quadm (p42, negative)

Percentage Accurate: 52.5% → 85.8%

Time: 12.1s

Alternatives: 8

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: 52.5% 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: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+128}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.6e-103)
   (/ (- c) b)
   (if (<= b 1.55e+128)
     (/ (- (- b) (sqrt (- (* b b) (* c (* 4.0 a))))) (* a 2.0))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.6e-103) {
		tmp = -c / b;
	} else if (b <= 1.55e+128) {
		tmp = (-b - sqrt(((b * b) - (c * (4.0 * a))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.6d-103)) then
        tmp = -c / b
    else if (b <= 1.55d+128) then
        tmp = (-b - sqrt(((b * b) - (c * (4.0d0 * a))))) / (a * 2.0d0)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.6e-103) {
		tmp = -c / b;
	} else if (b <= 1.55e+128) {
		tmp = (-b - Math.sqrt(((b * b) - (c * (4.0 * a))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.6e-103:
		tmp = -c / b
	elif b <= 1.55e+128:
		tmp = (-b - math.sqrt(((b * b) - (c * (4.0 * a))))) / (a * 2.0)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.6e-103)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.55e+128)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.6e-103)
		tmp = -c / b;
	elseif (b <= 1.55e+128)
		tmp = (-b - sqrt(((b * b) - (c * (4.0 * a))))) / (a * 2.0);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.6e-103], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.55e+128], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.60000000000000045e-103

   1. Initial program 14.5%

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

      1. *-commutative 14.5%

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

      2. sqr-neg 14.5%

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

      3. *-commutative 14.5%

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

      4. sqr-neg 14.5%

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

      5. associate-*r* 14.5%

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

      6. *-commutative 14.5%

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

   3. Simplified 14.5%

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

   4. Taylor expanded in b around -inf 86.7%

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

      1. mul-1-neg 86.7%

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

   6. Simplified 86.7%

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

   if -8.60000000000000045e-103 < b < 1.55000000000000002e128

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. sqr-neg 83.8%

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

      3. *-commutative 83.8%

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

      4. sqr-neg 83.8%

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

      5. associate-*r* 83.8%

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

      6. *-commutative 83.8%

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

   3. Simplified 83.8%

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

   if 1.55000000000000002e128 < b

   1. Initial program 45.9%

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

      1. *-commutative 45.9%

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

      2. sqr-neg 45.9%

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

      3. *-commutative 45.9%

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

      4. sqr-neg 45.9%

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

      5. associate-*r* 45.9%

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

      6. *-commutative 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in b around inf 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%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+128}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-96)
   (/ (- c) b)
   (if (<= b 5e+127)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-96) {
		tmp = -c / b;
	} else if (b <= 5e+127) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.5d-96)) then
        tmp = -c / b
    else if (b <= 5d+127) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-96) {
		tmp = -c / b;
	} else if (b <= 5e+127) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.5e-96:
		tmp = -c / b
	elif b <= 5e+127:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e-96)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 5e+127)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.5e-96)
		tmp = -c / b;
	elseif (b <= 5e+127)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e-96], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 5e+127], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.4999999999999997e-96

