The quadratic formula (r1)

Percentage Accurate: 52.0% → 84.9%

Time: 13.1s

Alternatives: 10

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: 52.0% 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: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+117}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-95} \lor \neg \left(b \leq 2.35 \cdot 10^{-80}\right) \land b \leq 0.135:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+117)
   (- (/ c b) (/ b a))
   (if (or (<= b 5e-95) (and (not (<= b 2.35e-80)) (<= b 0.135)))
     (/ (- (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 <= -4e+117) {
		tmp = (c / b) - (b / a);
	} else if ((b <= 5e-95) || (!(b <= 2.35e-80) && (b <= 0.135))) {
		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 <= (-4d+117)) then
        tmp = (c / b) - (b / a)
    else if ((b <= 5d-95) .or. (.not. (b <= 2.35d-80)) .and. (b <= 0.135d0)) 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 <= -4e+117) {
		tmp = (c / b) - (b / a);
	} else if ((b <= 5e-95) || (!(b <= 2.35e-80) && (b <= 0.135))) {
		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 <= -4e+117:
		tmp = (c / b) - (b / a)
	elif (b <= 5e-95) or (not (b <= 2.35e-80) and (b <= 0.135)):
		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 <= -4e+117)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif ((b <= 5e-95) || (!(b <= 2.35e-80) && (b <= 0.135)))
		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 <= -4e+117)
		tmp = (c / b) - (b / a);
	elseif ((b <= 5e-95) || (~((b <= 2.35e-80)) && (b <= 0.135)))
		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, -4e+117], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 5e-95], And[N[Not[LessEqual[b, 2.35e-80]], $MachinePrecision], LessEqual[b, 0.135]]], 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 < -4.0000000000000002e117

   1. Initial program 47.7%

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

   2. Taylor expanded in b around -inf 96.8%

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

      1. +-commutative 96.8%

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

      2. mul-1-neg 96.8%

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

      3. unsub-neg 96.8%

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

   4. Simplified 96.8%

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

   if -4.0000000000000002e117 < b < 4.9999999999999998e-95 or 2.34999999999999986e-80 < b < 0.13500000000000001

   1. Initial program 83.0%

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

   if 4.9999999999999998e-95 < b < 2.34999999999999986e-80 or 0.13500000000000001 < b

   1. Initial program 16.5%

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

   2. Taylor expanded in b around inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. neg-mul-1 83.2%

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

   4. Simplified 83.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+117}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-95} \lor \neg \left(b \leq 2.35 \cdot 10^{-80}\right) \land b \leq 0.135:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-75}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-95}:\\ \;\;\;\;\frac{{\left(\frac{c \cdot 4}{\frac{-1}{a}}\right)}^{0.5} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e-75)
   (- (/ c b) (/ b a))
   (if (<= b 8e-95)
     (/ (- (pow (/ (* c 4.0) (/ -1.0 a)) 0.5) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-75) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8e-95) {
		tmp = (pow(((c * 4.0) / (-1.0 / a)), 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 <= (-1.25d-75)) then
        tmp = (c / b) - (b / a)
    else if (b <= 8d-95) then
        tmp = ((((c * 4.0d0) / ((-1.0d0) / a)) ** 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 <= -1.25e-75) {
		tmp = (c / b) - (b / a);
	} else if (b <= 8e-95) {
		tmp = (Math.pow(((c * 4.0) / (-1.0 / a)), 0.5) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.25e-75:
		tmp = (c / b) - (b / a)
	elif b <= 8e-95:
		tmp = (math.pow(((c * 4.0) / (-1.0 / a)), 0.5) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.25e-75)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 8e-95)
		tmp = Float64(Float64((Float64(Float64(c * 4.0) / Float64(-1.0 / a)) ^ 0.5) - 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 <= -1.25e-75)
		tmp = (c / b) - (b / a);
	elseif (b <= 8e-95)
		tmp = ((((c * 4.0) / (-1.0 / a)) ^ 0.5) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e-75], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-95], N[(N[(N[Power[N[(N[(c * 4.0), $MachinePrecision] / N[(-1.0 / a), $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 < -1.24999999999999995e-75

