The quadratic formula (r1)

Percentage Accurate: 64.5% → 87.2%

Time: 16.9s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 64.5% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -24000000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -24000000.0)
   (- (/ c b) (/ b a))
   (if (<= b 5e+148)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ (- b b) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -24000000.0) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5e+148) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (b - b) / (a * 2.0);
	}
	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 <= (-24000000.0d0)) then
        tmp = (c / b) - (b / a)
    else if (b <= 5d+148) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (b - b) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -24000000.0) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5e+148) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (b - b) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -24000000.0:
		tmp = (c / b) - (b / a)
	elif b <= 5e+148:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (b - b) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -24000000.0)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5e+148)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(b - b) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -24000000.0)
		tmp = (c / b) - (b / a);
	elseif (b <= 5e+148)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (b - b) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -24000000.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+148], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(b - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4e7

   1. Initial program 64.8%

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

      1. *-commutative 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in b around -inf 94.5%

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

      1. +-commutative 94.5%

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

      2. mul-1-neg 94.5%

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

      3. unsub-neg 94.5%

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

   7. Simplified 94.5%

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

   if -2.4e7 < b < 5.00000000000000024e148

   1. Initial program 76.8%

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

   if 5.00000000000000024e148 < b

   1. Initial program 6.8%

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

      1. *-commutative 6.8%

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

   3. Simplified 6.8%

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

   5. Taylor expanded in b around inf 100.0%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{b}}{a \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -24000000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-56}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-60}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.85e-56)
   (- (/ c b) (/ b a))
   (if (<= b 1.05e-60)
     (* (/ 0.5 a) (+ b (sqrt (* a (* c -4.0)))))
     (/ (- b b) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e-56) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.05e-60) {
		tmp = (0.5 / a) * (b + sqrt((a * (c * -4.0))));
	} else {
		tmp = (b - b) / (a * 2.0);
	}
	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 <= (-1.85d-56)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.05d-60) then
        tmp = (0.5d0 / a) * (b + sqrt((a * (c * (-4.0d0)))))
    else
        tmp = (b - b) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e-56) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.05e-60) {
		tmp = (0.5 / a) * (b + Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = (b - b) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.85e-56:
		tmp = (c / b) - (b / a)
	elif b <= 1.05e-60:
		tmp = (0.5 / a) * (b + math.sqrt((a * (c * -4.0))))
	else:
		tmp = (b - b) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.85e-56)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.05e-60)
		tmp = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(b - b) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.85e-56)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.05e-60)
		tmp = (0.5 / a) * (b + sqrt((a * (c * -4.0))));
	else
		tmp = (b - b) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.85e-56], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-60], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8500000000000001e-56

   1. Initial program 70.7%

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

      1. *-commutative 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in b around -inf 92.2%

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

      1. +-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

   7. Simplified 92.2%

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

   if -1.8500000000000001e-56 < b < 1.04999999999999996e-60

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in b around 0 63.9%

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

      1. *-commutative 63.9%

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

   7. Simplified 63.9%

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

      1. expm1-log1p-u 50.0%

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

      2. expm1-udef 17.5%

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

      3. add-sqr-sqrt 10.8%

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

      4. sqrt-unprod 17.7%

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

      5. sqr-neg 17.7%

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

      6. sqrt-prod 7.0%

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

      7. add-sqr-sqrt 17.1%

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

      8. associate-*l* 17.1%

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

      9. *-commutative 17.1%

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

   9. Applied egg-rr 17.1%

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

       1. expm1-def 49.6%

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

       2. expm1-log1p 62.7%

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

       3. *-rgt-identity 62.7%

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

       4. associate-*r/ 62.7%

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

       5. *-commutative 62.7%

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

       6. associate-/r* 62.7%

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

       7. metadata-eval 62.7%

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

   11. Simplified 62.7%

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

   if 1.04999999999999996e-60 < b

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in b around inf 78.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{b}}{a \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-56}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-60}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{-57}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.22e-57)
   (- (/ c b) (/ b a))
   (if (<= b 8.6e-61)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- b b) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.22e-57) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.6e-61) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (b - b) / (a * 2.0);
	}
	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 <= (-1.22d-57)) then
        tmp = (c / b) - (b / a)
    else if (b <= 8.6d-61) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (b - b) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.22e-57) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.6e-61) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (b - b) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.22e-57:
		tmp = (c / b) - (b / a)
	elif b <= 8.6e-61:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = (b - b) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.22e-57)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 8.6e-61)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(b - b) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.22e-57)
		tmp = (c / b) - (b / a);
	elseif (b <= 8.6e-61)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = (b - b) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.22e-57], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-61], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(b - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2200000000000001e-57

