Quadratic roots, full range

Percentage Accurate: 51.1% → 85.5%

Time: 15.1s

Alternatives: 8

Speedup: 19.1×

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: 51.1% 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: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+139}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+139)
   (- (/ c b) (/ b a))
   (if (<= b 8.5e-81)
     (- (/ (sqrt (fma b b (* -4.0 (* c a)))) (* a 2.0)) (/ b (* a 2.0)))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+139) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8.5e-81) {
		tmp = (sqrt(fma(b, b, (-4.0 * (c * a)))) / (a * 2.0)) - (b / (a * 2.0));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+139)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 8.5e-81)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))) / Float64(a * 2.0)) - Float64(b / Float64(a * 2.0)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+139], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e-81], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] - N[(b / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000003e139

   1. Initial program 38.8%

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

   2. Taylor expanded in b around -inf 95.4%

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

      1. +-commutative 95.4%

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

      2. mul-1-neg 95.4%

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

      3. unsub-neg 95.4%

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

   4. Simplified 95.4%

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

   if -5.0000000000000003e139 < b < 8.5000000000000001e-81

   1. Initial program 79.2%

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

   3. Simplified 79.2%

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

      1. *-commutative 79.2%

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

      2. metadata-eval 79.2%

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

      3. distribute-lft-neg-in 79.2%

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

      4. distribute-rgt-neg-in 79.2%

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

      5. *-commutative 79.2%

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

      6. fma-neg 79.2%

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

      7. associate-*l* 79.2%

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

   5. Applied egg-rr 79.2%

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

      1. div-sub 79.2%

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

      2. cancel-sign-sub-inv 79.2%

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

      3. fma-def 79.2%

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

      4. metadata-eval 79.2%

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

   7. Applied egg-rr 79.2%

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

   if 8.5000000000000001e-81 < b

   1. Initial program 18.9%

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

   2. Taylor expanded in b around inf 88.7%

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

      1. mul-1-neg 88.7%

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

      2. distribute-neg-frac 88.7%

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

   4. Simplified 88.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+139}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+141}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+141)
   (- (/ c b) (/ b a))
   (if (<= b 1.7e-86)
     (/ (- (sqrt (- (* b b) (* (* c a) 4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+141) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.7e-86) {
		tmp = (sqrt(((b * b) - ((c * a) * 4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d+141)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.7d-86) then
        tmp = (sqrt(((b * b) - ((c * a) * 4.0d0))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+141) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.7e-86) {
		tmp = (Math.sqrt(((b * b) - ((c * a) * 4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+141:
		tmp = (c / b) - (b / a)
	elif b <= 1.7e-86:
		tmp = (math.sqrt(((b * b) - ((c * a) * 4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+141)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.7e-86)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(c * a) * 4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+141)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.7e-86)
		tmp = (sqrt(((b * b) - ((c * a) * 4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+141], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7e-86], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.00000000000000003e141

   1. Initial program 38.8%

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

   2. Taylor expanded in b around -inf 95.4%

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

      1. +-commutative 95.4%

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

      2. mul-1-neg 95.4%

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

      3. unsub-neg 95.4%

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

   4. Simplified 95.4%

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

   if -2.00000000000000003e141 < b < 1.7e-86

   1. Initial program 79.2%

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

   3. Simplified 79.2%

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

      1. *-commutative 79.2%

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

      2. metadata-eval 79.2%

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

      3. distribute-lft-neg-in 79.2%

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

      4. distribute-rgt-neg-in 79.2%

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

      5. *-commutative 79.2%

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

      6. fma-neg 79.2%

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

      7. associate-*l* 79.2%

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

   5. Applied egg-rr 79.2%

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

   if 1.7e-86 < b

   1. Initial program 18.9%

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

   2. Taylor expanded in b around inf 88.7%

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

      1. mul-1-neg 88.7%

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

      2. distribute-neg-frac 88.7%

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

   4. Simplified 88.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+141}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-82}:\\ \;\;\;\;\left(\sqrt{-4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e-34)
   (- (/ c b) (/ b a))
   (if (<= b 1.4e-82)
     (* (- (sqrt (* -4.0 (* c a))) b) (/ 1.0 (* a 2.0)))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-34) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.4e-82) {
		tmp = (sqrt((-4.0 * (c * a))) - b) * (1.0 / (a * 2.0));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.5d-34)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.4d-82) then
        tmp = (sqrt(((-4.0d0) * (c * a))) - b) * (1.0d0 / (a * 2.0d0))
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-34) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.4e-82) {
		tmp = (Math.sqrt((-4.0 * (c * a))) - b) * (1.0 / (a * 2.0));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.5e-34:
		tmp = (c / b) - (b / a)
	elif b <= 1.4e-82:
		tmp = (math.sqrt((-4.0 * (c * a))) - b) * (1.0 / (a * 2.0))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e-34)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.4e-82)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) - b) * Float64(1.0 / Float64(a * 2.0)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.5e-34)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.4e-82)
		tmp = (sqrt((-4.0 * (c * a))) - b) * (1.0 / (a * 2.0));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e-34], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4e-82], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5e-34