   1. Initial program 68.0%

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

   2. Taylor expanded in b around -inf 93.4%

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

      1. +-commutative 93.4%

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

      2. mul-1-neg 93.4%

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

      3. unsub-neg 93.4%

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

   4. Simplified 93.4%

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

   if -1.24999999999999995e-75 < b < 7.99999999999999992e-95

   1. Initial program 80.2%

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

      1. pow1/2 80.2%

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

      2. pow-to-exp 74.8%

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

      3. fma-neg 74.8%

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

      4. *-commutative 74.8%

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

      5. distribute-rgt-neg-in 74.8%

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

      6. *-commutative 74.8%

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

      7. distribute-rgt-neg-in 74.8%

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

      8. metadata-eval 74.8%

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

   3. Applied egg-rr 74.8%

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

   4. Taylor expanded in a around -inf 46.5%

      \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(4 \cdot c\right) + -1 \cdot \log \left(\frac{-1}{a}\right)\right)} \cdot 0.5}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. mul-1-neg 46.5%

         \[\leadsto \frac{\left(-b\right) + e^{\left(\log \left(4 \cdot c\right) + \color{blue}{\left(-\log \left(\frac{-1}{a}\right)\right)}\right) \cdot 0.5}}{2 \cdot a} \]

      2. unsub-neg 46.5%

         \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(4 \cdot c\right) - \log \left(\frac{-1}{a}\right)\right)} \cdot 0.5}}{2 \cdot a} \]

      3. *-commutative 46.5%

         \[\leadsto \frac{\left(-b\right) + e^{\left(\log \color{blue}{\left(c \cdot 4\right)} - \log \left(\frac{-1}{a}\right)\right) \cdot 0.5}}{2 \cdot a} \]

   6. Simplified 46.5%

      \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(c \cdot 4\right) - \log \left(\frac{-1}{a}\right)\right)} \cdot 0.5}}{2 \cdot a} \]
   7. Step-by-step derivation

      1. exp-prod 39.6%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(e^{\log \left(c \cdot 4\right) - \log \left(\frac{-1}{a}\right)}\right)}^{0.5}}}{2 \cdot a} \]

      2. diff-log 67.2%

         \[\leadsto \frac{\left(-b\right) + {\left(e^{\color{blue}{\log \left(\frac{c \cdot 4}{\frac{-1}{a}}\right)}}\right)}^{0.5}}{2 \cdot a} \]

      3. add-exp-log 71.8%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left(\frac{c \cdot 4}{\frac{-1}{a}}\right)}}^{0.5}}{2 \cdot a} \]

   8. Applied egg-rr 71.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\frac{c \cdot 4}{\frac{-1}{a}}\right)}^{0.5}}}{2 \cdot a} \]

   if 7.99999999999999992e-95 < b

   1. Initial program 23.4%

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

   2. Taylor expanded in b around inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   4. Simplified 74.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-75}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-95}:\\ \;\;\;\;\frac{{\left(\frac{c \cdot 4}{\frac{-1}{a}}\right)}^{0.5} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-94}:\\ \;\;\;\;\left(\sqrt{c \cdot \frac{a}{-0.25}} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e-69)
   (- (/ c b) (/ b a))
   (if (<= b 2.2e-94)
     (* (- (sqrt (* c (/ a -0.25))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-69) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.2e-94) {
		tmp = (sqrt((c * (a / -0.25))) - b) * (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 <= (-1.1d-69)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.2d-94) then
        tmp = (sqrt((c * (a / (-0.25d0)))) - b) * (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 <= -1.1e-69) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.2e-94) {
		tmp = (Math.sqrt((c * (a / -0.25))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.1e-69:
		tmp = (c / b) - (b / a)
	elif b <= 2.2e-94:
		tmp = (math.sqrt((c * (a / -0.25))) - b) * (0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e-69)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.2e-94)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a / -0.25))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.1e-69)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.2e-94)
		tmp = (sqrt((c * (a / -0.25))) - b) * (0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e-69], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.2e-94], N[(N[(N[Sqrt[N[(c * N[(a / -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1e-69

