Quadratic roots, full range

Percentage Accurate: 51.5% → 85.5%

Time: 12.7s

Alternatives: 9

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: 51.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119} \lor \neg \left(b \leq 6.5 \cdot 10^{-61}\right) \land b \leq 2.9 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+154)
   (/ (- b) a)
   (if (or (<= b 8.5e-119) (and (not (<= b 6.5e-61)) (<= b 2.9e-34)))
     (/ (- (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 <= -4e+154) {
		tmp = -b / a;
	} else if ((b <= 8.5e-119) || (!(b <= 6.5e-61) && (b <= 2.9e-34))) {
		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 <= (-4d+154)) then
        tmp = -b / a
    else if ((b <= 8.5d-119) .or. (.not. (b <= 6.5d-61)) .and. (b <= 2.9d-34)) 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 <= -4e+154) {
		tmp = -b / a;
	} else if ((b <= 8.5e-119) || (!(b <= 6.5e-61) && (b <= 2.9e-34))) {
		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 <= -4e+154:
		tmp = -b / a
	elif (b <= 8.5e-119) or (not (b <= 6.5e-61) and (b <= 2.9e-34)):
		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 <= -4e+154)
		tmp = Float64(Float64(-b) / a);
	elseif ((b <= 8.5e-119) || (!(b <= 6.5e-61) && (b <= 2.9e-34)))
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - 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 <= -4e+154)
		tmp = -b / a;
	elseif ((b <= 8.5e-119) || (~((b <= 6.5e-61)) && (b <= 2.9e-34)))
		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, -4e+154], N[((-b) / a), $MachinePrecision], If[Or[LessEqual[b, 8.5e-119], And[N[Not[LessEqual[b, 6.5e-61]], $MachinePrecision], LessEqual[b, 2.9e-34]]], 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 < -4.00000000000000015e154

   1. Initial program 41.2%

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

      1. *-commutative 41.2%

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

   3. Simplified 41.2%

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

   4. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -4.00000000000000015e154 < b < 8.49999999999999977e-119 or 6.4999999999999994e-61 < b < 2.9000000000000002e-34

   1. Initial program 82.3%

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

   if 8.49999999999999977e-119 < b < 6.4999999999999994e-61 or 2.9000000000000002e-34 < b

   1. Initial program 17.4%

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

      1. *-commutative 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in b around inf 86.6%

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

      1. mul-1-neg 86.6%

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

      2. distribute-neg-frac 86.6%

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

   6. Simplified 86.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119} \lor \neg \left(b \leq 6.5 \cdot 10^{-61}\right) \land b \leq 2.9 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e-82)
   (/ (- b) a)
   (if (<= b 4.7e-119)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-82) {
		tmp = -b / a;
	} else if (b <= 4.7e-119) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.6d-82)) then
        tmp = -b / a
    else if (b <= 4.7d-119) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-82) {
		tmp = -b / a;
	} else if (b <= 4.7e-119) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.6e-82:
		tmp = -b / a
	elif b <= 4.7e-119:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e-82)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.7e-119)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.6e-82)
		tmp = -b / a;
	elseif (b <= 4.7e-119)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e-82], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.7e-119], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.6e-82

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in b around -inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. mul-1-neg 87.4%

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

   6. Simplified 87.4%

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

   if -2.6e-82 < b < 4.70000000000000002e-119

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

   3. Simplified 73.9%

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

      1. add-sqr-sqrt 73.5%

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

      2. pow2 73.5%

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

      3. pow1/2 73.5%

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

      4. sqrt-pow1 73.5%

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

      5. fma-neg 73.5%

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

      6. *-commutative 73.5%

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

      7. distribute-rgt-neg-in 73.5%

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

      8. *-commutative 73.5%

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

      9. distribute-rgt-neg-in 73.5%

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

      10. metadata-eval 73.5%

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

      11. metadata-eval 73.5%

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

   5. Applied egg-rr 73.5%

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

   6. Taylor expanded in c around inf 46.8%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25 \cdot \left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)}\right)}^{2} - b}}{a \cdot 2} \]
   7. Step-by-step derivation

   8. Simplified 71.0%

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

   if 4.70000000000000002e-119 < b

   1. Initial program 22.1%

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

      1. *-commutative 22.1%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in b around inf 82.3%

