The quadratic formula (r1)

Percentage Accurate: 51.8% → 86.1%

Time: 10.4s

Alternatives: 9

Speedup: 2.5×

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: 51.8% 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: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+158}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.02e+158)
   (/ (- b) a)
   (if (<= b 8.4e-88)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e+158) {
		tmp = -b / a;
	} else if (b <= 8.4e-88) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * 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 <= (-1.02d+158)) then
        tmp = -b / a
    else if (b <= 8.4d-88) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (2.0d0 * 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 <= -1.02e+158) {
		tmp = -b / a;
	} else if (b <= 8.4e-88) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.02e+158:
		tmp = -b / a
	elif b <= 8.4e-88:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.02e+158)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 8.4e-88)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.02e+158)
		tmp = -b / a;
	elseif (b <= 8.4e-88)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (2.0 * a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.02e+158], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 8.4e-88], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.02e158

   1. Initial program 49.1%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\right)}}{a} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}} \]

      4. lower-neg.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -1.02e158 < b < 8.3999999999999998e-88

   1. Initial program 81.1%

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

   if 8.3999999999999998e-88 < b

   1. Initial program 17.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      4. lower-neg.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+158}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+144}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-88}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.7e+144)
   (/ (- b) a)
   (if (<= b 8.4e-88)
     (* (- (sqrt (fma (* -4.0 c) a (* b b))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.7e+144) {
		tmp = -b / a;
	} else if (b <= 8.4e-88) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.7e+144)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 8.4e-88)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.7e+144], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 8.4e-88], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.70000000000000015e144

   1. Initial program 53.6%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\right)}}{a} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}} \]

      4. lower-neg.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if -2.70000000000000015e144 < b < 8.3999999999999998e-88

   1. Initial program 80.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 80.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 80.3

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

   4. Applied rewrites 80.3%

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

   if 8.3999999999999998e-88 < b

   1. Initial program 17.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      4. lower-neg.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+144}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-88}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e-111)
   (- (/ c b) (/ b a))
   (if (<= b 5.8e-88)
     (/ (- (sqrt (* (* c a) -4.0)) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-111) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.8e-88) {
		tmp = (sqrt(((c * a) * -4.0)) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.5d-111)) then
        tmp = (c / b) - (b / a)
    else if (b <= 5.8d-88) then
        tmp = (sqrt(((c * a) * (-4.0d0))) - b) / (2.0d0 * a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-111) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.8e-88) {
		tmp = (Math.sqrt(((c * a) * -4.0)) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.5e-111:
		tmp = (c / b) - (b / a)
	elif b <= 5.8e-88:
		tmp = (math.sqrt(((c * a) * -4.0)) - b) / (2.0 * a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e-111)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5.8e-88)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -4.0)) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.5e-111)
		tmp = (c / b) - (b / a);
	elseif (b <= 5.8e-88)
		tmp = (sqrt(((c * a) * -4.0)) - b) / (2.0 * a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e-111], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e-88], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5e-111

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 85.8

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

   5. Applied rewrites 85.8%

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

   7. Applied rewrites 85.8%

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

   if -3.5e-111 < b < 5.8000000000000003e-88

   1. Initial program 69.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if 5.8000000000000003e-88 < b

   1. Initial program 17.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      4. lower-neg.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot a\right) \cdot -4} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e-111)
   (- (/ c b) (/ b a))
   (if (<= b 5.8e-88)
     (* (- (sqrt (* (* c a) -4.0)) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-111) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.8e-88) {
		tmp = (sqrt(((c * a) * -4.0)) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.5d-111)) then
        tmp = (c / b) - (b / a)
    else if (b <= 5.8d-88) then
        tmp = (sqrt(((c * a) * (-4.0d0))) - b) * (0.5d0 / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-111) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.8e-88) {
		tmp = (Math.sqrt(((c * a) * -4.0)) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.5e-111:
		tmp = (c / b) - (b / a)
	elif b <= 5.8e-88:
		tmp = (math.sqrt(((c * a) * -4.0)) - b) * (0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e-111)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5.8e-88)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -4.0)) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.5e-111)
		tmp = (c / b) - (b / a);
	elseif (b <= 5.8e-88)
		tmp = (sqrt(((c * a) * -4.0)) - b) * (0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e-111], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e-88], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5e-111

   1. Initial program 74.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 85.8

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

   5. Applied rewrites 85.8%

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

   7. Applied rewrites 85.8%

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

   if -3.5e-111 < b < 5.8000000000000003e-88

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 69.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 69.6

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 68.1

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

   7. Applied rewrites 68.1%

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

   if 5.8000000000000003e-88 < b

   1. Initial program 17.8%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      4. lower-neg.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\left(\sqrt{\left(c \cdot a\right) \cdot -4} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-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 < -4.999999999999985e-310

   1. Initial program 72.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 68.7

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

   5. Applied rewrites 68.7%

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

   7. Applied rewrites 71.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.5%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      4. lower-neg.f64 70.7

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

   5. Applied rewrites 70.7%

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

Alternative 6: 67.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-306}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2e-306) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-306) {
		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 <= 5.2d-306) 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 <= 5.2e-306) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2e-306:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2e-306)
		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 <= 5.2e-306)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2e-306], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2000000000000001e-306

   1. Initial program 72.4%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\right)}}{a} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}} \]

      4. lower-neg.f64 70.2

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

   5. Applied rewrites 70.2%

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

   if 5.2000000000000001e-306 < b

   1. Initial program 30.0%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      4. lower-neg.f64 71.2

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

   5. Applied rewrites 71.2%

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

Alternative 7: 43.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 135000000000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 135000000000.0) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 135000000000.0) {
		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 <= 135000000000.0d0) 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 <= 135000000000.0) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 135000000000.0:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 135000000000.0)
		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 <= 135000000000.0)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 135000000000.0], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35e11

   1. Initial program 67.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\right)}}{a} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\right)}{a}} \]

      4. lower-neg.f64 51.2

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

   5. Applied rewrites 51.2%

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

   if 1.35e11 < b

   1. Initial program 12.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 27.9%

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

Alternative 8: 11.0% accurate, 4.2× 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 51.5%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. distribute-neg-in N/A

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

   4. mul-1-neg N/A

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

   5. distribute-lft-neg-out N/A

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

   6. remove-double-neg N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. distribute-frac-neg N/A

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

   14. lower-/.f64 N/A

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

   15. lower-neg.f64 35.6

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

5. Applied rewrites 35.6%

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

6. Taylor expanded in c around inf

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

8. Applied rewrites 10.3%

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

Alternative 9: 2.6% accurate, 4.2× 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 51.5%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. distribute-neg-in N/A

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

   4. mul-1-neg N/A

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

   5. distribute-lft-neg-out N/A

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

   6. remove-double-neg N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. distribute-frac-neg N/A

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

   14. lower-/.f64 N/A

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

   15. lower-neg.f64 35.6

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

5. Applied rewrites 35.6%

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

7. Applied rewrites 2.3%

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

8. Taylor expanded in c around 0

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

10. Applied rewrites 2.5%

    \[\leadsto \frac{b}{\color{blue}{a}} \]
11. Add Preprocessing

Developer Target 1: 70.6% accurate, 0.7× 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 2024237 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (- (* b b) (* (* 4 a) c)))) (let ((r1 (/ (+ (- b) (sqrt d)) (* 2 a)))) (let ((r2 (/ (- (- b) (sqrt d)) (* 2 a)))) (if (< b 0) r1 (/ c (* a r2)))))))

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