   1. Initial program 68.0%

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

   2. Taylor expanded in b around -inf 93.4%

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

      1. +-commutative 93.4%

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

      2. mul-1-neg 93.4%

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

      3. unsub-neg 93.4%

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

   4. Simplified 93.4%

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

   if -1.1e-69 < b < 2.20000000000000001e-94

   1. Initial program 80.2%

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

      1. pow1/2 80.2%

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

      2. pow-to-exp 74.8%

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

      3. fma-neg 74.8%

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

      4. *-commutative 74.8%

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

      5. distribute-rgt-neg-in 74.8%

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

      6. *-commutative 74.8%

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

      7. distribute-rgt-neg-in 74.8%

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

      8. metadata-eval 74.8%

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

   3. Applied egg-rr 74.8%

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

   4. Taylor expanded in c around -inf 33.5%

      \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} \cdot 0.5}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. mul-1-neg 33.5%

         \[\leadsto \frac{\left(-b\right) + e^{\left(\log \left(4 \cdot a\right) + \color{blue}{\left(-\log \left(\frac{-1}{c}\right)\right)}\right) \cdot 0.5}}{2 \cdot a} \]

      2. unsub-neg 33.5%

         \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)} \cdot 0.5}}{2 \cdot a} \]

      3. *-commutative 33.5%

         \[\leadsto \frac{\left(-b\right) + e^{\left(\log \color{blue}{\left(a \cdot 4\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}}{2 \cdot a} \]

   6. Simplified 33.5%

      \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(a \cdot 4\right) - \log \left(\frac{-1}{c}\right)\right)} \cdot 0.5}}{2 \cdot a} \]

   7. Taylor expanded in a around 0 33.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{0.5 \cdot \left(\left(\log 4 + \log a\right) - \log \left(\frac{-1}{c}\right)\right)} - b}{a}} \]

   9. Simplified 71.7%

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

   if 2.20000000000000001e-94 < b

   1. Initial program 23.4%

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

   2. Taylor expanded in b around inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   4. Simplified 74.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-94}:\\ \;\;\;\;\left(\sqrt{c \cdot \frac{a}{-0.25}} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \frac{a}{-0.25}} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-69)
   (- (/ c b) (/ b a))
   (if (<= b 3.9e-95)
     (/ (- (sqrt (* c (/ a -0.25))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-69) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.9e-95) {
		tmp = (sqrt((c * (a / -0.25))) - 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 <= (-4.5d-69)) then
        tmp = (c / b) - (b / a)
    else if (b <= 3.9d-95) then
        tmp = (sqrt((c * (a / (-0.25d0)))) - 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 <= -4.5e-69) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.9e-95) {
		tmp = (Math.sqrt((c * (a / -0.25))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e-69:
		tmp = (c / b) - (b / a)
	elif b <= 3.9e-95:
		tmp = (math.sqrt((c * (a / -0.25))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-69)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 3.9e-95)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a / -0.25))) - 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 <= -4.5e-69)
		tmp = (c / b) - (b / a);
	elseif (b <= 3.9e-95)
		tmp = (sqrt((c * (a / -0.25))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-69], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e-95], N[(N[(N[Sqrt[N[(c * N[(a / -0.25), $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 < -4.50000000000000009e-69