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

      1. mul-1-neg 82.3%

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

      2. distribute-neg-frac 82.3%

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

   6. Simplified 82.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-82}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-82)
   (/ (- b) a)
   (if (<= b 8.5e-119)
     (/ (- (sqrt (* -4.0 (* a c))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-82) {
		tmp = -b / a;
	} else if (b <= 8.5e-119) {
		tmp = (sqrt((-4.0 * (a * 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 <= (-5d-82)) then
        tmp = -b / a
    else if (b <= 8.5d-119) then
        tmp = (sqrt(((-4.0d0) * (a * 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 <= -5e-82) {
		tmp = -b / a;
	} else if (b <= 8.5e-119) {
		tmp = (Math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-82:
		tmp = -b / a
	elif b <= 8.5e-119:
		tmp = (math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-82)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 8.5e-119)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) - 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 <= -5e-82)
		tmp = -b / a;
	elseif (b <= 8.5e-119)
		tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-82], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 8.5e-119], N[(N[(N[Sqrt[N[(-4.0 * N[(a * 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 < -4.9999999999999998e-82

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in b around -inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. mul-1-neg 87.4%

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

   6. Simplified 87.4%

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

   if -4.9999999999999998e-82 < b < 8.49999999999999977e-119

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in b around 0 71.0%

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

   if 8.49999999999999977e-119 < b

   1. Initial program 22.1%

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

      1. *-commutative 22.1%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in b around inf 82.3%

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

      1. mul-1-neg 82.3%

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

      2. distribute-neg-frac 82.3%

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

   6. Simplified 82.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-82}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 68.2% 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 71.2%

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

      1. *-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in b around -inf 65.6%

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

      1. +-commutative 65.6%

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

      2. mul-1-neg 65.6%

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

      3. unsub-neg 65.6%

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

   6. Simplified 65.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.8%

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

      1. *-commutative 31.8%

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

   3. Simplified 31.8%

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

   4. Taylor expanded in b around inf 67.1%

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

      1. mul-1-neg 67.1%

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

      2. distribute-neg-frac 67.1%

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

   6. Simplified 67.1%

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

4. Final simplification 66.4%

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

Alternative 5: 42.7% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 4.8e+44) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8e+44) {
		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 <= 4.8d+44) 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 <= 4.8e+44) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.8e+44:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.8e+44)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.8e+44)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.8e+44], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.80000000000000026e44

   1. Initial program 63.3%

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

      1. *-commutative 63.3%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in b around -inf 43.9%

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

      1. associate-*r/ 43.9%

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

      2. mul-1-neg 43.9%

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

   6. Simplified 43.9%

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

   if 4.80000000000000026e44 < b

   1. Initial program 15.9%

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

      1. *-commutative 15.9%

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

   3. Simplified 15.9%

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

   4. Taylor expanded in b around -inf 2.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]

   5. Taylor expanded in b around 0 32.3%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6: 68.0% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.5e-299) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.5e-299) {
		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 <= 6.5d-299) 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 <= 6.5e-299) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.5e-299:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.5e-299)
		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 <= 6.5e-299)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.5e-299], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.4999999999999997e-299

   1. Initial program 70.1%

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

      1. *-commutative 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in b around -inf 64.5%

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

      1. associate-*r/ 64.5%

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

      2. mul-1-neg 64.5%

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

   6. Simplified 64.5%

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

   if 6.4999999999999997e-299 < b

   1. Initial program 32.2%

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

      1. *-commutative 32.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in b around inf 68.1%

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

      1. mul-1-neg 68.1%

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

      2. distribute-neg-frac 68.1%

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

   6. Simplified 68.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 13.2% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+56}:\\ \;\;\;\;\frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.15e+56) (/ 0.5 a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.15e+56) {
		tmp = 0.5 / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.15d+56) then
        tmp = 0.5d0 / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.15e+56) {
		tmp = 0.5 / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.15e+56:
		tmp = 0.5 / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.15e+56)
		tmp = Float64(0.5 / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.15e+56)
		tmp = 0.5 / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.15e+56], N[(0.5 / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15000000000000007e56

