The quadratic formula (r1)

Percentage Accurate: 63.7% → 87.9%

Time: 18.0s

Alternatives: 8

Speedup: 11.6×

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: 63.7% 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: 87.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e+141)
   (- (/ c b) (/ b a))
   (if (<= b 1.08e+77)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ 0.0 (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e+141) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.08e+77) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 / (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 <= (-7.8d+141)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.08d+77) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = 0.0d0 / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e+141) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.08e+77) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.8e+141:
		tmp = (c / b) - (b / a)
	elif b <= 1.08e+77:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = 0.0 / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.8e+141)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.08e+77)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.8e+141)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.08e+77)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = 0.0 / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.8e+141], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.08e+77], 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[(0.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.79999999999999983e141

   1. Initial program 54.3%

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

   3. Simplified 54.4%

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

   5. Taylor expanded in b around -inf 96.3%

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

      1. +-commutative 96.3%

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

      2. mul-1-neg 96.3%

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

      3. unsub-neg 96.3%

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

   7. Simplified 96.3%

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

   if -7.79999999999999983e141 < b < 1.07999999999999996e77

   1. Initial program 84.1%

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

   if 1.07999999999999996e77 < b

   1. Initial program 27.0%

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

      1. *-commutative 27.0%

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

   3. Simplified 27.0%

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

      1. add-cube-cbrt 27.0%

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

      2. pow3 27.0%

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

      3. associate-*l* 27.0%

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

   6. Applied egg-rr 27.0%

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

   7. Taylor expanded in a around 0 95.3%

      \[\leadsto \frac{\color{blue}{0}}{a \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-182}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-87)
   (- (/ c b) (/ b a))
   (if (<= b 1.26e-182)
     (/ (+ b (sqrt (* c (* a -4.0)))) (* a 2.0))
     (/ 0.0 (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.26e-182) {
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = 0.0 / (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 <= (-8.5d-87)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.26d-182) then
        tmp = (b + sqrt((c * (a * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = 0.0d0 / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.26e-182) {
		tmp = (b + Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = 0.0 / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e-87:
		tmp = (c / b) - (b / a)
	elif b <= 1.26e-182:
		tmp = (b + math.sqrt((c * (a * -4.0)))) / (a * 2.0)
	else:
		tmp = 0.0 / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-87)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.26e-182)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e-87)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.26e-182)
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * 2.0);
	else
		tmp = 0.0 / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-87], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.26e-182], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5000000000000001e-87

   1. Initial program 74.4%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b around -inf 88.5%

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

      1. +-commutative 88.5%

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

      2. mul-1-neg 88.5%

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

      3. unsub-neg 88.5%

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

   7. Simplified 88.5%

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

   if -8.5000000000000001e-87 < b < 1.26e-182

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in b around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. *-commutative 72.9%

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

      3. associate-*r* 72.9%

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

   7. Simplified 72.9%

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

      1. expm1-log1p-u 51.5%

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

      2. expm1-udef 19.4%

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

      3. add-sqr-sqrt 14.0%

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

      4. sqrt-unprod 19.1%

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

      5. sqr-neg 19.1%

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

      6. sqrt-prod 5.3%

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

      7. add-sqr-sqrt 18.7%

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

   9. Applied egg-rr 18.7%

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

       1. expm1-def 50.8%

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

       2. expm1-log1p 70.7%

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

   11. Simplified 70.7%

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

   if 1.26e-182 < b

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

      1. add-cube-cbrt 48.7%

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

      2. pow3 48.7%

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

      3. associate-*l* 48.7%

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

   6. Applied egg-rr 48.7%

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

   7. Taylor expanded in a around 0 80.3%

      \[\leadsto \frac{\color{blue}{0}}{a \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-182}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.65e-87)
   (- (/ c b) (/ b a))
   (if (<= b 1.26e-182)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ 0.0 (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.65e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.26e-182) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 / (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 <= (-1.65d-87)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.26d-182) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = 0.0d0 / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.65e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.26e-182) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = 0.0 / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.65e-87:
		tmp = (c / b) - (b / a)
	elif b <= 1.26e-182:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = 0.0 / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.65e-87)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.26e-182)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(0.0 / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.65e-87)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.26e-182)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = 0.0 / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.65e-87], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.26e-182], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.65e-87

   1. Initial program 74.4%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b around -inf 88.5%

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

      1. +-commutative 88.5%

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

      2. mul-1-neg 88.5%

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

      3. unsub-neg 88.5%

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

   7. Simplified 88.5%

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

   if -1.65e-87 < b < 1.26e-182

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in b around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. *-commutative 72.9%