   1. Initial program 68.0%

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

   2. Taylor expanded in b around -inf 93.4%

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

      1. +-commutative 93.4%

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

      2. mul-1-neg 93.4%

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

      3. unsub-neg 93.4%

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

   4. Simplified 93.4%

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

   if -4.50000000000000009e-69 < b < 3.9e-95

   1. Initial program 80.2%

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

      1. pow1/2 80.2%

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

      2. pow-to-exp 74.8%

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

      3. fma-neg 74.8%

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

      4. *-commutative 74.8%

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

      5. distribute-rgt-neg-in 74.8%

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

      6. *-commutative 74.8%

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

      7. distribute-rgt-neg-in 74.8%

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

      8. metadata-eval 74.8%

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

   3. Applied egg-rr 74.8%

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

   4. Taylor expanded in c around -inf 33.5%

      \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} \cdot 0.5}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. mul-1-neg 33.5%

         \[\leadsto \frac{\left(-b\right) + e^{\left(\log \left(4 \cdot a\right) + \color{blue}{\left(-\log \left(\frac{-1}{c}\right)\right)}\right) \cdot 0.5}}{2 \cdot a} \]

      2. unsub-neg 33.5%

         \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)} \cdot 0.5}}{2 \cdot a} \]

      3. *-commutative 33.5%

         \[\leadsto \frac{\left(-b\right) + e^{\left(\log \color{blue}{\left(a \cdot 4\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}}{2 \cdot a} \]

   6. Simplified 33.5%

      \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(a \cdot 4\right) - \log \left(\frac{-1}{c}\right)\right)} \cdot 0.5}}{2 \cdot a} \]

   7. Taylor expanded in b around 0 33.5%

      \[\leadsto \frac{\color{blue}{e^{0.5 \cdot \left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)} + -1 \cdot b}}{2 \cdot a} \]
   8. Step-by-step derivation

      1. neg-mul-1 33.5%

         \[\leadsto \frac{e^{0.5 \cdot \left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)} + \color{blue}{\left(-b\right)}}{2 \cdot a} \]

      2. sub-neg 33.5%

         \[\leadsto \frac{\color{blue}{e^{0.5 \cdot \left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)} - b}}{2 \cdot a} \]

      3. *-commutative 33.5%

         \[\leadsto \frac{e^{\color{blue}{\left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}} - b}{2 \cdot a} \]

      4. log-prod 33.5%

         \[\leadsto \frac{e^{\left(\color{blue}{\left(\log 4 + \log a\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5} - b}{2 \cdot a} \]

      5. +-commutative 33.5%

         \[\leadsto \frac{e^{\left(\color{blue}{\left(\log a + \log 4\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5} - b}{2 \cdot a} \]

      6. log-prod 33.5%

         \[\leadsto \frac{e^{\left(\color{blue}{\log \left(a \cdot 4\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5} - b}{2 \cdot a} \]

      7. log-div 67.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{a \cdot 4}{\frac{-1}{c}}\right)} \cdot 0.5} - b}{2 \cdot a} \]

      8. exp-to-pow 71.8%

         \[\leadsto \frac{\color{blue}{{\left(\frac{a \cdot 4}{\frac{-1}{c}}\right)}^{0.5}} - b}{2 \cdot a} \]

      9. unpow1/2 71.8%

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

      10. associate-/r/ 71.8%

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

      11. *-commutative 71.8%

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

      12. associate-/l* 71.8%

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

      13. metadata-eval 71.8%

          \[\leadsto \frac{\sqrt{c \cdot \frac{a}{\color{blue}{-0.25}}} - b}{2 \cdot a} \]