   1. Initial program 14.5%

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

      1. *-commutative 14.5%

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

      2. sqr-neg 14.5%

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

      3. *-commutative 14.5%

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

      4. sqr-neg 14.5%

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

      5. associate-*r* 14.5%

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

      6. *-commutative 14.5%

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

   3. Simplified 14.5%

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

   4. Taylor expanded in b around -inf 86.7%

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

      1. mul-1-neg 86.7%

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

   6. Simplified 86.7%

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

   if -5.4999999999999997e-96 < b < 5.0000000000000004e127

   1. Initial program 83.8%

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

   if 5.0000000000000004e127 < b

   1. Initial program 45.9%

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

      1. *-commutative 45.9%

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

      2. sqr-neg 45.9%

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

      3. *-commutative 45.9%

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

      4. sqr-neg 45.9%

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

      5. associate-*r* 45.9%

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

      6. *-commutative 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in b around inf 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%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-103)
   (/ (- c) b)
   (if (<= b 2.4e-77)
     (/ (- (- b) (sqrt (* (* c a) -4.0))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-103) {
		tmp = -c / b;
	} else if (b <= 2.4e-77) {
		tmp = (-b - sqrt(((c * a) * -4.0))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.5d-103)) then
        tmp = -c / b
    else if (b <= 2.4d-77) then
        tmp = (-b - sqrt(((c * a) * (-4.0d0)))) / (a * 2.0d0)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-103) {
		tmp = -c / b;
	} else if (b <= 2.4e-77) {
		tmp = (-b - Math.sqrt(((c * a) * -4.0))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e-103:
		tmp = -c / b
	elif b <= 2.4e-77:
		tmp = (-b - math.sqrt(((c * a) * -4.0))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-103)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 2.4e-77)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(c * a) * -4.0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e-103)
		tmp = -c / b;
	elseif (b <= 2.4e-77)
		tmp = (-b - sqrt(((c * a) * -4.0))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-103], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 2.4e-77], N[(N[((-b) - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.50000000000000032e-103

   1. Initial program 14.5%

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

      1. *-commutative 14.5%

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

      2. sqr-neg 14.5%

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

      3. *-commutative 14.5%

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

      4. sqr-neg 14.5%

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

      5. associate-*r* 14.5%

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

      6. *-commutative 14.5%

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

   3. Simplified 14.5%

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

   4. Taylor expanded in b around -inf 86.7%

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

      1. mul-1-neg 86.7%

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

   6. Simplified 86.7%

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

   if -8.50000000000000032e-103 < b < 2.3999999999999999e-77

   1. Initial program 75.4%

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

      1. *-commutative 75.4%

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

      2. sqr-neg 75.4%

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

      3. *-commutative 75.4%

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

      4. sqr-neg 75.4%

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

      5. associate-*r* 75.4%

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

      6. *-commutative 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in b around 0 69.9%

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

   if 2.3999999999999999e-77 < b

   1. Initial program 73.3%

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

      1. *-commutative 73.3%

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

      2. sqr-neg 73.3%

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

      3. *-commutative 73.3%

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

      4. sqr-neg 73.3%

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

      5. associate-*r* 73.3%

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

      6. *-commutative 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in b around inf 85.1%

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

      1. +-commutative 85.1%

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

      2. mul-1-neg 85.1%

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

      3. unsub-neg 85.1%

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

   6. Simplified 85.1%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e-101)
   (/ (- c) b)
   (if (<= b 5.4e-77)
     (* (/ -0.5 a) (+ b (sqrt (* a (* c -4.0)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-101) {
		tmp = -c / b;
	} else if (b <= 5.4e-77) {
		tmp = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.2d-101)) then
        tmp = -c / b
    else if (b <= 5.4d-77) then
        tmp = ((-0.5d0) / a) * (b + sqrt((a * (c * (-4.0d0)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-101) {
		tmp = -c / b;
	} else if (b <= 5.4e-77) {
		tmp = (-0.5 / a) * (b + Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.2e-101:
		tmp = -c / b
	elif b <= 5.4e-77:
		tmp = (-0.5 / a) * (b + math.sqrt((a * (c * -4.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e-101)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 5.4e-77)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.2e-101)
		tmp = -c / b;
	elseif (b <= 5.4e-77)
		tmp = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e-101], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 5.4e-77], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.2000000000000002e-101

