Quadratic roots, full range

Percentage Accurate: 51.2% → 85.5%

Time: 11.3s

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: 51.2% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+158}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e+158)
   (- (/ c b) (/ b a))
   (if (<= b 1.9e-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 <= -8e+158) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.9e-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 <= (-8d+158)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.9d-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 <= -8e+158) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.9e-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 <= -8e+158:
		tmp = (c / b) - (b / a)
	elif b <= 1.9e-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 <= -8e+158)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.9e-86)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - 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 <= -8e+158)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.9e-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, -8e+158], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e-86], 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[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.99999999999999962e158

   1. Initial program 39.4%

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

      1. *-commutative 39.4%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in b around -inf 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   6. Simplified 94.8%

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

   if -7.99999999999999962e158 < b < 1.9e-86

   1. Initial program 78.2%

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

   if 1.9e-86 < b

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

   3. Simplified 17.6%

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

   4. Taylor expanded in b around inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. distribute-neg-frac 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+158}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+14}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.18e+14)
   (/ (- b) a)
   (if (<= b 6.8e-82)
     (* (/ 0.5 a) (+ b (sqrt (* -4.0 (* c a)))))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.18e+14) {
		tmp = -b / a;
	} else if (b <= 6.8e-82) {
		tmp = (0.5 / a) * (b + sqrt((-4.0 * (c * 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 <= (-1.18d+14)) then
        tmp = -b / a
    else if (b <= 6.8d-82) then
        tmp = (0.5d0 / a) * (b + sqrt(((-4.0d0) * (c * 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 <= -1.18e+14) {
		tmp = -b / a;
	} else if (b <= 6.8e-82) {
		tmp = (0.5 / a) * (b + Math.sqrt((-4.0 * (c * a))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.18e+14:
		tmp = -b / a
	elif b <= 6.8e-82:
		tmp = (0.5 / a) * (b + math.sqrt((-4.0 * (c * a))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.18e+14)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 6.8e-82)
		tmp = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.18e+14)
		tmp = -b / a;
	elseif (b <= 6.8e-82)
		tmp = (0.5 / a) * (b + sqrt((-4.0 * (c * a))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.18e+14], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 6.8e-82], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.18e14

   1. Initial program 66.9%

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

      1. *-commutative 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in b around -inf 90.4%

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

      1. associate-*r/ 90.4%

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

      2. mul-1-neg 90.4%

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

   6. Simplified 90.4%

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

   if -1.18e14 < b < 6.7999999999999995e-82

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

   3. Simplified 70.5%

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

      1. clear-num 70.2%

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

      2. associate-/r/ 70.3%

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

      3. *-commutative 70.3%

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

      4. associate-/r* 70.3%

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

      5. metadata-eval 70.3%

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

      6. add-sqr-sqrt 47.5%

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

      7. sqrt-unprod 69.6%

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

      8. sqr-neg 69.6%

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

      9. sqrt-prod 23.0%

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

      10. add-sqr-sqrt 62.1%

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

      11. fma-neg 62.1%

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

      12. *-commutative 62.1%

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

      13. distribute-rgt-neg-in 62.1%

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

      14. *-commutative 62.1%

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

      15. distribute-rgt-neg-in 62.1%

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

      16. metadata-eval 62.1%

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

   5. Applied egg-rr 62.1%

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

   6. Taylor expanded in b around 0 62.0%

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

   if 6.7999999999999995e-82 < b

   1. Initial program 16.9%

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

      1. *-commutative 16.9%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in b around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. distribute-neg-frac 86.9%

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

   6. Simplified 86.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+14}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+14}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.18e+14)
   (/ (- b) a)
   (if (<= b 4.5e-86)
     (/ (+ b (sqrt (* a (* c -4.0)))) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.18e+14) {
		tmp = -b / a;
	} else if (b <= 4.5e-86) {
		tmp = (b + sqrt((a * (c * -4.0)))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.18d+14)) then
        tmp = -b / a
    else if (b <= 4.5d-86) then
        tmp = (b + sqrt((a * (c * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.18e+14) {
		tmp = -b / a;
	} else if (b <= 4.5e-86) {
		tmp = (b + Math.sqrt((a * (c * -4.0)))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.18e+14:
		tmp = -b / a
	elif b <= 4.5e-86:
		tmp = (b + math.sqrt((a * (c * -4.0)))) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.18e+14)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.5e-86)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(c * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.18e+14)
		tmp = -b / a;
	elseif (b <= 4.5e-86)
		tmp = (b + sqrt((a * (c * -4.0)))) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.18e+14], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.5e-86], N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.18e14