   9. Simplified 71.8%

      \[\leadsto \frac{\color{blue}{\sqrt{c \cdot \frac{a}{-0.25}} - b}}{2 \cdot a} \]

   if 3.9e-95 < b

   1. Initial program 23.4%

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

   2. Taylor expanded in b around inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   4. Simplified 74.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \frac{a}{-0.25}} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-94}:\\ \;\;\;\;\frac{0.5}{a} \cdot \sqrt{a \cdot \frac{c}{-0.25}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e-95)
   (- (/ c b) (/ b a))
   (if (<= b 2.15e-94) (* (/ 0.5 a) (sqrt (* a (/ c -0.25)))) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-95) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.15e-94) {
		tmp = (0.5 / a) * sqrt((a * (c / -0.25)));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.6d-95)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.15d-94) then
        tmp = (0.5d0 / a) * sqrt((a * (c / (-0.25d0))))
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-95) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.15e-94) {
		tmp = (0.5 / a) * Math.sqrt((a * (c / -0.25)));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.6e-95:
		tmp = (c / b) - (b / a)
	elif b <= 2.15e-94:
		tmp = (0.5 / a) * math.sqrt((a * (c / -0.25)))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e-95)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.15e-94)
		tmp = Float64(Float64(0.5 / a) * sqrt(Float64(a * Float64(c / -0.25))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.6e-95)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.15e-94)
		tmp = (0.5 / a) * sqrt((a * (c / -0.25)));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e-95], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.15e-94], N[(N[(0.5 / a), $MachinePrecision] * N[Sqrt[N[(a * N[(c / -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.5999999999999995e-95

   1. Initial program 69.3%

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

   2. Taylor expanded in b around -inf 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

   4. Simplified 91.8%

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

   if -7.5999999999999995e-95 < b < 2.1499999999999999e-94

   1. Initial program 78.9%

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

      1. pow1/2 78.9%

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

      2. pow-to-exp 73.7%

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

      3. fma-neg 73.7%

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

      4. *-commutative 73.7%

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

      5. distribute-rgt-neg-in 73.7%

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

      6. *-commutative 73.7%

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

      7. distribute-rgt-neg-in 73.7%

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

      8. metadata-eval 73.7%

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

   3. Applied egg-rr 73.7%

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

   4. Taylor expanded in a around -inf 49.6%

      \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(4 \cdot c\right) + -1 \cdot \log \left(\frac{-1}{a}\right)\right)} \cdot 0.5}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. mul-1-neg 49.6%

         \[\leadsto \frac{\left(-b\right) + e^{\left(\log \left(4 \cdot c\right) + \color{blue}{\left(-\log \left(\frac{-1}{a}\right)\right)}\right) \cdot 0.5}}{2 \cdot a} \]

      2. unsub-neg 49.6%

         \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(4 \cdot c\right) - \log \left(\frac{-1}{a}\right)\right)} \cdot 0.5}}{2 \cdot a} \]

      3. *-commutative 49.6%

         \[\leadsto \frac{\left(-b\right) + e^{\left(\log \color{blue}{\left(c \cdot 4\right)} - \log \left(\frac{-1}{a}\right)\right) \cdot 0.5}}{2 \cdot a} \]

   6. Simplified 49.6%

      \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(c \cdot 4\right) - \log \left(\frac{-1}{a}\right)\right)} \cdot 0.5}}{2 \cdot a} \]

   7. Taylor expanded in b around 0 49.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{0.5 \cdot \left(\log \left(4 \cdot c\right) - \log \left(\frac{-1}{a}\right)\right)}}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/ 49.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot e^{0.5 \cdot \left(\log \left(4 \cdot c\right) - \log \left(\frac{-1}{a}\right)\right)}}{a}} \]

      2. *-lft-identity 49.8%

         \[\leadsto \frac{0.5 \cdot e^{0.5 \cdot \left(\log \left(4 \cdot c\right) - \log \left(\frac{-1}{a}\right)\right)}}{\color{blue}{1 \cdot a}} \]

      3. *-commutative 49.8%

         \[\leadsto \frac{0.5 \cdot e^{\color{blue}{\left(\log \left(4 \cdot c\right) - \log \left(\frac{-1}{a}\right)\right) \cdot 0.5}}}{1 \cdot a} \]

      4. *-commutative 49.8%

         \[\leadsto \frac{0.5 \cdot e^{\left(\log \color{blue}{\left(c \cdot 4\right)} - \log \left(\frac{-1}{a}\right)\right) \cdot 0.5}}{1 \cdot a} \]

