Quadratic roots, full range

Percentage Accurate: 52.3% → 85.2%

Time: 22.2s

Alternatives: 8

Speedup: 12.9×

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: 52.3% 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: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+118}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e+118)
   (/ b (- a))
   (if (<= b 1.6e-141)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e+118) {
		tmp = b / -a;
	} else if (b <= 1.6e-141) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - 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 <= (-1.35d+118)) then
        tmp = b / -a
    else if (b <= 1.6d-141) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - 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 <= -1.35e+118) {
		tmp = b / -a;
	} else if (b <= 1.6e-141) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.35e+118:
		tmp = b / -a
	elif b <= 1.6e-141:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e+118)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.6e-141)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.35e+118)
		tmp = b / -a;
	elseif (b <= 1.6e-141)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e+118], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.6e-141], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $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 < -1.35e118

   1. Initial program 50.5%

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

      1. *-commutative 50.5%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in b around -inf 96.9%

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

      1. mul-1-neg 96.9%

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

      2. distribute-neg-frac2 96.9%

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

   7. Simplified 96.9%

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

   if -1.35e118 < b < 1.6000000000000001e-141

   1. Initial program 87.4%

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

   if 1.6000000000000001e-141 < b

   1. Initial program 19.3%

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

      1. *-commutative 19.3%

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

   3. Simplified 19.3%

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

   5. Taylor expanded in b around inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. mul-1-neg 84.3%

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

   7. Simplified 84.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+118}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-144}:\\ \;\;\;\;\frac{{\left(\frac{-0.25}{a \cdot c}\right)}^{-0.5} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-38)
   (/ b (- a))
   (if (<= b 1.05e-144)
     (/ (- (pow (/ -0.25 (* a c)) -0.5) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-38) {
		tmp = b / -a;
	} else if (b <= 1.05e-144) {
		tmp = (pow((-0.25 / (a * c)), -0.5) - 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 <= (-3.8d-38)) then
        tmp = b / -a
    else if (b <= 1.05d-144) then
        tmp = ((((-0.25d0) / (a * c)) ** (-0.5d0)) - 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 <= -3.8e-38) {
		tmp = b / -a;
	} else if (b <= 1.05e-144) {
		tmp = (Math.pow((-0.25 / (a * c)), -0.5) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-38:
		tmp = b / -a
	elif b <= 1.05e-144:
		tmp = (math.pow((-0.25 / (a * c)), -0.5) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-38)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.05e-144)
		tmp = Float64(Float64((Float64(-0.25 / Float64(a * c)) ^ -0.5) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.8e-38)
		tmp = b / -a;
	elseif (b <= 1.05e-144)
		tmp = (((-0.25 / (a * c)) ^ -0.5) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-38], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.05e-144], N[(N[(N[Power[N[(-0.25 / N[(a * c), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.8e-38

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in b around -inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-neg-frac2 87.7%

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

   7. Simplified 87.7%

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

   if -3.8e-38 < b < 1.0500000000000001e-144

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

   3. Simplified 83.9%

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

      1. flip-- 55.7%

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

      2. clear-num 55.8%

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

      3. fma-define 55.8%

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

      4. *-commutative 55.8%

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

      5. associate-*l* 55.8%

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

      6. pow2 55.8%

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

      7. pow2 55.8%

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

      8. pow-prod-up 55.8%

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

      9. metadata-eval 55.8%

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

      10. pow2 55.8%

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

      11. *-commutative 55.8%

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

      12. associate-*l* 55.8%

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

   6. Applied egg-rr 55.8%

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

   7. Taylor expanded in b around 0 73.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{1}{\color{blue}{\frac{-0.25}{a \cdot c}}}}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. inv-pow 73.0%

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

      2. sqrt-pow1 73.1%

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

      3. metadata-eval 73.1%

         \[\leadsto \frac{\left(-b\right) + {\left(\frac{-0.25}{a \cdot c}\right)}^{\color{blue}{-0.5}}}{a \cdot 2} \]

   9. Applied egg-rr 73.1%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\frac{-0.25}{a \cdot c}\right)}^{-0.5}}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. *-commutative 73.1%

          \[\leadsto \frac{\left(-b\right) + {\left(\frac{-0.25}{\color{blue}{c \cdot a}}\right)}^{-0.5}}{a \cdot 2} \]