   1. Initial program 66.9%

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

      1. *-commutative 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in b around -inf 90.4%

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

      1. associate-*r/ 90.4%

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

      2. mul-1-neg 90.4%

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

   6. Simplified 90.4%

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

   if -1.18e14 < b < 4.4999999999999998e-86

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

   3. Simplified 71.0%

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

      1. add-cbrt-cube 47.9%

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

      2. pow3 47.9%

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

      3. add-sqr-sqrt 32.0%

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

      4. sqrt-unprod 47.9%

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

      5. sqr-neg 47.9%

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

      6. sqrt-prod 15.9%

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

      7. add-sqr-sqrt 43.2%

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

      8. fma-neg 43.2%

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

      9. *-commutative 43.2%

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

      10. distribute-rgt-neg-in 43.2%

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

      11. *-commutative 43.2%

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

      12. distribute-rgt-neg-in 43.2%

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

      13. metadata-eval 43.2%

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

   5. Applied egg-rr 43.2%

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

   6. Taylor expanded in b around 0 43.1%

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

      1. rem-cbrt-cube 62.3%

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

      2. +-commutative 62.3%

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

      3. *-commutative 62.3%

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

      4. associate-*l* 62.3%

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

   8. Applied egg-rr 62.3%

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

   if 4.4999999999999998e-86 < b

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

   3. Simplified 17.6%

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

   4. Taylor expanded in b around inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. distribute-neg-frac 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+14}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+14}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.18e+14)
   (/ (- b) a)
   (if (<= b 4.6e-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 <= -1.18e+14) {
		tmp = -b / a;
	} else if (b <= 4.6e-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 <= (-1.18d+14)) then
        tmp = -b / a
    else if (b <= 4.6d-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 <= -1.18e+14) {
		tmp = -b / a;
	} else if (b <= 4.6e-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 <= -1.18e+14:
		tmp = -b / a
	elif b <= 4.6e-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 <= -1.18e+14)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.6e-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 <= -1.18e+14)
		tmp = -b / a;
	elseif (b <= 4.6e-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, -1.18e+14], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.6e-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 < -1.18e14

   1. Initial program 66.9%

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

      1. *-commutative 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in b around -inf 90.4%

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

      1. associate-*r/ 90.4%

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

      2. mul-1-neg 90.4%

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

   6. Simplified 90.4%

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

   if -1.18e14 < b < 4.59999999999999992e-86

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in b around 0 64.0%

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

      1. *-commutative 64.0%

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

      2. *-commutative 64.0%

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

      3. associate-*r* 64.0%

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

   6. Simplified 64.0%

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

   if 4.59999999999999992e-86 < b

   1. Initial program 17.6%

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

      1. *-commutative 17.6%

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

   3. Simplified 17.6%

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

   4. Taylor expanded in b around inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. distribute-neg-frac 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+14}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 68.4% 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 72.3%

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

      1. *-commutative 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in b around -inf 61.6%

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

      1. +-commutative 61.6%

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

      2. mul-1-neg 61.6%

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

      3. unsub-neg 61.6%

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

   6. Simplified 61.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.8%

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

      1. *-commutative 26.8%

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

   3. Simplified 26.8%

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

   4. Taylor expanded in b around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. distribute-neg-frac 72.7%

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

   6. Simplified 72.7%

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

4. Final simplification 67.6%

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

Alternative 6: 43.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-309}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -9e-309) (/ (- b) a) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-309) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / 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) :: tmp
    if (b <= (-9d-309)) then
        tmp = -b / a
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-309) {
		tmp = -b / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9e-309:
		tmp = -b / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-309)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9e-309)
		tmp = -b / a;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-309], N[((-b) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.0000000000000021e-309

