The quadratic formula (r2)

Percentage Accurate: 57.9% → 90.0%

Time: 31.2s

Alternatives: 8

Speedup: 19.1×

• could not determine a ground truth

  1. a = -6.345523484422262e+49
  2. b = 4.1334891586105655e+249
  3. c = 6.759880953206116e+186

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: 57.9% 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: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-192}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 120000000:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-192)
   (/ (- c) b)
   (if (<= b 120000000.0)
     (/ (- (- b) (sqrt (- (* b b) (* c (* a 4.0))))) (* a 2.0))
     (- (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-192) {
		tmp = -c / b;
	} else if (b <= 120000000.0) {
		tmp = (-b - sqrt(((b * b) - (c * (a * 4.0))))) / (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 <= (-9d-192)) then
        tmp = -c / b
    else if (b <= 120000000.0d0) then
        tmp = (-b - sqrt(((b * b) - (c * (a * 4.0d0))))) / (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 <= -9e-192) {
		tmp = -c / b;
	} else if (b <= 120000000.0) {
		tmp = (-b - Math.sqrt(((b * b) - (c * (a * 4.0))))) / (a * 2.0);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9e-192:
		tmp = -c / b
	elif b <= 120000000.0:
		tmp = (-b - math.sqrt(((b * b) - (c * (a * 4.0))))) / (a * 2.0)
	else:
		tmp = -(b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-192)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 120000000.0)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))) / 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 <= -9e-192)
		tmp = -c / b;
	elseif (b <= 120000000.0)
		tmp = (-b - sqrt(((b * b) - (c * (a * 4.0))))) / (a * 2.0);
	else
		tmp = -(b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-192], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 120000000.0], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $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 < -9.00000000000000048e-192

   1. Initial program 21.3%

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

   2. Taylor expanded in b around -inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-neg-frac 87.7%

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

   4. Simplified 87.7%

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

   if -9.00000000000000048e-192 < b < 1.2e8

   1. Initial program 90.9%

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

   2. Taylor expanded in a around 0 90.9%

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

      1. *-commutative 90.9%

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

      2. associate-*r* 90.9%

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

   4. Simplified 90.9%

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

   if 1.2e8 < b

   1. Initial program 86.9%

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

   2. Taylor expanded in b around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-192}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 120000000:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 2: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-192}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 120000000:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-192)
   (/ (- c) b)
   (if (<= b 120000000.0)
     (/ (- (- 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 <= -9e-192) {
		tmp = -c / b;
	} else if (b <= 120000000.0) {
		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 <= (-9d-192)) then
        tmp = -c / b
    else if (b <= 120000000.0d0) 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 <= -9e-192) {
		tmp = -c / b;
	} else if (b <= 120000000.0) {
		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 <= -9e-192:
		tmp = -c / b
	elif b <= 120000000.0:
		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 <= -9e-192)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 120000000.0)
		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 <= -9e-192)
		tmp = -c / b;
	elseif (b <= 120000000.0)
		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, -9e-192], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 120000000.0], 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 < -9.00000000000000048e-192

   1. Initial program 21.3%

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

   2. Taylor expanded in b around -inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-neg-frac 87.7%

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

   4. Simplified 87.7%

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

   if -9.00000000000000048e-192 < b < 1.2e8

   1. Initial program 90.9%

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

   if 1.2e8 < b

   1. Initial program 86.9%

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

   2. Taylor expanded in b around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-192}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 120000000:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 3: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-192}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{0.5}{\frac{a}{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-192)
   (/ (- c) b)
   (if (<= b 1.55e-108)
     (/ 0.5 (/ a (- b (sqrt (- (* b b) (* c (* a 4.0)))))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-192) {
		tmp = -c / b;
	} else if (b <= 1.55e-108) {
		tmp = 0.5 / (a / (b - sqrt(((b * b) - (c * (a * 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 <= (-9d-192)) then
        tmp = -c / b
    else if (b <= 1.55d-108) then
        tmp = 0.5d0 / (a / (b - sqrt(((b * b) - (c * (a * 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 <= -9e-192) {
		tmp = -c / b;
	} else if (b <= 1.55e-108) {
		tmp = 0.5 / (a / (b - Math.sqrt(((b * b) - (c * (a * 4.0))))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9e-192:
		tmp = -c / b
	elif b <= 1.55e-108:
		tmp = 0.5 / (a / (b - math.sqrt(((b * b) - (c * (a * 4.0))))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-192)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.55e-108)
		tmp = Float64(0.5 / Float64(a / Float64(b - sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 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 <= -9e-192)
		tmp = -c / b;
	elseif (b <= 1.55e-108)
		tmp = 0.5 / (a / (b - sqrt(((b * b) - (c * (a * 4.0))))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-192], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.55e-108], N[(0.5 / N[(a / N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $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 < -9.00000000000000048e-192

   1. Initial program 21.3%

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

   2. Taylor expanded in b around -inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-neg-frac 87.7%

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

   4. Simplified 87.7%

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

   if -9.00000000000000048e-192 < b < 1.55000000000000007e-108

   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. add-sqr-sqrt 5.7%

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

      2. sqrt-unprod 9.0%

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

      3. *-commutative 9.0%

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

      4. *-commutative 9.0%

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

      5. swap-sqr 9.0%

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

      6. metadata-eval 9.0%

         \[\leadsto \frac{\left(-b\right) - \sqrt{b \cdot b - \sqrt{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{16}}}}{2 \cdot a} \]

