quadp (p42, positive)

Percentage Accurate: 51.5% → 85.5%

Time: 13.1s

Alternatives: 10

Speedup: 19.1×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+154)
   (- (/ b a))
   (if (<= b 8.5e-119)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (if (or (<= b 6.5e-61) (not (<= b 2.9e-34)))
       (/ (- c) b)
       (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+154) {
		tmp = -(b / a);
	} else if (b <= 8.5e-119) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else if ((b <= 6.5e-61) || !(b <= 2.9e-34)) {
		tmp = -c / b;
	} else {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	}
	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)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	elseif ((b <= 6.5e-61) || !(b <= 2.9e-34))
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e+154], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 8.5e-119], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 6.5e-61], N[Not[LessEqual[b, 2.9e-34]], $MachinePrecision]], N[((-c) / b), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.00000000000000015e154

   1. Initial program 41.2%

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

      1. *-commutative 41.2%

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

   3. Simplified 41.2%

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

   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

   1. Initial program 82.0%

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

      1. +-commutative 82.0%

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

      2. unsub-neg 82.0%

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

      3. fma-neg 82.0%

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

      4. distribute-lft-neg-in 82.0%

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

      5. *-commutative 82.0%

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

      6. *-commutative 82.0%

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

      7. associate-*l* 82.0%

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

      8. metadata-eval 82.0%

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

      9. *-commutative 82.0%

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

   3. Simplified 82.0%

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

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

   1. Initial program 17.4%

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

      1. *-commutative 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in b around inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. neg-mul-1 86.6%

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

   6. Simplified 86.6%

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

   if 6.4999999999999994e-61 < b < 2.9000000000000002e-34

   1. Initial program 86.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

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

Alternative 2: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+152}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119} \lor \neg \left(b \leq 5.5 \cdot 10^{-61}\right) \land b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+152)
   (- (/ b a))
   (if (or (<= b 8.5e-119) (and (not (<= b 5.5e-61)) (<= b 3.1e-34)))
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+152) {
		tmp = -(b / a);
	} else if ((b <= 8.5e-119) || (!(b <= 5.5e-61) && (b <= 3.1e-34))) {
		tmp = (sqrt(((b * b) - (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 <= (-4d+152)) then
        tmp = -(b / a)
    else if ((b <= 8.5d-119) .or. (.not. (b <= 5.5d-61)) .and. (b <= 3.1d-34)) then
        tmp = (sqrt(((b * b) - (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 <= -4e+152) {
		tmp = -(b / a);
	} else if ((b <= 8.5e-119) || (!(b <= 5.5e-61) && (b <= 3.1e-34))) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e+152:
		tmp = -(b / a)
	elif (b <= 8.5e-119) or (not (b <= 5.5e-61) and (b <= 3.1e-34)):
		tmp = (math.sqrt(((b * b) - (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 <= -4e+152)
		tmp = Float64(-Float64(b / a));
	elseif ((b <= 8.5e-119) || (!(b <= 5.5e-61) && (b <= 3.1e-34)))
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - 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 <= -4e+152)
		tmp = -(b / a);
	elseif ((b <= 8.5e-119) || (~((b <= 5.5e-61)) && (b <= 3.1e-34)))
		tmp = (sqrt(((b * b) - (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, -4e+152], (-N[(b / a), $MachinePrecision]), If[Or[LessEqual[b, 8.5e-119], And[N[Not[LessEqual[b, 5.5e-61]], $MachinePrecision], LessEqual[b, 3.1e-34]]], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $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 < -4.0000000000000002e152

   1. Initial program 41.2%

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

      1. *-commutative 41.2%

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

   3. Simplified 41.2%

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

   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 -4.0000000000000002e152 < b < 8.49999999999999977e-119 or 5.4999999999999997e-61 < b < 3.0999999999999998e-34

   1. Initial program 82.2%

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

   if 8.49999999999999977e-119 < b < 5.4999999999999997e-61 or 3.0999999999999998e-34 < b

   1. Initial program 17.4%

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

      1. *-commutative 17.4%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in b around inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. neg-mul-1 86.6%