   1. Initial program 70.7%

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

      1. *-commutative 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in b around -inf 92.2%

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

      1. +-commutative 92.2%

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

      2. mul-1-neg 92.2%

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

      3. unsub-neg 92.2%

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

   7. Simplified 92.2%

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

   if -1.2200000000000001e-57 < b < 8.6000000000000007e-61

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 71.9%

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

      1. prod-diff 71.3%

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

      2. *-commutative 71.3%

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

      3. fma-def 71.3%

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

      4. associate-+l+ 71.3%

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

      5. pow2 71.3%

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

      6. distribute-lft-neg-in 71.3%

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

      7. *-commutative 71.3%

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

      8. distribute-rgt-neg-in 71.3%

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

      9. metadata-eval 71.3%

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

      10. associate-*r* 71.3%

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

      11. *-commutative 71.3%

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

      12. *-commutative 71.3%

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

      13. fma-udef 71.3%

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

   6. Applied egg-rr 71.3%

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

      1. fma-def 71.3%

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

      2. fma-def 71.3%

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

      3. *-commutative 71.3%

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

   8. Simplified 71.3%

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

   9. Taylor expanded in b around 0 62.1%

      \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. distribute-rgt-out 63.9%

          \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 + 4\right)}} + -1 \cdot b}{a \cdot 2} \]

       2. metadata-eval 63.9%

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

       3. associate-*r* 63.9%

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

       4. mul-1-neg 63.9%

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

       5. unsub-neg 63.9%

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

   11. Simplified 63.9%

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

   if 8.6000000000000007e-61 < b

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in b around inf 78.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{b}}{a \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{-57}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+250} \lor \neg \left(b \leq 5.2 \cdot 10^{+284}\right):\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{-a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4e-308)
   (/ (- b) a)
   (if (or (<= b 6.2e+250) (not (<= b 5.2e+284)))
     (/ (* c (/ a b)) (- a))
     (* a (/ c (* b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4e-308) {
		tmp = -b / a;
	} else if ((b <= 6.2e+250) || !(b <= 5.2e+284)) {
		tmp = (c * (a / b)) / -a;
	} else {
		tmp = a * (c / (b * a));
	}
	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-308) then
        tmp = -b / a
    else if ((b <= 6.2d+250) .or. (.not. (b <= 5.2d+284))) then
        tmp = (c * (a / b)) / -a
    else
        tmp = a * (c / (b * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4e-308) {
		tmp = -b / a;
	} else if ((b <= 6.2e+250) || !(b <= 5.2e+284)) {
		tmp = (c * (a / b)) / -a;
	} else {
		tmp = a * (c / (b * a));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4e-308:
		tmp = -b / a
	elif (b <= 6.2e+250) or not (b <= 5.2e+284):
		tmp = (c * (a / b)) / -a
	else:
		tmp = a * (c / (b * a))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4e-308)
		tmp = Float64(Float64(-b) / a);
	elseif ((b <= 6.2e+250) || !(b <= 5.2e+284))
		tmp = Float64(Float64(c * Float64(a / b)) / Float64(-a));
	else
		tmp = Float64(a * Float64(c / Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4e-308)
		tmp = -b / a;
	elseif ((b <= 6.2e+250) || ~((b <= 5.2e+284)))
		tmp = (c * (a / b)) / -a;
	else
		tmp = a * (c / (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4e-308], N[((-b) / a), $MachinePrecision], If[Or[LessEqual[b, 6.2e+250], N[Not[LessEqual[b, 5.2e+284]], $MachinePrecision]], N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(a * N[(c / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.00000000000000013e-308

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. mul-1-neg 67.1%

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

   7. Simplified 67.1%

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

   if 4.00000000000000013e-308 < b < 6.2000000000000001e250 or 5.1999999999999997e284 < b

   1. Initial program 59.3%

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

      1. *-commutative 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in b around inf 33.9%

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

      1. *-commutative 33.9%

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

      2. associate-/l* 37.4%

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

   7. Simplified 37.4%

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

      1. *-commutative 37.4%

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

      2. *-commutative 37.4%

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

      3. times-frac 37.4%

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

   9. Applied egg-rr 37.4%

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

       1. expm1-log1p-u 35.6%

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

       2. expm1-udef 48.4%

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

       3. associate-*r/ 48.4%

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

       4. div-inv 48.4%

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

       5. associate-*l* 48.4%

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

       6. div-inv 48.4%

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

       7. clear-num 47.6%

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

       8. metadata-eval 47.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\left(c \cdot \frac{a}{b}\right) \cdot \left(\color{blue}{0.5} \cdot -2\right)}{a}\right)} - 1 \]