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in b around -inf 62.5%

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

      1. associate-*r/ 62.5%

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

      2. mul-1-neg 62.5%

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

   6. Simplified 62.5%

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

   if -9.0000000000000021e-309 < b

   1. Initial program 27.9%

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

      1. *-commutative 27.9%

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

   3. Simplified 27.9%

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

      1. add-sqr-sqrt 25.4%

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

      2. pow2 25.4%

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

      3. pow1/2 25.4%

         \[\leadsto \frac{\left(-b\right) + {\left(\sqrt{\color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{0.5}}}\right)}^{2}}{a \cdot 2} \]

      4. sqrt-pow1 25.4%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a \cdot 2} \]

      5. fma-neg 25.4%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

      6. *-commutative 25.4%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(4 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

      7. distribute-rgt-neg-in 25.4%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

      8. *-commutative 25.4%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

      9. distribute-rgt-neg-in 25.4%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

      10. metadata-eval 25.4%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

      11. metadata-eval 25.4%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a \cdot 2} \]

   5. Applied egg-rr 25.4%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}\right)}^{2}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. div-inv 25.4%

         \[\leadsto \color{blue}{\left(\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{a \cdot 2}} \]

   7. Applied egg-rr 28.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot \frac{0.5}{a}} \]

   8. Taylor expanded in c around 0 16.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
   9. Step-by-step derivation

      1. distribute-rgt1-in 16.7%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot b}}{a} \]

      2. metadata-eval 16.7%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{0} \cdot b}{a} \]

      3. mul0-lft 16.7%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{0}}{a} \]

      4. associate-*r/ 16.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot 0}{a}} \]

      5. metadata-eval 16.7%

         \[\leadsto \frac{\color{blue}{0}}{a} \]

   10. Simplified 16.7%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-309}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 7: 68.2% 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 72.3%

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

      1. *-commutative 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in b around -inf 61.5%

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

      1. associate-*r/ 61.5%

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

      2. mul-1-neg 61.5%

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

   6. Simplified 61.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.8%

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

      1. *-commutative 26.8%

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

   3. Simplified 26.8%

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

   4. Taylor expanded in b around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. distribute-neg-frac 72.7%

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

   6. Simplified 72.7%

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

4. Final simplification 67.6%

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

Alternative 8: 10.9% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 47.8%

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

   1. *-commutative 47.8%

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

3. Simplified 47.8%

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

   1. add-sqr-sqrt 46.3%

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

   2. pow2 46.3%

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

   3. pow1/2 46.3%

      \[\leadsto \frac{\left(-b\right) + {\left(\sqrt{\color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{0.5}}}\right)}^{2}}{a \cdot 2} \]

   4. sqrt-pow1 46.3%

      \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a \cdot 2} \]

   5. fma-neg 46.4%

      \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

   6. *-commutative 46.4%

      \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(4 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

   7. distribute-rgt-neg-in 46.4%

      \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

   8. *-commutative 46.4%

      \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

   9. distribute-rgt-neg-in 46.4%

      \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

   10. metadata-eval 46.4%

       \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a \cdot 2} \]

   11. metadata-eval 46.4%

       \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a \cdot 2} \]

5. Applied egg-rr 46.4%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}\right)}^{2}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. div-inv 46.3%

      \[\leadsto \color{blue}{\left(\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{a \cdot 2}} \]

7. Applied egg-rr 45.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot \frac{0.5}{a}} \]

8. Taylor expanded in c around 0 10.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
9. Step-by-step derivation

   1. distribute-rgt1-in 10.4%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot b}}{a} \]

   2. metadata-eval 10.4%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{0} \cdot b}{a} \]

   3. mul0-lft 10.4%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{0}}{a} \]

   4. associate-*r/ 10.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot 0}{a}} \]

   5. metadata-eval 10.4%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

10. Simplified 10.4%

    \[\leadsto \color{blue}{\frac{0}{a}} \]

11. Final simplification 10.4%

    \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023308 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))