quadp (p42, positive)

Percentage Accurate: 51.8% → 85.7%

Time: 40.9s

Alternatives: 6

Speedup: 12.9×

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: 51.8% 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: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+151}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+151)
   (/ b (- a))
   (if (<= b 7.6e-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 <= -3e+151) {
		tmp = b / -a;
	} else if (b <= 7.6e-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 <= (-3d+151)) then
        tmp = b / -a
    else if (b <= 7.6d-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 <= -3e+151) {
		tmp = b / -a;
	} else if (b <= 7.6e-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 <= -3e+151:
		tmp = b / -a
	elif b <= 7.6e-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 <= -3e+151)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 7.6e-34)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e+151)
		tmp = b / -a;
	elseif (b <= 7.6e-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, -3e+151], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 7.6e-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 < -2.9999999999999999e151

   1. Initial program 52.3%

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

      1. *-commutative 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in b around -inf 96.3%

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

      1. mul-1-neg 96.3%

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

      2. distribute-neg-frac2 96.3%

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

   7. Simplified 96.3%

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

   if -2.9999999999999999e151 < b < 7.6000000000000002e-34

   1. Initial program 83.1%

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

   if 7.6000000000000002e-34 < b

   1. Initial program 15.2%

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

      1. *-commutative 15.2%

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

   3. Simplified 15.2%

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

   5. Taylor expanded in b around inf 89.5%

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

      1. associate-*r/ 89.5%

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

      2. neg-mul-1 89.5%

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

   7. Simplified 89.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+151}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-70)
   (/ b (- a))
   (if (<= b 1.3e-38)
     (/ (- b (sqrt (* c (* a -4.0)))) (* a -2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-70) {
		tmp = b / -a;
	} else if (b <= 1.3e-38) {
		tmp = (b - sqrt((c * (a * -4.0)))) / (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 <= (-5.5d-70)) then
        tmp = b / -a
    else if (b <= 1.3d-38) then
        tmp = (b - sqrt((c * (a * (-4.0d0))))) / (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 <= -5.5e-70) {
		tmp = b / -a;
	} else if (b <= 1.3e-38) {
		tmp = (b - Math.sqrt((c * (a * -4.0)))) / (a * -2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.5e-70:
		tmp = b / -a
	elif b <= 1.3e-38:
		tmp = (b - math.sqrt((c * (a * -4.0)))) / (a * -2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e-70)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.3e-38)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * -2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.5e-70)
		tmp = b / -a;
	elseif (b <= 1.3e-38)
		tmp = (b - sqrt((c * (a * -4.0)))) / (a * -2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e-70], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.3e-38], N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5000000000000001e-70

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in b around -inf 88.1%

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

      1. mul-1-neg 88.1%

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

      2. distribute-neg-frac2 88.1%

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

   7. Simplified 88.1%

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

   if -5.5000000000000001e-70 < b < 1.30000000000000005e-38

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in b around 0 72.6%

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

      1. associate-*r* 72.6%

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

      2. *-commutative 72.6%

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

   7. Simplified 72.6%

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

      1. associate-*l* 72.6%

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

      2. sqrt-prod 49.7%

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

      3. *-commutative 49.7%

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

   9. Applied egg-rr 49.7%

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

       1. frac-2neg 49.7%

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

       2. div-inv 49.8%

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

   11. Applied egg-rr 72.4%

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

       1. associate-*r/ 72.6%

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

       2. *-rgt-identity 72.6%

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

       3. *-commutative 72.6%

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

       4. *-commutative 72.6%

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

   13. Simplified 72.6%

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

   if 1.30000000000000005e-38 < b

   1. Initial program 15.2%

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

      1. *-commutative 15.2%

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

   3. Simplified 15.2%

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

   5. Taylor expanded in b around inf 89.5%

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

      1. associate-*r/ 89.5%

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

      2. neg-mul-1 89.5%

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

   7. Simplified 89.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-297}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.9e-297) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.9e-297) {
		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 <= 1.9d-297) 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 <= 1.9e-297) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.9e-297:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.9e-297)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.9e-297)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.9e-297], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.90000000000000002e-297

