The quadratic formula (r1)

Percentage Accurate: 51.3% → 85.7%

Time: 14.3s

Alternatives: 7

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+100}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+100)
   (/ b (- a))
   (if (<= b 2.2e-78)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+100) {
		tmp = b / -a;
	} else if (b <= 2.2e-78) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d+100)) then
        tmp = b / -a
    else if (b <= 2.2d-78) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+100) {
		tmp = b / -a;
	} else if (b <= 2.2e-78) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e+100:
		tmp = b / -a
	elif b <= 2.2e-78:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e+100)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2.2e-78)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * 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 <= -4e+100)
		tmp = b / -a;
	elseif (b <= 2.2e-78)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e+100], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2.2e-78], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.00000000000000006e100

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in b around -inf 96.1%

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

      1. associate-*r/ 96.1%

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

      2. mul-1-neg 96.1%

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

   7. Simplified 96.1%

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

   if -4.00000000000000006e100 < b < 2.1999999999999999e-78

   1. Initial program 83.9%

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

   if 2.1999999999999999e-78 < b

   1. Initial program 20.1%

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

      1. *-commutative 20.1%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in a around 0 85.9%

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

      1. associate-*r/ 85.9%

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

      2. mul-1-neg 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+100}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.4e-74)
   (/ b (- a))
   (if (<= b 1.45e-80)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e-74) {
		tmp = b / -a;
	} else if (b <= 1.45e-80) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.4d-74)) then
        tmp = b / -a
    else if (b <= 1.45d-80) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e-74) {
		tmp = b / -a;
	} else if (b <= 1.45e-80) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.4e-74:
		tmp = b / -a
	elif b <= 1.45e-80:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.4e-74)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.45e-80)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - 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 <= -3.4e-74)
		tmp = b / -a;
	elseif (b <= 1.45e-80)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.4e-74], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.45e-80], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4000000000000001e-74

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in b around -inf 87.7%

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

      1. associate-*r/ 87.7%

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

      2. mul-1-neg 87.7%

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

   7. Simplified 87.7%

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

   if -3.4000000000000001e-74 < b < 1.44999999999999999e-80

   1. Initial program 77.0%

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

      1. *-commutative 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in a around inf 73.8%

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

      1. *-commutative 73.8%

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

      2. associate-*r* 73.8%

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

   7. Simplified 73.8%

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

   if 1.44999999999999999e-80 < b

   1. Initial program 20.1%

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

      1. *-commutative 20.1%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in a around 0 85.9%

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

      1. associate-*r/ 85.9%

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

      2. mul-1-neg 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.15 \cdot 10^{-75}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.15e-75)
   (/ b (- a))
   (if (<= b 6.5e-7) (* 0.5 (/ (sqrt (* a (* c -4.0))) a)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.15e-75) {
		tmp = b / -a;
	} else if (b <= 6.5e-7) {
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.15d-75)) then
        tmp = b / -a
    else if (b <= 6.5d-7) then
        tmp = 0.5d0 * (sqrt((a * (c * (-4.0d0)))) / a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.15e-75) {
		tmp = b / -a;
	} else if (b <= 6.5e-7) {
		tmp = 0.5 * (Math.sqrt((a * (c * -4.0))) / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.15e-75:
		tmp = b / -a
	elif b <= 6.5e-7:
		tmp = 0.5 * (math.sqrt((a * (c * -4.0))) / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.15e-75)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 6.5e-7)
		tmp = Float64(0.5 * Float64(sqrt(Float64(a * Float64(c * -4.0))) / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.15e-75)
		tmp = b / -a;
	elseif (b <= 6.5e-7)
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.15e-75], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 6.5e-7], N[(0.5 * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.14999999999999987e-75

