5e+27))) {
		tmp = (b + Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e-74:
		tmp = -b / a
	elif (b <= 9.2e-88) or (not (b <= 1.42e-71) and (b <= 6.5e+27)):
		tmp = (b + math.sqrt((c * (a * -4.0)))) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-74)
		tmp = Float64(Float64(-b) / a);
	elseif ((b <= 9.2e-88) || (!(b <= 1.42e-71) && (b <= 6.5e+27)))
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / 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 <= -3.6e-74)
		tmp = -b / a;
	elseif ((b <= 9.2e-88) || (~((b <= 1.42e-71)) && (b <= 6.5e+27)))
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-74], N[((-b) / a), $MachinePrecision], If[Or[LessEqual[b, 9.2e-88], And[N[Not[LessEqual[b, 1.42e-71]], $MachinePrecision], LessEqual[b, 6.5e+27]]], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.6000000000000002e-74

   1. Initial program 68.9%

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

      1. *-commutative 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in b around -inf 87.7%

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

      1. associate-*r/ 87.7%

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

      2. mul-1-neg 87.7%

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

   7. Simplified 87.7%

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

   if -3.6000000000000002e-74 < b < 9.19999999999999945e-88 or 1.4199999999999999e-71 < b < 6.5000000000000005e27

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in b around 0 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*r* 76.4%

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

   7. Simplified 76.4%

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

      1. expm1-log1p-u 57.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\right)\right)} \]

      2. expm1-udef 21.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\right)} - 1} \]

   9. Applied egg-rr 20.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 56.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{a}\right)\right)} \]

       2. expm1-log1p 75.0%

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

       3. metadata-eval 75.0%

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

       4. times-frac 75.0%

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

       5. *-lft-identity 75.0%

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

       6. *-commutative 75.0%

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

       7. associate-*r* 75.1%

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

       8. *-commutative 75.1%

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

       9. *-commutative 75.1%

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

   11. Simplified 75.1%

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

   if 9.19999999999999945e-88 < b < 1.4199999999999999e-71 or 6.5000000000000005e27 < 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 97.2%

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

      1. associate-*r/ 97.2%

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

      2. neg-mul-1 97.2%

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

   7. Simplified 97.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-88} \lor \neg \left(b \leq 1.42 \cdot 10^{-71}\right) \land b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-88} \lor \neg \left(b \leq 4.6 \cdot 10^{-70}\right) \land b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-75)
   (/ (- b) a)
   (if (or (<= b 9.4e-88) (and (not (<= b 4.6e-70)) (<= b 6.5e+27)))
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-75) {
		tmp = -b / a;
	} else if ((b <= 9.4e-88) || (!(b <= 4.6e-70) && (b <= 6.5e+27))) {
		tmp = (sqrt((a * (c * -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 <= (-8.5d-75)) then
        tmp = -b / a
    else if ((b <= 9.4d-88) .or. (.not. (b <= 4.6d-70)) .and. (b <= 6.5d+27)) then
        tmp = (sqrt((a * (c * (-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 <= -8.5e-75) {
		tmp = -b / a;
	} else if ((b <= 9.4e-88) || (!(b <= 4.6e-70) && (b <= 6.5e+27))) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e-75:
		tmp = -b / a
	elif (b <= 9.4e-88) or (not (b <= 4.6e-70) and (b <= 6.5e+27)):
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-75)
		tmp = Float64(Float64(-b) / a);
	elseif ((b <= 9.4e-88) || (!(b <= 4.6e-70) && (b <= 6.5e+27)))
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -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 <= -8.5e-75)
		tmp = -b / a;
	elseif ((b <= 9.4e-88) || (~((b <= 4.6e-70)) && (b <= 6.5e+27)))
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-75], N[((-b) / a), $MachinePrecision], If[Or[LessEqual[b, 9.4e-88], And[N[Not[LessEqual[b, 4.6e-70]], $MachinePrecision], LessEqual[b, 6.5e+27]]], N[(N[(N[Sqrt[N[(a * N[(c * -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 < -8.5000000000000001e-75

   1. Initial program 68.9%

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

      1. *-commutative 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in b around -inf 87.7%

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

      1. associate-*r/ 87.7%

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

      2. mul-1-neg 87.7%

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

   7. Simplified 87.7%

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

   if -8.5000000000000001e-75 < b < 9.4e-88 or 4.60000000000000001e-70 < b < 6.5000000000000005e27

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in b around 0 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*r* 76.4%

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

   7. Simplified 76.4%

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

   if 9.4e-88 < b < 4.60000000000000001e-70 or 6.5000000000000005e27 < 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 97.2%

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

      1. associate-*r/ 97.2%

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

      2. neg-mul-1 97.2%

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

   7. Simplified 97.2%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 9.4 \cdot 10^{-88} \lor \neg \left(b \leq 4.6 \cdot 10^{-70}\right) \land b \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-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 <= (-4d-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 <= -4e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-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 <= -4e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-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 < -3.999999999999988e-310

   1. Initial program 74.6%

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

      1. *-commutative 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in b around -inf 71.1%