       9. metadata-eval 47.6%

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

   11. Applied egg-rr 47.6%

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

       1. expm1-def 34.7%

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

       2. expm1-log1p 36.6%

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

       3. /-rgt-identity 36.6%

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

       4. associate-/l* 36.6%

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

       5. metadata-eval 36.6%

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

       6. associate-/r* 36.6%

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

       7. neg-mul-1 36.6%

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

   13. Simplified 36.6%

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

   if 6.2000000000000001e250 < b < 5.1999999999999997e284

   1. Initial program 1.4%

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

      1. *-commutative 1.4%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in b around inf 48.1%

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

      1. *-commutative 48.1%

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

      2. associate-/l* 48.4%

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

   7. Simplified 48.4%

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

      1. *-commutative 48.4%

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

      2. *-commutative 48.4%

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

      3. times-frac 48.4%

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

   9. Applied egg-rr 48.4%

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

       1. frac-2neg 48.4%

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

       2. metadata-eval 48.4%

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

       3. clear-num 48.4%

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

       4. frac-times 48.4%

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

       5. distribute-neg-frac 48.4%

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

       6. add-sqr-sqrt 10.6%

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

       7. sqrt-unprod 47.9%

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

       8. sqr-neg 47.9%

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

       9. sqrt-prod 37.8%

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

       10. add-sqr-sqrt 48.4%

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

       11. *-commutative 48.4%

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

       12. *-un-lft-identity 48.4%

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

       13. div-inv 48.4%

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

       14. clear-num 48.4%

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

       15. div-inv 48.4%

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

       16. metadata-eval 48.4%

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

   11. Applied egg-rr 48.4%

       \[\leadsto \color{blue}{\frac{c \cdot \frac{a}{b}}{-2 \cdot \left(a \cdot -0.5\right)}} \]
   12. Step-by-step derivation

       1. *-commutative 48.4%

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

       2. associate-*l* 48.4%

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

       3. metadata-eval 48.4%

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

       4. *-rgt-identity 48.4%

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

       5. *-rgt-identity 48.4%

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

       6. *-commutative 48.4%

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

       7. metadata-eval 48.4%

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

       8. times-frac 48.4%

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

       9. neg-mul-1 48.4%

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

       10. associate-*r/ 48.1%

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

       11. distribute-neg-frac 48.1%

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

       12. distribute-rgt-neg-out 48.1%

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

       13. neg-mul-1 48.1%

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

       14. associate-/r* 73.3%

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

       15. distribute-rgt-neg-in 73.3%

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

       16. neg-mul-1 73.3%

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

       17. *-commutative 73.3%

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

       18. times-frac 91.5%

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

   13. Simplified 91.5%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+250} \lor \neg \left(b \leq 5.2 \cdot 10^{+284}\right):\\ \;\;\;\;\frac{c \cdot \frac{a}{b}}{-a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c}{b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.1e-308)
   (/ (- b) a)
   (if (<= b 4.8e+134) (/ (- c) b) (* a (/ c (* b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.1e-308) {
		tmp = -b / a;
	} else if (b <= 4.8e+134) {
		tmp = -c / b;
	} else {
		tmp = a * (c / (b * a));
	}
	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 <= 1.1d-308) then
        tmp = -b / a
    else if (b <= 4.8d+134) then
        tmp = -c / b
    else
        tmp = a * (c / (b * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.1e-308) {
		tmp = -b / a;
	} else if (b <= 4.8e+134) {
		tmp = -c / b;
	} else {
		tmp = a * (c / (b * a));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.1e-308:
		tmp = -b / a
	elif b <= 4.8e+134:
		tmp = -c / b
	else:
		tmp = a * (c / (b * a))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.1e-308)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.8e+134)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(a * Float64(c / Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.1e-308)
		tmp = -b / a;
	elseif (b <= 4.8e+134)
		tmp = -c / b;
	else
		tmp = a * (c / (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.1e-308], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.8e+134], N[((-c) / b), $MachinePrecision], N[(a * N[(c / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.1000000000000001e-308

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. mul-1-neg 67.1%

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

   7. Simplified 67.1%

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

   if 1.1000000000000001e-308 < b < 4.80000000000000011e134

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in b around inf 18.3%

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

      1. mul-1-neg 18.3%

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

      2. distribute-neg-frac 18.3%

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

   7. Simplified 18.3%

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

   if 4.80000000000000011e134 < b

   1. Initial program 16.9%

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

      1. *-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in b around inf 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-/l* 62.3%