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in b around -inf 87.7%

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

      1. associate-*r/ 87.7%

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

      2. mul-1-neg 87.7%

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

   7. Simplified 87.7%

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

   if -4.14999999999999987e-75 < b < 6.50000000000000024e-7

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

   3. Simplified 70.6%

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

      1. div-sub 70.6%

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

      2. sub-neg 70.6%

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

      3. div-inv 70.6%

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

      4. pow2 70.6%

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

      5. *-commutative 70.6%

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

      6. associate-/r* 70.6%

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

      7. metadata-eval 70.6%

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

      8. div-inv 70.5%

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

      9. *-commutative 70.5%

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

      10. associate-/r* 70.5%

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

      11. metadata-eval 70.5%

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

   6. Applied egg-rr 70.5%

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

      1. sub-neg 70.5%

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

      2. distribute-rgt-out-- 70.5%

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

   8. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 70.5%

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

      2. unpow2 70.5%

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

      3. sqr-neg 70.5%

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

      4. neg-mul-1 70.5%

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

      5. associate-*l* 70.5%

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

      6. metadata-eval 70.5%

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

      7. pow1 70.5%

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

      8. metadata-eval 70.5%

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

      9. sqrt-pow1 67.9%

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

      10. add-sqr-sqrt 20.6%

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

      11. sqrt-unprod 67.3%

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

      12. sqr-neg 67.3%

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

      13. unpow2 67.3%

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

      14. add-sqr-sqrt 67.3%

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

      15. cancel-sign-sub-inv 67.3%

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

      16. add-sqr-sqrt 67.3%

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

      17. *-un-lft-identity 67.3%

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

      18. unpow2 67.3%

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

      19. difference-of-squares 67.3%

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

   10. Applied egg-rr 32.5%

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

       1. *-commutative 32.5%

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

       2. clear-num 32.5%

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

       3. un-div-inv 32.5%

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

       4. fma-undefine 32.5%

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

       5. sqrt-prod 67.5%

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

       6. associate--l+ 68.0%

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

       7. div-inv 68.0%

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

       8. metadata-eval 68.0%

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

   12. Applied egg-rr 68.0%

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

       1. *-lft-identity 68.0%

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

       2. *-commutative 68.0%

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

       3. times-frac 68.0%

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

       4. metadata-eval 68.0%

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

       5. +-inverses 68.0%

          \[\leadsto 0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)} + \color{blue}{0}}{a} \]

       6. +-rgt-identity 68.0%

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

   14. Simplified 68.0%

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

   if 6.50000000000000024e-7 < b

   1. Initial program 14.6%

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

      1. *-commutative 14.6%

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

   3. Simplified 14.6%

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

   5. Taylor expanded in a around 0 92.7%

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

      1. associate-*r/ 92.7%

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

      2. mul-1-neg 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.15 \cdot 10^{-75}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{-248}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.3e-248) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.3e-248) {
		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 <= 1.3d-248) 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 <= 1.3e-248) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.3e-248:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.3e-248)
		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 <= 1.3e-248)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.3e-248], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.30000000000000003e-248

   1. Initial program 73.7%

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

      1. *-commutative 73.7%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in b around -inf 68.9%

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

      1. associate-*r/ 68.9%

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

      2. mul-1-neg 68.9%

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

   7. Simplified 68.9%

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

   if 1.30000000000000003e-248 < b

   1. Initial program 31.2%

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

      1. *-commutative 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%

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

   5. Taylor expanded in a around 0 72.9%

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

      1. associate-*r/ 72.9%

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

      2. mul-1-neg 72.9%

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

   7. Simplified 72.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{-248}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 55000000:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 55000000.0) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 55000000.0) {
		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 <= 55000000.0d0) 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 <= 55000000.0) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 55000000.0:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 55000000.0)
		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 <= 55000000.0)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 55000000.0], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5e7

   1. Initial program 70.1%

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

      1. *-commutative 70.1%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in b around -inf 50.7%

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

      1. associate-*r/ 50.7%

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

      2. mul-1-neg 50.7%

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

   7. Simplified 50.7%

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

   if 5.5e7 < b

   1. Initial program 15.0%

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

      1. *-commutative 15.0%

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

   3. Simplified 15.0%

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

   6. Applied egg-rr 6.8%

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

      1. unpow-1 6.8%

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

      2. *-commutative 6.8%

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

      3. *-lft-identity 6.8%

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

      4. times-frac 6.8%

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

      5. metadata-eval 6.8%

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

   8. Simplified 6.8%

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

   9. Taylor expanded in b around -inf 30.7%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 55000000:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 10.7% 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 51.6%

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

   1. *-commutative 51.6%

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

3. Simplified 51.6%

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

6. Applied egg-rr 32.5%

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

   1. unpow-1 32.5%

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

   2. *-commutative 32.5%

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

   3. *-lft-identity 32.5%

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

   4. times-frac 32.5%

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

   5. metadata-eval 32.5%

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

8. Simplified 32.5%

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

9. Taylor expanded in b around -inf 12.2%

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

Alternative 7: 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 51.6%

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

   1. *-commutative 51.6%

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

3. Simplified 51.6%

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

6. Applied egg-rr 32.5%

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

   1. unpow-1 32.5%

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

   2. *-commutative 32.5%

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

   3. *-lft-identity 32.5%

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

   4. times-frac 32.5%

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

   5. metadata-eval 32.5%

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

8. Simplified 32.5%

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

9. Taylor expanded in a around 0 2.7%

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

Developer Target 1: 69.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - t\_0}{2 \cdot a}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024147 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (- (* b b) (* (* 4 a) c)))) (let ((r1 (/ (+ (- b) (sqrt d)) (* 2 a)))) (let ((r2 (/ (- (- b) (sqrt d)) (* 2 a)))) (if (< b 0) r1 (/ c (* a r2)))))))

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