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in b around -inf 69.5%

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

      1. mul-1-neg 69.5%

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

      2. distribute-neg-frac2 69.5%

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

   7. Simplified 69.5%

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

   if 1.90000000000000002e-297 < b

   1. Initial program 33.0%

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

      1. *-commutative 33.0%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in b around inf 64.1%

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

      1. associate-*r/ 64.1%

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

      2. neg-mul-1 64.1%

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

   7. Simplified 64.1%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-297}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 5.8e+27) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.8e+27) {
		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 <= 5.8d+27) 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 <= 5.8e+27) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.8e+27:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.8e+27)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.8e+27)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.8e+27], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.8000000000000002e27

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in b around -inf 51.6%

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

      1. mul-1-neg 51.6%

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

      2. distribute-neg-frac2 51.6%

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

   7. Simplified 51.6%

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

   if 5.8000000000000002e27 < b

   1. Initial program 13.1%

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

      1. *-commutative 13.1%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in b around inf 82.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-/l* 85.9%

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

   7. Simplified 85.9%

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

      1. associate-*r/ 82.3%

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

      2. add-cube-cbrt 81.7%

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

      3. unpow3 81.7%

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

      4. frac-2neg 81.7%

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

      5. unpow3 81.7%

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

      6. add-cube-cbrt 82.3%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 41.0%

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

      9. sqr-neg 41.0%

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

      10. sqrt-prod 40.7%

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

      11. add-sqr-sqrt 40.7%

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

   9. Applied egg-rr 40.7%

      \[\leadsto \frac{-2 \cdot \color{blue}{\frac{-a \cdot c}{b}}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. distribute-rgt-neg-in 40.7%

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

   11. Simplified 40.7%

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

   12. Taylor expanded in a around 0 40.7%

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

Alternative 5: 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 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in b around inf 26.0%

   \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. associate-/l* 27.7%

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

7. Simplified 27.7%

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

   1. associate-*r/ 26.0%

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

   2. add-cube-cbrt 25.8%

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

   3. unpow3 25.8%

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

   4. frac-2neg 25.8%

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

   5. unpow3 25.8%

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

   6. add-cube-cbrt 26.0%

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

   7. add-sqr-sqrt 1.2%

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

   8. sqrt-unprod 11.8%

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

   9. sqr-neg 11.8%

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

   10. sqrt-prod 10.7%

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

   11. add-sqr-sqrt 12.5%

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

9. Applied egg-rr 12.5%

   \[\leadsto \frac{-2 \cdot \color{blue}{\frac{-a \cdot c}{b}}}{a \cdot 2} \]
10. Step-by-step derivation

    1. distribute-rgt-neg-in 12.5%

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

11. Simplified 12.5%

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

12. Taylor expanded in a around 0 12.6%

    \[\leadsto \color{blue}{\frac{c}{b}} \]
13. Add Preprocessing

Alternative 6: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

5. Taylor expanded in b around -inf 39.1%

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

   1. mul-1-neg 39.1%

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

   2. distribute-neg-frac2 39.1%

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

7. Simplified 39.1%

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

   1. add-sqr-sqrt 22.1%

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

   2. sqrt-unprod 20.2%

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

   3. sqr-neg 20.2%

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

   4. sqrt-unprod 1.3%

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

   5. add-sqr-sqrt 2.5%

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

   6. *-un-lft-identity 2.5%

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

9. Applied egg-rr 2.5%

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

    1. *-lft-identity 2.5%

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

11. Simplified 2.5%

    \[\leadsto \frac{b}{\color{blue}{a}} \]
12. Add Preprocessing

Developer Target 1: 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 2024116 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

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