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

      3. associate-*r* 72.9%

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

   7. Simplified 72.9%

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

   if 1.26e-182 < b

   1. Initial program 48.8%

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

      1. *-commutative 48.8%

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

   3. Simplified 48.8%

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

      1. add-cube-cbrt 48.7%

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

      2. pow3 48.7%

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

      3. associate-*l* 48.7%

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

   6. Applied egg-rr 48.7%

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

   7. Taylor expanded in a around 0 80.3%

      \[\leadsto \frac{\color{blue}{0}}{a \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-296) (- (/ c b) (/ b a)) (/ 0.0 (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-296) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 0.0 / (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 <= (-2d-296)) then
        tmp = (c / b) - (b / a)
    else
        tmp = 0.0d0 / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-296) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 0.0 / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-296:
		tmp = (c / b) - (b / a)
	else:
		tmp = 0.0 / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-296)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(0.0 / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-296)
		tmp = (c / b) - (b / a);
	else
		tmp = 0.0 / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-296], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2e-296

   1. Initial program 73.6%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in b around -inf 70.7%

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

      1. +-commutative 70.7%

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

      2. mul-1-neg 70.7%

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

      3. unsub-neg 70.7%

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

   7. Simplified 70.7%

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

   if -2e-296 < b

   1. Initial program 54.3%

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

      1. *-commutative 54.3%

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

   3. Simplified 54.3%

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

      1. add-cube-cbrt 54.1%

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

      2. pow3 54.1%

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

      3. associate-*l* 54.1%

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

   6. Applied egg-rr 54.1%

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

   7. Taylor expanded in a around 0 68.3%

      \[\leadsto \frac{\color{blue}{0}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-301}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e-301) (/ (- b) a) (/ 0.0 (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-301) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / (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 <= (-9.5d-301)) then
        tmp = -b / a
    else
        tmp = 0.0d0 / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-301) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.5e-301:
		tmp = -b / a
	else:
		tmp = 0.0 / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e-301)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(0.0 / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.5e-301)
		tmp = -b / a;
	else
		tmp = 0.0 / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e-301], N[((-b) / a), $MachinePrecision], N[(0.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.50000000000000032e-301

   1. Initial program 73.8%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in b around -inf 69.4%

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

      1. associate-*r/ 69.4%

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

      2. mul-1-neg 69.4%

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

   7. Simplified 69.4%

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

   if -9.50000000000000032e-301 < b

   1. Initial program 54.0%

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

      1. *-commutative 54.0%

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

   3. Simplified 54.0%

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

      1. add-cube-cbrt 53.8%

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

      2. pow3 53.8%

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

      3. associate-*l* 53.8%

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

   6. Applied egg-rr 53.8%

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

   7. Taylor expanded in a around 0 68.8%

      \[\leadsto \frac{\color{blue}{0}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-301}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 7.2e-46) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.2e-46) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 7.2d-46) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.2e-46) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.2e-46:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.2e-46)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.2e-46)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.2e-46], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.2e-46

   1. Initial program 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in b around -inf 55.5%

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

      1. associate-*r/ 55.5%

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

      2. mul-1-neg 55.5%

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

   7. Simplified 55.5%

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

   if 7.2e-46 < b

   1. Initial program 45.5%

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

   3. Simplified 45.5%

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

   5. Taylor expanded in b around -inf 2.5%

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

      1. +-commutative 2.5%

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

      2. mul-1-neg 2.5%

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

      3. unsub-neg 2.5%

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

   7. Simplified 2.5%

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

   8. Taylor expanded in c around inf 28.2%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-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 <= (-2d-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 <= -2e-310) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-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 <= -2e-310)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 74.2%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in b around -inf 68.5%

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

      1. associate-*r/ 68.5%

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

      2. mul-1-neg 68.5%

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

   7. Simplified 68.5%

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

   if -1.999999999999994e-310 < b

   1. Initial program 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in a around 0 22.6%

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

      1. mul-1-neg 22.6%

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

      2. distribute-neg-frac 22.6%

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

   7. Simplified 22.6%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.7% 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 64.4%

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

3. Simplified 64.5%

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

5. Taylor expanded in b around -inf 38.1%

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

   1. +-commutative 38.1%

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

   2. mul-1-neg 38.1%

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

   3. unsub-neg 38.1%

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

7. Simplified 38.1%

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

8. Taylor expanded in c around inf 11.4%

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

9. Final simplification 11.4%

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

Developer target: 71.1% 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 2024023 
(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)))