      5. log-div 67.9%

         \[\leadsto \frac{0.5 \cdot e^{\color{blue}{\log \left(\frac{c \cdot 4}{\frac{-1}{a}}\right)} \cdot 0.5}}{1 \cdot a} \]

      6. exp-to-pow 72.6%

         \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\frac{c \cdot 4}{\frac{-1}{a}}\right)}^{0.5}}}{1 \cdot a} \]

      7. *-commutative 72.6%

         \[\leadsto \frac{0.5 \cdot {\left(\frac{c \cdot 4}{\frac{-1}{a}}\right)}^{0.5}}{\color{blue}{a \cdot 1}} \]

      8. times-frac 72.6%

         \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \frac{{\left(\frac{c \cdot 4}{\frac{-1}{a}}\right)}^{0.5}}{1}} \]

      9. /-rgt-identity 72.6%

         \[\leadsto \frac{0.5}{a} \cdot \color{blue}{{\left(\frac{c \cdot 4}{\frac{-1}{a}}\right)}^{0.5}} \]

      10. unpow1/2 72.6%

          \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\sqrt{\frac{c \cdot 4}{\frac{-1}{a}}}} \]

      11. associate-/r/ 72.5%

          \[\leadsto \frac{0.5}{a} \cdot \sqrt{\color{blue}{\frac{c \cdot 4}{-1} \cdot a}} \]

      12. *-commutative 72.5%

          \[\leadsto \frac{0.5}{a} \cdot \sqrt{\color{blue}{a \cdot \frac{c \cdot 4}{-1}}} \]

      13. associate-/l* 72.5%

          \[\leadsto \frac{0.5}{a} \cdot \sqrt{a \cdot \color{blue}{\frac{c}{\frac{-1}{4}}}} \]

      14. metadata-eval 72.5%

          \[\leadsto \frac{0.5}{a} \cdot \sqrt{a \cdot \frac{c}{\color{blue}{-0.25}}} \]

   9. Simplified 72.5%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \sqrt{a \cdot \frac{c}{-0.25}}} \]

   if 2.1499999999999999e-94 < b

   1. Initial program 23.4%

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

   2. Taylor expanded in b around inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   4. Simplified 74.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-94}:\\ \;\;\;\;\frac{0.5}{a} \cdot \sqrt{a \cdot \frac{c}{-0.25}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-94}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{c \cdot \frac{a}{-0.25}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.3e-95)
   (- (/ c b) (/ b a))
   (if (<= b 2.15e-94) (/ (* 0.5 (sqrt (* c (/ a -0.25)))) a) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-95) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.15e-94) {
		tmp = (0.5 * sqrt((c * (a / -0.25)))) / 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 <= (-4.3d-95)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.15d-94) then
        tmp = (0.5d0 * sqrt((c * (a / (-0.25d0))))) / 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 <= -4.3e-95) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.15e-94) {
		tmp = (0.5 * Math.sqrt((c * (a / -0.25)))) / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.3e-95:
		tmp = (c / b) - (b / a)
	elif b <= 2.15e-94:
		tmp = (0.5 * math.sqrt((c * (a / -0.25)))) / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.3e-95)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.15e-94)
		tmp = Float64(Float64(0.5 * sqrt(Float64(c * Float64(a / -0.25)))) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.3e-95)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.15e-94)
		tmp = (0.5 * sqrt((c * (a / -0.25)))) / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.3e-95], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.15e-94], N[(N[(0.5 * N[Sqrt[N[(c * N[(a / -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.29999999999999997e-95