   1. Initial program 14.5%

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

      1. *-commutative 14.5%

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

      2. sqr-neg 14.5%

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

      3. *-commutative 14.5%

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

      4. sqr-neg 14.5%

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

      5. associate-*r* 14.5%

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

      6. *-commutative 14.5%

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

   3. Simplified 14.5%

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

   4. Taylor expanded in b around -inf 86.7%

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

      1. mul-1-neg 86.7%

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

   6. Simplified 86.7%

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

   if -5.2000000000000002e-101 < b < 5.4000000000000001e-77

   1. Initial program 75.4%

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

      1. *-commutative 75.4%

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

      2. sqr-neg 75.4%

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

      3. *-commutative 75.4%

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

      4. sqr-neg 75.4%

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

      5. associate-*r* 75.4%

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

      6. *-commutative 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in b around 0 69.9%

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

      1. frac-2neg 69.9%

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

      2. div-inv 69.8%

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

      3. neg-sub0 69.8%

         \[\leadsto \color{blue}{\left(0 - \left(\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\right)} \cdot \frac{1}{-a \cdot 2} \]

      4. add-sqr-sqrt 36.1%

         \[\leadsto \left(0 - \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

      5. sqrt-unprod 67.9%

         \[\leadsto \left(0 - \left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

      6. sqr-neg 67.9%

         \[\leadsto \left(0 - \left(\sqrt{\color{blue}{b \cdot b}} - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

      7. sqrt-prod 32.1%

         \[\leadsto \left(0 - \left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

      8. add-sqr-sqrt 67.9%

         \[\leadsto \left(0 - \left(\color{blue}{b} - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

      9. associate-+l- 67.9%

         \[\leadsto \color{blue}{\left(\left(0 - b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}\right)} \cdot \frac{1}{-a \cdot 2} \]

      10. neg-sub0 67.9%

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

      11. add-sqr-sqrt 35.8%

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

      12. sqrt-unprod 68.6%

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

      13. sqr-neg 68.6%

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

      14. sqrt-prod 33.6%

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

      15. add-sqr-sqrt 69.8%

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

      16. *-commutative 69.8%

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

      17. associate-*r* 69.7%

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

      18. distribute-rgt-neg-in 69.7%

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

      19. metadata-eval 69.7%

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

      20. metadata-eval 69.7%

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

      21. div-inv 69.7%

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

   6. Applied egg-rr 69.7%

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

      1. *-commutative 69.7%

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

   8. Simplified 69.7%

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

   if 5.4000000000000001e-77 < b

   1. Initial program 73.3%

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

      1. *-commutative 73.3%

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

      2. sqr-neg 73.3%

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

      3. *-commutative 73.3%

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

      4. sqr-neg 73.3%

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

      5. associate-*r* 73.3%

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

      6. *-commutative 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in b around inf 85.1%

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

      1. +-commutative 85.1%

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

      2. mul-1-neg 85.1%

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

      3. unsub-neg 85.1%

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

   6. Simplified 85.1%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 68.2% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[((-c) / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 31.1%

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

      1. *-commutative 31.1%

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

      2. sqr-neg 31.1%

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

      3. *-commutative 31.1%

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

      4. sqr-neg 31.1%

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

      5. associate-*r* 31.2%

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

      6. *-commutative 31.2%

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

   3. Simplified 31.2%

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

   4. Taylor expanded in b around -inf 65.4%

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

      1. mul-1-neg 65.4%

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

   6. Simplified 65.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

      2. sqr-neg 73.4%

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

      3. *-commutative 73.4%

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

      4. sqr-neg 73.4%

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

      5. associate-*r* 73.4%

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

      6. *-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in b around inf 67.5%

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

      1. +-commutative 67.5%

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

      2. mul-1-neg 67.5%

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

      3. unsub-neg 67.5%

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

   6. Simplified 67.5%

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

4. Final simplification 66.5%

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

Alternative 6: 68.0% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-304}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-304) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-304) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-304)) then
        tmp = -c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-304) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-304:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-304)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-304)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-304], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.99999999999999988e-304