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

      1. +-commutative 71.1%

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

      2. mul-1-neg 71.1%

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

      3. unsub-neg 71.1%

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

   7. Simplified 71.1%

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

   if -3.999999999999988e-310 < b

   1. Initial program 40.5%

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

      1. *-commutative 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in b around inf 62.6%

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

      1. associate-*r/ 62.6%

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

      2. neg-mul-1 62.6%

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

   7. Simplified 62.6%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+27}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 8e+27) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e+27) {
		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 <= 8d+27) 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 <= 8e+27) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e+27:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e+27)
		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 <= 8e+27)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e+27], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.0000000000000001e27

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in b around -inf 49.4%

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

      1. associate-*r/ 49.4%

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

      2. mul-1-neg 49.4%

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

   7. Simplified 49.4%

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

   if 8.0000000000000001e27 < b

   1. Initial program 11.5%

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

      1. *-commutative 11.5%

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

   3. Simplified 11.5%

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

   5. Taylor expanded in b around inf 78.7%

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

      1. frac-2neg 78.7%

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

      2. div-inv 78.6%

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

      3. *-commutative 78.6%

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

      4. distribute-lft-neg-in 78.6%

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

      5. add-sqr-sqrt 78.5%

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

      6. sqrt-unprod 63.8%

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

      7. sqr-neg 63.8%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 33.4%

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

      10. distribute-frac-neg 33.4%

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

      11. frac-2neg 33.4%

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

      12. associate-/l* 33.5%

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

      13. associate-*l/ 33.5%

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

      14. distribute-rgt-neg-in 33.5%

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

      15. metadata-eval 33.5%

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

   7. Applied egg-rr 33.5%

      \[\leadsto \color{blue}{\frac{a \cdot -2}{\frac{b}{c}} \cdot \frac{1}{a \cdot -2}} \]
   8. Step-by-step derivation

      1. associate-*l/ 33.2%

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

      2. rgt-mult-inverse 33.2%

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

      3. associate-/r/ 33.2%

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

      4. *-commutative 33.2%

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

      5. associate-*r/ 33.2%

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

      6. *-rgt-identity 33.2%

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

   9. Simplified 33.2%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+27}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		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 <= (-4d-310)) 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 <= -4e-310) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		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 <= -4e-310)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 74.6%

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

      1. *-commutative 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in b around -inf 71.0%

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

      1. associate-*r/ 71.0%

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

      2. mul-1-neg 71.0%

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

   7. Simplified 71.0%

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

   if -3.999999999999988e-310 < b

   1. Initial program 40.5%

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

      1. *-commutative 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in b around inf 62.6%

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

      1. associate-*r/ 62.6%

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

      2. neg-mul-1 62.6%

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

   7. Simplified 62.6%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 2.5% 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 57.9%

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

   1. +-commutative 57.9%

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

   2. unsub-neg 57.9%

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

   3. fma-neg 57.9%

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

   4. distribute-lft-neg-in 57.9%

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

   5. *-commutative 57.9%

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

   6. *-commutative 57.9%

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

   7. associate-*l* 58.0%

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

   8. metadata-eval 58.0%

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

   9. *-commutative 58.0%

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

3. Simplified 58.0%

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

   1. sub-neg 58.0%

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

   2. fma-udef 58.0%

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

   3. add-sqr-sqrt 52.1%

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

   4. hypot-def 60.4%

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

   5. *-commutative 60.4%

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

   6. associate-*l* 60.4%

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

   7. add-sqr-sqrt 37.8%

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

   8. sqrt-unprod 50.3%

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

   9. sqr-neg 50.3%

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

   10. sqrt-prod 17.5%

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

   11. add-sqr-sqrt 31.2%

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

6. Applied egg-rr 31.2%

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

7. Taylor expanded in b around inf 2.5%

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

8. Final simplification 2.5%

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

Alternative 10: 10.6% 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 57.9%

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

   1. *-commutative 57.9%

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

3. Simplified 57.9%

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

5. Taylor expanded in b around inf 24.6%

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

   1. frac-2neg 24.6%

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

   2. div-inv 24.6%

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

   3. *-commutative 24.6%

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

   4. distribute-lft-neg-in 24.6%

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

   5. add-sqr-sqrt 23.6%

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

   6. sqrt-unprod 21.5%

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

   7. sqr-neg 21.5%

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

   8. sqrt-unprod 1.7%

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

   9. add-sqr-sqrt 10.4%

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

   10. distribute-frac-neg 10.4%

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

   11. frac-2neg 10.4%

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

   12. associate-/l* 10.5%

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

   13. associate-*l/ 10.5%

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

   14. distribute-rgt-neg-in 10.5%

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

   15. metadata-eval 10.5%

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

7. Applied egg-rr 10.5%

   \[\leadsto \color{blue}{\frac{a \cdot -2}{\frac{b}{c}} \cdot \frac{1}{a \cdot -2}} \]
8. Step-by-step derivation

   1. associate-*l/ 10.5%

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

   2. rgt-mult-inverse 10.5%

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

   3. associate-/r/ 10.5%

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

   4. *-commutative 10.5%

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

   5. associate-*r/ 10.5%

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

   6. *-rgt-identity 10.5%

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

9. Simplified 10.5%

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

10. Final simplification 10.5%

    \[\leadsto \frac{c}{b} \]
11. 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 2024026 
(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)))