   1. Initial program 69.3%

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

   2. Taylor expanded in b around -inf 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

   4. Simplified 91.8%

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

   if -4.29999999999999997e-95 < b < 2.1499999999999999e-94

   1. Initial program 78.9%

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

      1. pow1/2 78.9%

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

      2. pow-to-exp 73.7%

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

      3. fma-neg 73.7%

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

      4. *-commutative 73.7%

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

      5. distribute-rgt-neg-in 73.7%

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

      6. *-commutative 73.7%

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

      7. distribute-rgt-neg-in 73.7%

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

      8. metadata-eval 73.7%

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

   3. Applied egg-rr 73.7%

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

   4. Taylor expanded in c around -inf 32.4%

      \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(4 \cdot a\right) + -1 \cdot \log \left(\frac{-1}{c}\right)\right)} \cdot 0.5}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. mul-1-neg 32.4%

         \[\leadsto \frac{\left(-b\right) + e^{\left(\log \left(4 \cdot a\right) + \color{blue}{\left(-\log \left(\frac{-1}{c}\right)\right)}\right) \cdot 0.5}}{2 \cdot a} \]

      2. unsub-neg 32.4%

         \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)} \cdot 0.5}}{2 \cdot a} \]

      3. *-commutative 32.4%

         \[\leadsto \frac{\left(-b\right) + e^{\left(\log \color{blue}{\left(a \cdot 4\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}}{2 \cdot a} \]

   6. Simplified 32.4%

      \[\leadsto \frac{\left(-b\right) + e^{\color{blue}{\left(\log \left(a \cdot 4\right) - \log \left(\frac{-1}{c}\right)\right)} \cdot 0.5}}{2 \cdot a} \]

   7. Taylor expanded in b around 0 32.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{0.5 \cdot \left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)}}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/ 32.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot e^{0.5 \cdot \left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right)}}{a}} \]

      2. *-commutative 32.4%

         \[\leadsto \frac{0.5 \cdot e^{\color{blue}{\left(\log \left(4 \cdot a\right) - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}}}{a} \]

      3. log-prod 32.4%

         \[\leadsto \frac{0.5 \cdot e^{\left(\color{blue}{\left(\log 4 + \log a\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}}{a} \]

      4. +-commutative 32.4%

         \[\leadsto \frac{0.5 \cdot e^{\left(\color{blue}{\left(\log a + \log 4\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}}{a} \]

      5. log-prod 32.4%

         \[\leadsto \frac{0.5 \cdot e^{\left(\color{blue}{\log \left(a \cdot 4\right)} - \log \left(\frac{-1}{c}\right)\right) \cdot 0.5}}{a} \]

      6. log-div 67.9%

         \[\leadsto \frac{0.5 \cdot e^{\color{blue}{\log \left(\frac{a \cdot 4}{\frac{-1}{c}}\right)} \cdot 0.5}}{a} \]

      7. exp-to-pow 72.5%

         \[\leadsto \frac{0.5 \cdot \color{blue}{{\left(\frac{a \cdot 4}{\frac{-1}{c}}\right)}^{0.5}}}{a} \]

      8. unpow1/2 72.5%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\sqrt{\frac{a \cdot 4}{\frac{-1}{c}}}}}{a} \]

      9. associate-/r/ 72.5%

         \[\leadsto \frac{0.5 \cdot \sqrt{\color{blue}{\frac{a \cdot 4}{-1} \cdot c}}}{a} \]

      10. *-commutative 72.5%

          \[\leadsto \frac{0.5 \cdot \sqrt{\color{blue}{c \cdot \frac{a \cdot 4}{-1}}}}{a} \]

      11. associate-/l* 72.5%

          \[\leadsto \frac{0.5 \cdot \sqrt{c \cdot \color{blue}{\frac{a}{\frac{-1}{4}}}}}{a} \]

      12. metadata-eval 72.5%

          \[\leadsto \frac{0.5 \cdot \sqrt{c \cdot \frac{a}{\color{blue}{-0.25}}}}{a} \]

   9. Simplified 72.5%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \sqrt{c \cdot \frac{a}{-0.25}}}{a}} \]

   if 2.1499999999999999e-94 < b

   1. Initial program 23.4%

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

   2. Taylor expanded in b around inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   4. Simplified 74.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-94}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{c \cdot \frac{a}{-0.25}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 68.2% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-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 < -1.999999999999994e-310