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

   6. Simplified 86.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+152}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119} \lor \neg \left(b \leq 5.5 \cdot 10^{-61}\right) \land b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-81}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-81)
   (- (/ b a))
   (if (<= b 3.6e-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 <= -1.15e-81) {
		tmp = -(b / a);
	} else if (b <= 3.6e-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 <= (-1.15d-81)) then
        tmp = -(b / a)
    else if (b <= 3.6d-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 <= -1.15e-81) {
		tmp = -(b / a);
	} else if (b <= 3.6e-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 <= -1.15e-81:
		tmp = -(b / a)
	elif b <= 3.6e-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 <= -1.15e-81)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 3.6e-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 <= -1.15e-81)
		tmp = -(b / a);
	elseif (b <= 3.6e-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, -1.15e-81], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 3.6e-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 < -1.14999999999999996e-81

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

   3. Simplified 67.6%

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

   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 -1.14999999999999996e-81 < b < 3.6e-119

   1. Initial program 73.8%

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

      1. *-commutative 73.8%

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

   3. Simplified 73.8%

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

   5. Applied egg-rr 72.9%

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

      1. +-commutative 72.9%

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

      2. fma-udef 72.9%

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

      3. associate-+l+ 72.9%

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

      4. *-commutative 72.9%

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

      5. *-commutative 72.9%

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

      6. +-commutative 72.9%

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

      7. fma-udef 72.9%

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

      8. unpow2 72.9%

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

      9. associate-+l+ 72.9%

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

      10. distribute-lft-out 72.9%

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

   7. Simplified 72.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right) \cdot {\left(\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right)}^{2} + \left({b}^{2} + a \cdot \left(c \cdot -8\right)\right)}}}{a \cdot 2} \]

   8. Taylor expanded in b around 0 70.6%

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

      1. neg-mul-1 70.6%

         \[\leadsto \frac{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + \color{blue}{\left(-b\right)}}{a \cdot 2} \]

      2. unsub-neg 70.6%

         \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} - b}}{a \cdot 2} \]

      3. distribute-rgt-out 71.0%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 + 4\right)}} - b}{a \cdot 2} \]

      4. metadata-eval 71.0%

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

      5. associate-*r* 71.0%

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

   10. Simplified 71.0%

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

   if 3.6e-119 < b

   1. Initial program 22.1%

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

      1. *-commutative 22.1%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in b around inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. neg-mul-1 82.3%

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

   6. Simplified 82.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-81}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-81}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e-81)
   (- (/ b a))
   (if (<= b 8e-121) (/ (- (sqrt (* -4.0 (* a c))) b) (* a 2.0)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e-81) {
		tmp = -(b / a);
	} else if (b <= 8e-121) {
		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 <= (-1.2d-81)) then
        tmp = -(b / a)
    else if (b <= 8d-121) 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 <= -1.2e-81) {
		tmp = -(b / a);
	} else if (b <= 8e-121) {
		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 <= -1.2e-81:
		tmp = -(b / a)
	elif b <= 8e-121:
		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 <= -1.2e-81)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 8e-121)
		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 <= -1.2e-81)
		tmp = -(b / a);
	elseif (b <= 8e-121)
		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, -1.2e-81], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 8e-121], 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 < -1.2e-81

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

   3. Simplified 67.6%

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

   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 -1.2e-81 < b < 7.9999999999999998e-121

   1. Initial program 73.8%

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

      1. *-commutative 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in b around 0 71.0%

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

      1. *-commutative 71.0%

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

   6. Simplified 71.0%

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

   if 7.9999999999999998e-121 < b

   1. Initial program 22.1%

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

      1. *-commutative 22.1%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in b around inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. neg-mul-1 82.3%

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

   6. Simplified 82.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-81}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-95}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-95)
   (- (/ b a))
   (if (<= b 8.5e-119) (* 0.5 (/ (sqrt (* a (* c -4.0))) a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-95) {
		tmp = -(b / a);
	} else if (b <= 8.5e-119) {
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / 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 <= (-2.4d-95)) then
        tmp = -(b / a)
    else if (b <= 8.5d-119) then
        tmp = 0.5d0 * (sqrt((a * (c * (-4.0d0)))) / 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 <= -2.4e-95) {
		tmp = -(b / a);
	} else if (b <= 8.5e-119) {
		tmp = 0.5 * (Math.sqrt((a * (c * -4.0))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-95:
		tmp = -(b / a)
	elif b <= 8.5e-119:
		tmp = 0.5 * (math.sqrt((a * (c * -4.0))) / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-95)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 8.5e-119)
		tmp = Float64(0.5 * Float64(sqrt(Float64(a * Float64(c * -4.0))) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e-95)
		tmp = -(b / a);
	elseif (b <= 8.5e-119)
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-95], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 8.5e-119], N[(0.5 * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4e-95