   1. Initial program 63.1%

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

      1. *-commutative 63.1%

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

   3. Simplified 63.1%

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

      1. clear-num 63.0%

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

      2. inv-pow 63.0%

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

   5. Applied egg-rr 44.4%

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

   6. Taylor expanded in b around inf 2.7%

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

      1. expm1-log1p-u 2.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{2}{\frac{b + b}{a}}\right)}^{-1}\right)\right)} \]

      2. expm1-udef 2.2%

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

      3. unpow-1 2.2%

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

      4. clear-num 2.2%

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

      5. div-inv 2.2%

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

      6. flip-+ 0.0%

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

      7. +-inverses 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0}}{b - b}}{a} \cdot \frac{1}{2}\right)} - 1 \]

      8. +-inverses 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{0}{\color{blue}{0}}}{a} \cdot \frac{1}{2}\right)} - 1 \]

      9. metadata-eval 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a} \cdot \color{blue}{0.5}\right)} - 1 \]

   8. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a} \cdot 0.5\right)} - 1} \]

   10. Simplified 5.6%

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

   if 1.15000000000000007e56 < b

   1. Initial program 14.8%

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

      1. *-commutative 14.8%

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

   3. Simplified 14.8%

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

   4. Taylor expanded in b around -inf 2.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]

   5. Taylor expanded in b around 0 33.2%

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

4. Final simplification 12.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+56}:\\ \;\;\;\;\frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 8: 2.3% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ -0.5 a))

C

double code(double a, double b, double c) {
	return -0.5 / 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 = (-0.5d0) / a
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 / a;
}

Python

def code(a, b, c):
	return -0.5 / a

Julia

function code(a, b, c)
	return Float64(-0.5 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 / a;
end

Wolfram

code[a_, b_, c_] := N[(-0.5 / a), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. *-commutative 50.9%

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

3. Simplified 50.9%

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

   1. clear-num 50.8%

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

   2. inv-pow 50.8%

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

5. Applied egg-rr 34.8%

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

6. Taylor expanded in b around inf 2.7%

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

   1. unpow-1 2.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{\frac{b + b}{a}}}} \]

   2. frac-2neg 2.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{-2}{-\frac{b + b}{a}}}} \]

   3. metadata-eval 2.7%

      \[\leadsto \frac{1}{\frac{\color{blue}{-2}}{-\frac{b + b}{a}}} \]

   4. associate-/r/ 2.7%

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

   5. metadata-eval 2.7%

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

   6. distribute-neg-frac 2.7%

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

   7. flip-+ 0.0%

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

   8. +-inverses 0.0%

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

   9. +-inverses 0.0%

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

   10. distribute-neg-frac 0.0%

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

   11. metadata-eval 0.0%

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

8. Applied egg-rr 0.0%

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

10. Simplified 2.3%

    \[\leadsto \color{blue}{\frac{-0.5}{a}} \]

11. Final simplification 2.3%

    \[\leadsto \frac{-0.5}{a} \]

Alternative 9: 5.3% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.5 a))

C

double code(double a, double b, double c) {
	return 0.5 / 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 = 0.5d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.5 / a;
}

Python

def code(a, b, c):
	return 0.5 / a

Julia

function code(a, b, c)
	return Float64(0.5 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.5 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.5 / a), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. *-commutative 50.9%

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

3. Simplified 50.9%

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

   1. clear-num 50.8%

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

   2. inv-pow 50.8%

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

5. Applied egg-rr 34.8%

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

6. Taylor expanded in b around inf 2.7%

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

   1. expm1-log1p-u 2.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{2}{\frac{b + b}{a}}\right)}^{-1}\right)\right)} \]

   2. expm1-udef 2.5%

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

   3. unpow-1 2.5%

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

   4. clear-num 2.5%

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

   5. div-inv 2.5%

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

   6. flip-+ 0.0%

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

   7. +-inverses 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0}}{b - b}}{a} \cdot \frac{1}{2}\right)} - 1 \]

   8. +-inverses 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{0}{\color{blue}{0}}}{a} \cdot \frac{1}{2}\right)} - 1 \]

   9. metadata-eval 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a} \cdot \color{blue}{0.5}\right)} - 1 \]

8. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a} \cdot 0.5\right)} - 1} \]

10. Simplified 5.2%

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

11. Final simplification 5.2%

    \[\leadsto \frac{0.5}{a} \]

Reproduce

herbie shell --seed 2023332 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))