   1. Initial program 30.6%

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

      1. *-commutative 30.6%

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

      2. sqr-neg 30.6%

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

      3. *-commutative 30.6%

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

      4. sqr-neg 30.6%

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

      5. associate-*r* 30.6%

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

      6. *-commutative 30.6%

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

   3. Simplified 30.6%

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

   4. Taylor expanded in b around -inf 65.9%

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

      1. mul-1-neg 65.9%

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

   6. Simplified 65.9%

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

   if -3.99999999999999988e-304 < b

   1. Initial program 73.6%

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

      1. *-commutative 73.6%

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

      2. sqr-neg 73.6%

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

      3. *-commutative 73.6%

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

      4. sqr-neg 73.6%

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

      5. associate-*r* 73.6%

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

      6. *-commutative 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in b around inf 66.6%

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

      1. associate-*r/ 66.6%

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

      2. mul-1-neg 66.6%

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

   6. Simplified 66.6%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-304}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7: 34.1% accurate, 29.0× 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(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 53.0%

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

   1. *-commutative 53.0%

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

   2. sqr-neg 53.0%

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

   3. *-commutative 53.0%

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

   4. sqr-neg 53.0%

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

   5. associate-*r* 53.0%

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

   6. *-commutative 53.0%

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

3. Simplified 53.0%

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

4. Taylor expanded in b around -inf 32.9%

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

   1. mul-1-neg 32.9%

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

6. Simplified 32.9%

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

7. Final simplification 32.9%

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

Alternative 8: 10.9% 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 53.0%

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

   1. *-commutative 53.0%

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

   2. sqr-neg 53.0%

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

   3. *-commutative 53.0%

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

   4. sqr-neg 53.0%

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

   5. associate-*r* 53.0%

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

   6. *-commutative 53.0%

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

3. Simplified 53.0%

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

5. Applied egg-rr 19.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a \cdot -2}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 33.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a \cdot -2}\right)\right)} \]

   2. expm1-log1p 53.0%

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

   3. *-rgt-identity 53.0%

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

   4. associate-*r/ 52.9%

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

   5. *-commutative 52.9%

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

   6. associate-/r* 52.9%

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

   7. metadata-eval 52.9%

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

7. Simplified 52.9%

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

8. Taylor expanded in b around -inf 25.5%

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

   1. associate-*r/ 25.5%

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

   2. clear-num 25.4%

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

   3. associate-*r/ 25.4%

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

10. Applied egg-rr 25.4%

    \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{2 \cdot \left(a \cdot c\right)}{b} \cdot -0.5}}} \]
11. Step-by-step derivation

    1. associate-/r/ 25.5%

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

    2. associate-*l/ 25.5%

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

    3. *-commutative 25.5%

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

    4. associate-*r* 25.5%

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

    5. metadata-eval 25.5%

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

    6. neg-mul-1 25.5%

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

    7. distribute-neg-frac 25.5%

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

    8. associate-*r/ 27.8%

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

    9. distribute-rgt-neg-in 27.8%

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

    10. distribute-frac-neg 27.8%

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

12. Simplified 27.8%

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

    1. expm1-log1p-u 24.8%

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

    2. expm1-udef 13.7%

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

    3. associate-*r* 13.8%

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

    4. lft-mult-inverse 13.8%

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

    5. *-un-lft-identity 13.8%

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

    6. add-sqr-sqrt 6.0%

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

    7. sqrt-unprod 12.3%

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

    8. sqr-neg 12.3%

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

    9. sqrt-unprod 5.9%

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

    10. add-sqr-sqrt 11.6%

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

14. Applied egg-rr 11.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c}{b}\right)} - 1} \]
15. Step-by-step derivation

    1. expm1-def 11.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{b}\right)\right)} \]

    2. expm1-log1p 11.9%

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

16. Simplified 11.9%

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

17. Final simplification 11.9%

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

Developer target: 99.7% accurate, 0.1× speedup

Math

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

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) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	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(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	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 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	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[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

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

  :herbie-target
  (if (< b 0.0) (/ c (- (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))) (/ (+ (/ 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)))))) (- a)))

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