   11. Simplified 73.1%

       \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\frac{-0.25}{c \cdot a}\right)}^{-0.5}}}{a \cdot 2} \]

   if 1.0500000000000001e-144 < b

   1. Initial program 20.0%

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

      1. *-commutative 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around inf 83.8%

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

      1. associate-*r/ 83.8%

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

      2. mul-1-neg 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-144}:\\ \;\;\;\;\frac{{\left(\frac{-0.25}{a \cdot c}\right)}^{-0.5} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-144}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-36)
   (/ b (- a))
   (if (<= b 1.35e-144)
     (* (/ 0.5 a) (- (sqrt (* a (* c -4.0))) b))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-36) {
		tmp = b / -a;
	} else if (b <= 1.35e-144) {
		tmp = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	} 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 <= (-2.9d-36)) then
        tmp = b / -a
    else if (b <= 1.35d-144) then
        tmp = (0.5d0 / a) * (sqrt((a * (c * (-4.0d0)))) - b)
    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 <= -2.9e-36) {
		tmp = b / -a;
	} else if (b <= 1.35e-144) {
		tmp = (0.5 / a) * (Math.sqrt((a * (c * -4.0))) - b);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e-36:
		tmp = b / -a
	elif b <= 1.35e-144:
		tmp = (0.5 / a) * (math.sqrt((a * (c * -4.0))) - b)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-36)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.35e-144)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(a * Float64(c * -4.0))) - b));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.9e-36)
		tmp = b / -a;
	elseif (b <= 1.35e-144)
		tmp = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-36], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.35e-144], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.90000000000000013e-36

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in b around -inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-neg-frac2 87.7%

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

   7. Simplified 87.7%

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

   if -2.90000000000000013e-36 < b < 1.34999999999999988e-144

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

   3. Simplified 83.9%

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

   6. Applied egg-rr 83.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a} + \left(-b \cdot \frac{0.5}{a}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 83.8%

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

      2. distribute-rgt-out-- 83.8%

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

   8. Simplified 83.8%

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

   9. Taylor expanded in a around inf 73.0%

      \[\leadsto \frac{0.5}{a} \cdot \left(\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}} - b\right) \]
   10. Step-by-step derivation

       1. *-commutative 73.0%

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

       2. associate-*r* 73.0%

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

   11. Simplified 73.0%

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

   if 1.34999999999999988e-144 < b

   1. Initial program 20.0%

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

      1. *-commutative 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around inf 83.8%

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

      1. associate-*r/ 83.8%

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

      2. mul-1-neg 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-144}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e-37)
   (/ b (- a))
   (if (<= b 1.35e-144)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-37) {
		tmp = b / -a;
	} else if (b <= 1.35e-144) {
		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 <= (-3.3d-37)) then
        tmp = b / -a
    else if (b <= 1.35d-144) 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 <= -3.3e-37) {
		tmp = b / -a;
	} else if (b <= 1.35e-144) {
		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 <= -3.3e-37:
		tmp = b / -a
	elif b <= 1.35e-144:
		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 <= -3.3e-37)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.35e-144)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.3e-37)
		tmp = b / -a;
	elseif (b <= 1.35e-144)
		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, -3.3e-37], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.35e-144], 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 < -3.29999999999999982e-37

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in b around -inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-neg-frac2 87.7%

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

   7. Simplified 87.7%

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

   if -3.29999999999999982e-37 < b < 1.34999999999999988e-144

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in b around 0 73.0%

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

      1. *-commutative 73.0%

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

      2. *-commutative 73.0%

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

      3. associate-*r* 73.0%

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

   7. Simplified 73.0%

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

   if 1.34999999999999988e-144 < b

   1. Initial program 20.0%

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

      1. *-commutative 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in b around inf 83.8%

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

      1. associate-*r/ 83.8%

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

      2. mul-1-neg 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 8e-6) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-6) {
		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 <= 8d-6) 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 <= 8e-6) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e-6:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e-6)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8e-6)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e-6], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.99999999999999964e-6

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 48.7%

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

      1. mul-1-neg 48.7%

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

      2. distribute-neg-frac2 48.7%

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

   7. Simplified 48.7%

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

   if 7.99999999999999964e-6 < b

   1. Initial program 16.9%

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

      1. *-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in c around 0 89.1%

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. fma-neg 89.1%

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{a \cdot c}{{b}^{3}}, -\frac{1}{b}\right)} \]