   1. Initial program 72.7%

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

   2. Taylor expanded in b around -inf 76.4%

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

      1. +-commutative 76.4%

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

      2. mul-1-neg 76.4%

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

      3. unsub-neg 76.4%

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

   4. Simplified 76.4%

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

   if -1.999999999999994e-310 < b

   1. Initial program 36.2%

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

   2. Taylor expanded in b around inf 58.1%

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

      1. associate-*r/ 58.1%

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

      2. neg-mul-1 58.1%

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

   4. Simplified 58.1%

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

4. Final simplification 67.2%

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

Alternative 8: 42.9% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.1 \cdot 10^{+72}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 7.1e+72) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.1e+72) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 7.1d+72) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.1e+72) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.1e+72:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.1e+72)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.1e+72)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.1e+72], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.0999999999999993e72

   1. Initial program 66.0%

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

   2. Taylor expanded in b around -inf 49.5%

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

      1. associate-*r/ 49.5%

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

      2. mul-1-neg 49.5%

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

   4. Simplified 49.5%

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

   if 7.0999999999999993e72 < b

   1. Initial program 11.4%

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

      1. pow1/2 11.4%

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

      2. pow-to-exp 6.4%

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

      3. fma-neg 6.4%

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

      4. *-commutative 6.4%

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

      5. distribute-rgt-neg-in 6.4%

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

      6. *-commutative 6.4%

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

      7. distribute-rgt-neg-in 6.4%

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

      8. metadata-eval 6.4%

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

   3. Applied egg-rr 6.4%

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

   4. Taylor expanded in b around -inf 2.5%

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

      1. neg-mul-1 2.5%

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

      2. +-commutative 2.5%

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

      3. unsub-neg 2.5%

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

      4. associate-/l* 2.6%

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

      5. *-commutative 2.6%

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

      6. associate-*l/ 2.6%

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

   6. Simplified 2.6%

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

   7. Taylor expanded in b around 0 37.2%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.1 \cdot 10^{+72}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 9: 68.0% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 72.7%

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

   2. Taylor expanded in b around -inf 76.3%

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

      1. associate-*r/ 76.3%

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

      2. mul-1-neg 76.3%

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

   4. Simplified 76.3%

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

   if -1.999999999999994e-310 < b

   1. Initial program 36.2%

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

   2. Taylor expanded in b around inf 58.1%

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

      1. associate-*r/ 58.1%

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

      2. neg-mul-1 58.1%

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

   4. Simplified 58.1%

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

4. Final simplification 67.2%

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

Alternative 10: 10.6% 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 54.5%

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

   1. pow1/2 54.5%

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

   2. pow-to-exp 50.9%

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

   3. fma-neg 50.9%

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

   4. *-commutative 50.9%

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

   5. distribute-rgt-neg-in 50.9%

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

   6. *-commutative 50.9%

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

   7. distribute-rgt-neg-in 50.9%

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

   8. metadata-eval 50.9%

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

3. Applied egg-rr 50.9%

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

4. Taylor expanded in b around -inf 37.0%

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

   1. neg-mul-1 37.0%

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

   2. +-commutative 37.0%

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

   3. unsub-neg 37.0%

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

   4. associate-/l* 39.3%

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

   5. *-commutative 39.3%

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

   6. associate-*l/ 39.3%

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

6. Simplified 39.3%

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

7. Taylor expanded in b around 0 10.1%

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

8. Final simplification 10.1%

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

Developer target: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + t_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - t_0}{2 \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* a (/ (- (- b) t_0) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b < 0.0d0) then
        tmp = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-b - t_0) / (2.0d0 * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b < 0.0:
		tmp = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-b - t_0) / (2.0 * a)))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b < 0.0)
		tmp = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023279 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))