   1. Initial program 68.4%

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

      1. *-commutative 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in b around -inf 86.7%

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

      1. associate-*r/ 86.7%

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

      2. mul-1-neg 86.7%

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

   6. Simplified 86.7%

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

   if -2.4e-95 < b < 8.49999999999999977e-119

   1. Initial program 73.0%

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

      1. *-commutative 73.0%

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

   3. Simplified 73.0%

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

   5. Applied egg-rr 72.0%

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

      1. +-commutative 72.0%

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

      2. fma-udef 72.0%

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

      3. associate-+l+ 72.0%

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

      4. *-commutative 72.0%

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

      5. *-commutative 72.0%

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

      6. +-commutative 72.0%

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

      7. fma-udef 72.0%

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

      8. unpow2 72.0%

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

      9. associate-+l+ 72.0%

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

      10. distribute-lft-out 72.0%

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

   7. Simplified 72.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right) \cdot {\left(\sqrt[3]{a \cdot \left(c \cdot -4\right)}\right)}^{2} + \left({b}^{2} + a \cdot \left(c \cdot -8\right)\right)}}}{a \cdot 2} \]

   8. Taylor expanded in b around 0 70.3%

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

      1. associate-*l/ 70.4%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}}{a}} \]

      2. *-lft-identity 70.4%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}}}{a} \]

      3. distribute-rgt-out 70.8%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 + 4\right)}}}{a} \]

      4. metadata-eval 70.8%

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

      5. associate-*r* 70.8%

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

   10. Simplified 70.8%

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

   if 8.49999999999999977e-119 < b

   1. Initial program 22.1%

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

      1. *-commutative 22.1%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in b around inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. neg-mul-1 82.3%

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

   6. Simplified 82.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-95}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 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%

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

      1. *-commutative 71.2%

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

   3. Simplified 71.2%

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

   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%

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

      1. *-commutative 31.8%

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

   3. Simplified 31.8%

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

   4. Taylor expanded in b around inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. neg-mul-1 67.1%

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

   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 7: 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 63.2%

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

      1. *-commutative 63.2%

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

   3. Simplified 63.2%

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

   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%

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

      1. +-commutative 15.9%

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

      2. unsub-neg 15.9%

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

      3. fma-neg 15.9%

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

      4. distribute-lft-neg-in 15.9%

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

      5. *-commutative 15.9%

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

      6. *-commutative 15.9%

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

      7. associate-*l* 15.9%

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

      8. metadata-eval 15.9%

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

      9. *-commutative 15.9%

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

   3. Simplified 15.9%

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

      1. sub-neg 15.9%

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

   5. Applied egg-rr 0.8%

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

   6. Taylor expanded in b around inf 2.3%

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

   7. 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 8: 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%

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

      1. *-commutative 70.1%

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

   3. Simplified 70.1%

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

   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%

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

      1. *-commutative 32.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in b around inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. neg-mul-1 68.1%

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

   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 9: 2.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.8%

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

   1. +-commutative 50.8%

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

   2. unsub-neg 50.8%

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

   3. fma-neg 50.8%

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

   4. distribute-lft-neg-in 50.8%

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

   5. *-commutative 50.8%

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

   6. *-commutative 50.8%

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

   7. associate-*l* 50.9%

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

   8. metadata-eval 50.9%

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

   9. *-commutative 50.9%

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

3. Simplified 50.9%

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

   1. sub-neg 50.9%

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

5. Applied egg-rr 1.1%

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

6. Taylor expanded in b around inf 2.7%

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

7. Final simplification 2.7%

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

Alternative 10: 11.4% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.8%

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

   1. +-commutative 50.8%

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

   2. unsub-neg 50.8%

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

   3. fma-neg 50.8%

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

   4. distribute-lft-neg-in 50.8%

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

   5. *-commutative 50.8%

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

   6. *-commutative 50.8%

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

   7. associate-*l* 50.9%

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

   8. metadata-eval 50.9%

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

   9. *-commutative 50.9%

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

3. Simplified 50.9%

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

   1. sub-neg 50.9%

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

5. Applied egg-rr 1.1%

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

6. Taylor expanded in b around inf 2.3%

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

7. Taylor expanded in b around 0 10.6%

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

8. Final simplification 10.6%

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

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t_0 - t_1} \cdot \sqrt{t_0 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2023332 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))))))

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