      2. associate-/l* 91.1%

         \[\leadsto c \cdot \mathsf{fma}\left(-1, \color{blue}{a \cdot \frac{c}{{b}^{3}}}, -\frac{1}{b}\right) \]

      3. distribute-neg-frac 91.1%

         \[\leadsto c \cdot \mathsf{fma}\left(-1, a \cdot \frac{c}{{b}^{3}}, \color{blue}{\frac{-1}{b}}\right) \]

      4. metadata-eval 91.1%

         \[\leadsto c \cdot \mathsf{fma}\left(-1, a \cdot \frac{c}{{b}^{3}}, \frac{\color{blue}{-1}}{b}\right) \]

   7. Simplified 91.1%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-1, a \cdot \frac{c}{{b}^{3}}, \frac{-1}{b}\right)} \]

   8. Taylor expanded in a around 0 91.5%

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

      1. frac-2neg 91.5%

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

      2. metadata-eval 91.5%

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

      3. un-div-inv 91.6%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 31.0%

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

      6. sqr-neg 31.0%

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

      7. sqrt-unprod 30.5%

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

      8. add-sqr-sqrt 30.5%

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

   10. Applied egg-rr 30.5%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-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 75.9%

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

      1. *-commutative 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in b around -inf 68.4%

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

      1. mul-1-neg 68.4%

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

      2. distribute-neg-frac2 68.4%

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

   7. Simplified 68.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 32.7%

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

      1. *-commutative 32.7%

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

   3. Simplified 32.7%

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

   5. Taylor expanded in b around inf 67.4%

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

      1. associate-*r/ 67.4%

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

      2. mul-1-neg 67.4%

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

   7. Simplified 67.4%

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

4. Final simplification 67.9%

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

Alternative 7: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. *-commutative 54.2%

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

3. Simplified 54.2%

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

6. Applied egg-rr 53.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a} + \left(-b \cdot \frac{0.5}{a}\right)} \]
7. Step-by-step derivation

   1. sub-neg 53.4%

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

   2. distribute-rgt-out-- 54.0%

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

8. Simplified 54.0%

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

9. Taylor expanded in b around -inf 35.2%

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

    1. *-commutative 35.2%

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

11. Simplified 35.2%

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

    1. associate-*l/ 35.3%

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

    2. *-commutative 35.3%

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

    3. associate-*r* 35.3%

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

    4. metadata-eval 35.3%

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

    5. neg-mul-1 35.3%

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

    6. add-sqr-sqrt 33.7%

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

    7. sqrt-unprod 27.0%

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

    8. sqr-neg 27.0%

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

    9. sqrt-unprod 1.8%

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

    10. add-sqr-sqrt 2.4%

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

13. Applied egg-rr 2.4%

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

14. Final simplification 2.4%

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

Alternative 8: 11.0% 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 54.2%

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

   1. *-commutative 54.2%

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

3. Simplified 54.2%

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

5. Taylor expanded in c around 0 31.5%

   \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
6. Step-by-step derivation

   1. fma-neg 31.5%

      \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{a \cdot c}{{b}^{3}}, -\frac{1}{b}\right)} \]

   2. associate-/l* 32.2%

      \[\leadsto c \cdot \mathsf{fma}\left(-1, \color{blue}{a \cdot \frac{c}{{b}^{3}}}, -\frac{1}{b}\right) \]

   3. distribute-neg-frac 32.2%

      \[\leadsto c \cdot \mathsf{fma}\left(-1, a \cdot \frac{c}{{b}^{3}}, \color{blue}{\frac{-1}{b}}\right) \]

   4. metadata-eval 32.2%

      \[\leadsto c \cdot \mathsf{fma}\left(-1, a \cdot \frac{c}{{b}^{3}}, \frac{\color{blue}{-1}}{b}\right) \]

7. Simplified 32.2%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-1, a \cdot \frac{c}{{b}^{3}}, \frac{-1}{b}\right)} \]

8. Taylor expanded in a around 0 34.8%

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

   1. frac-2neg 34.8%

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

   2. metadata-eval 34.8%

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

   3. un-div-inv 34.9%

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

   4. add-sqr-sqrt 1.0%

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

   5. sqrt-unprod 10.2%

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

   6. sqr-neg 10.2%

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

   7. sqrt-unprod 9.2%

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

   8. add-sqr-sqrt 11.2%

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

10. Applied egg-rr 11.2%

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

11. Final simplification 11.2%

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

Reproduce

herbie shell --seed 2024074 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))