   1. Initial program 58.4%

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

   2. Taylor expanded in b around -inf 91.2%

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

      1. +-commutative 91.2%

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

      2. mul-1-neg 91.2%

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

      3. unsub-neg 91.2%

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

   4. Simplified 91.2%

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

   if -3.5e-34 < b < 1.40000000000000012e-82

   1. Initial program 72.7%

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

   3. Simplified 72.7%

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

      1. *-commutative 72.7%

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

      2. metadata-eval 72.7%

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

      3. distribute-lft-neg-in 72.7%

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

      4. distribute-rgt-neg-in 72.7%

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

      5. *-commutative 72.7%

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

      6. fma-neg 72.7%

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

      7. associate-*l* 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. div-inv 72.7%

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

      2. cancel-sign-sub-inv 72.7%

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

      3. fma-def 72.7%

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

      4. metadata-eval 72.7%

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

   7. Applied egg-rr 72.7%

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

   8. Taylor expanded in b around 0 61.4%

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

      1. *-commutative 61.4%

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

   10. Simplified 61.4%

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

   if 1.40000000000000012e-82 < b

   1. Initial program 18.9%

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

   2. Taylor expanded in b around inf 88.7%

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

      1. mul-1-neg 88.7%

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

      2. distribute-neg-frac 88.7%

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

   4. Simplified 88.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-82}:\\ \;\;\;\;\left(\sqrt{-4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e-32)
   (- (/ c b) (/ b a))
   (if (<= b 4.1e-86)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-32) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.1e-86) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.2d-32)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4.1d-86) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-32) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.1e-86) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e-32:
		tmp = (c / b) - (b / a)
	elif b <= 4.1e-86:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e-32)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4.1e-86)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.2e-32)
		tmp = (c / b) - (b / a);
	elseif (b <= 4.1e-86)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e-32], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.1e-86], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2e-32

   1. Initial program 58.4%

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

   2. Taylor expanded in b around -inf 91.2%

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

      1. +-commutative 91.2%

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

      2. mul-1-neg 91.2%

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

      3. unsub-neg 91.2%

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

   4. Simplified 91.2%

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

   if -2.2e-32 < b < 4.09999999999999979e-86

   1. Initial program 72.7%

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

   3. Simplified 72.7%

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

      1. *-commutative 72.7%

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

      2. metadata-eval 72.7%

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

      3. distribute-lft-neg-in 72.7%

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

      4. distribute-rgt-neg-in 72.7%

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

      5. *-commutative 72.7%

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

      6. fma-neg 72.7%

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

      7. associate-*l* 72.7%

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

   5. Applied egg-rr 72.7%

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

   6. Taylor expanded in b around 0 61.4%

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

      1. associate-*r* 61.4%

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

   8. Simplified 61.4%

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

   if 4.09999999999999979e-86 < b

   1. Initial program 18.9%

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

   2. Taylor expanded in b around inf 88.7%

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

      1. mul-1-neg 88.7%

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

      2. distribute-neg-frac 88.7%

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

   4. Simplified 88.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 68.6% accurate, 12.8× 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 62.7%

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

   2. Taylor expanded in b around -inf 74.0%

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

      1. +-commutative 74.0%

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

      2. mul-1-neg 74.0%

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

      3. unsub-neg 74.0%

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

   4. Simplified 74.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.0%

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

   2. Taylor expanded in b around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. distribute-neg-frac 72.7%

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

   4. Simplified 72.7%

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

4. Final simplification 73.4%

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

Alternative 6: 42.9% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 6e+23) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6e+23) {
		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 <= 6d+23) 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 <= 6e+23) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6e+23:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6e+23)
		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 <= 6e+23)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6e+23], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.0000000000000002e23

   1. Initial program 61.7%

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

   2. Taylor expanded in b around -inf 55.2%

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

      1. associate-*r/ 55.2%

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

      2. mul-1-neg 55.2%

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

   4. Simplified 55.2%

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

   if 6.0000000000000002e23 < b

   1. Initial program 14.3%

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

   2. Taylor expanded in b around -inf 2.4%

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

      1. +-commutative 2.4%

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

      2. mul-1-neg 2.4%

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

      3. unsub-neg 2.4%

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

   4. Simplified 2.4%

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

   5. Taylor expanded in c around inf 40.4%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 7: 68.4% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 62.7%

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

   2. Taylor expanded in b around -inf 73.3%

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

      1. associate-*r/ 73.3%

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

      2. mul-1-neg 73.3%

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

   4. Simplified 73.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.0%

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

   2. Taylor expanded in b around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. distribute-neg-frac 72.7%

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

   4. Simplified 72.7%

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

4. Final simplification 73.1%

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

Alternative 8: 10.4% 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 47.5%

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

2. Taylor expanded in b around -inf 39.5%

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

   1. +-commutative 39.5%

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

   2. mul-1-neg 39.5%

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

   3. unsub-neg 39.5%

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

4. Simplified 39.5%

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

5. Taylor expanded in c around inf 14.3%

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

6. Final simplification 14.3%

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

Reproduce

herbie shell --seed 2023278 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))