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

   7. Simplified 62.3%

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

      1. *-commutative 62.3%

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

      2. *-commutative 62.3%

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

      3. times-frac 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. frac-2neg 62.3%

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

       2. metadata-eval 62.3%

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

       3. clear-num 62.3%

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

       4. frac-times 62.3%

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

       5. distribute-neg-frac 62.3%

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

       6. add-sqr-sqrt 26.6%

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

       7. sqrt-unprod 56.6%

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

       8. sqr-neg 56.6%

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

       9. sqrt-prod 35.7%

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

       10. add-sqr-sqrt 60.9%

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

       11. *-commutative 60.9%

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

       12. *-un-lft-identity 60.9%

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

       13. div-inv 60.9%

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

       14. clear-num 60.9%

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

       15. div-inv 60.9%

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

       16. metadata-eval 60.9%

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

   11. Applied egg-rr 60.9%

       \[\leadsto \color{blue}{\frac{c \cdot \frac{a}{b}}{-2 \cdot \left(a \cdot -0.5\right)}} \]
   12. Step-by-step derivation

       1. *-commutative 60.9%

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

       2. associate-*l* 60.9%

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

       3. metadata-eval 60.9%

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

       4. *-rgt-identity 60.9%

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

       5. *-rgt-identity 60.9%

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

       6. *-commutative 60.9%

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

       7. metadata-eval 60.9%

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

       8. times-frac 60.9%

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

       9. neg-mul-1 60.9%

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

       10. associate-*r/ 54.1%

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

       11. distribute-neg-frac 54.1%

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

       12. distribute-rgt-neg-out 54.1%

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

       13. neg-mul-1 54.1%

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

       14. associate-/r* 54.0%

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

       15. distribute-rgt-neg-in 54.0%

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

       16. neg-mul-1 54.0%

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

       17. *-commutative 54.0%

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

       18. times-frac 62.2%

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

   13. Simplified 62.2%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c}{b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-300}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.4e-300) (/ (- b) a) (/ (- b b) (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-300) {
		tmp = -b / a;
	} else {
		tmp = (b - b) / (a * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.4d-300)) then
        tmp = -b / a
    else
        tmp = (b - b) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-300) {
		tmp = -b / a;
	} else {
		tmp = (b - b) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.4e-300:
		tmp = -b / a
	else:
		tmp = (b - b) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.4e-300)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(b - b) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.4e-300)
		tmp = -b / a;
	else
		tmp = (b - b) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.4e-300], N[((-b) / a), $MachinePrecision], N[(N[(b - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.4000000000000003e-300

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in b around -inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. mul-1-neg 68.1%

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

   7. Simplified 68.1%

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

   if -7.4000000000000003e-300 < b

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in b around inf 58.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{b}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-300}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 4.1e+14) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.1e+14) {
		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 <= 4.1d+14) 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 <= 4.1e+14) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.1e+14:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.1e+14)
		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 <= 4.1e+14)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.1e+14], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.1e14

   1. Initial program 71.1%

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

      1. *-commutative 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in b around -inf 49.8%

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

      1. associate-*r/ 49.8%

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

      2. mul-1-neg 49.8%

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

   7. Simplified 49.8%

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

   if 4.1e14 < b

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

   3. Simplified 47.5%

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

   6. Applied egg-rr 6.5%

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

   7. Taylor expanded in b around -inf 29.7%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.3% accurate, 12.9× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-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 <= (-5d-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 <= -5e-310) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-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 <= -5e-310)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. mul-1-neg 67.1%

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

   7. Simplified 67.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 54.4%

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

      1. *-commutative 54.4%

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

   3. Simplified 54.4%

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

   5. Taylor expanded in b around inf 27.1%

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

      1. mul-1-neg 27.1%

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

      2. distribute-neg-frac 27.1%

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

   7. Simplified 27.1%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
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 63.5%

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

   1. *-commutative 63.5%

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

3. Simplified 63.5%

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

6. Applied egg-rr 29.3%

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

7. Taylor expanded in a around 0 2.6%

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

8. Final simplification 2.6%

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

Alternative 10: 10.9% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

   1. *-commutative 63.5%

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

3. Simplified 63.5%

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

6. Applied egg-rr 29.3%

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

7. Taylor expanded in b around -inf 11.6%

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

8. Final simplification 11.6%

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

Developer target: 70.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - t\_0}{2 \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* a (/ (- (- b) t_0) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	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 * a) * c)))
    if (b < 0.0d0) then
        tmp = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-b - t_0) / (2.0d0 * a)))
    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 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b < 0.0:
		tmp = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-b - t_0) / (2.0 * a)))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b < 0.0)
		tmp = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024031 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

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