      7. metadata-eval 9.0%

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

      8. swap-sqr 9.0%

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

      9. associate-*r* 9.0%

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

      10. associate-*r* 9.0%

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

      11. sqrt-unprod 7.9%

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

      12. add-sqr-sqrt 7.9%

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

      13. add-cube-cbrt 7.9%

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

      14. pow3 7.9%

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

   3. Applied egg-rr 83.6%

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

      1. associate-*r* 83.5%

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

      2. cbrt-prod 83.3%

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

   5. Applied egg-rr 83.3%

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

      1. clear-num 83.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{c \cdot a} \cdot \sqrt[3]{4}\right)}^{3}}}}} \]

      2. inv-pow 83.2%

         \[\leadsto \color{blue}{{\left(\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{c \cdot a} \cdot \sqrt[3]{4}\right)}^{3}}}\right)}^{-1}} \]

   7. Applied egg-rr 83.8%

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

      1. unpow-1 83.7%

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

      2. *-commutative 83.7%

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

      3. *-un-lft-identity 83.7%

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

      4. times-frac 83.7%

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

      5. metadata-eval 83.7%

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

      6. add-sqr-sqrt 24.1%

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

      7. sqrt-unprod 76.0%

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

      8. sqr-neg 76.0%

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

      9. sqrt-prod 51.8%

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

      10. add-sqr-sqrt 76.1%

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

      11. associate-*l* 76.2%

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

   9. Applied egg-rr 76.2%

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

       1. associate-/r* 76.2%

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

       2. metadata-eval 76.2%

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

   11. Simplified 76.2%

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

   if 1.55000000000000007e-108 < b

   1. Initial program 91.5%

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

   2. Taylor expanded in b around inf 88.6%

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

      1. mul-1-neg 88.6%

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

      2. unsub-neg 88.6%

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

   4. Simplified 88.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-192}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{0.5}{\frac{a}{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-192}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-192)
   (/ (- c) b)
   (if (<= b 1.7e-109)
     (/ (/ (- b (sqrt (- (* b b) (* c (* a 4.0))))) a) 2.0)
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-192) {
		tmp = -c / b;
	} else if (b <= 1.7e-109) {
		tmp = ((b - sqrt(((b * b) - (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 <= (-9d-192)) then
        tmp = -c / b
    else if (b <= 1.7d-109) then
        tmp = ((b - sqrt(((b * b) - (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 <= -9e-192) {
		tmp = -c / b;
	} else if (b <= 1.7e-109) {
		tmp = ((b - Math.sqrt(((b * b) - (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 <= -9e-192:
		tmp = -c / b
	elif b <= 1.7e-109:
		tmp = ((b - math.sqrt(((b * b) - (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 <= -9e-192)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.7e-109)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))) / 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 <= -9e-192)
		tmp = -c / b;
	elseif (b <= 1.7e-109)
		tmp = ((b - sqrt(((b * b) - (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, -9e-192], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.7e-109], N[(N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.00000000000000048e-192

   1. Initial program 21.3%

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

   2. Taylor expanded in b around -inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-neg-frac 87.7%

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

   4. Simplified 87.7%

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

   if -9.00000000000000048e-192 < b < 1.70000000000000006e-109

   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. add-sqr-sqrt 5.7%

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

      2. sqrt-unprod 9.0%

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

      3. *-commutative 9.0%

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

      4. *-commutative 9.0%

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

      5. swap-sqr 9.0%

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

      6. metadata-eval 9.0%

         \[\leadsto \frac{\left(-b\right) - \sqrt{b \cdot b - \sqrt{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{16}}}}{2 \cdot a} \]

      7. metadata-eval 9.0%

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

      8. swap-sqr 9.0%

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

      9. associate-*r* 9.0%

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

      10. associate-*r* 9.0%

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

      11. sqrt-unprod 7.9%

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

      12. add-sqr-sqrt 7.9%

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

      13. add-cube-cbrt 7.9%

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

      14. pow3 7.9%

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

   3. Applied egg-rr 83.6%

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

      1. associate-*r* 83.5%

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

      2. cbrt-prod 83.3%

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

   5. Applied egg-rr 83.3%

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

      1. clear-num 83.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{c \cdot a} \cdot \sqrt[3]{4}\right)}^{3}}}}} \]

      2. inv-pow 83.2%

         \[\leadsto \color{blue}{{\left(\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{c \cdot a} \cdot \sqrt[3]{4}\right)}^{3}}}\right)}^{-1}} \]

   7. Applied egg-rr 83.8%

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

      1. unpow-1 83.7%

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

      2. clear-num 83.8%

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

      3. associate-/r* 83.8%

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

      4. add-sqr-sqrt 24.2%

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

      5. sqrt-unprod 76.1%

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

      6. sqr-neg 76.1%

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

      7. sqrt-prod 51.8%

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

      8. add-sqr-sqrt 76.2%

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

      9. associate-*l* 76.3%

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

   9. Applied egg-rr 76.3%

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

   if 1.70000000000000006e-109 < b

   1. Initial program 91.5%

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

   2. Taylor expanded in b around inf 88.6%

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

      1. mul-1-neg 88.6%

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

      2. unsub-neg 88.6%

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

   4. Simplified 88.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-192}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 77.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310)
   (/ (- c) b)
   (/ (+ (* 2.0 (/ (* c a) b)) (* b -2.0)) (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -c / b;
	} else {
		tmp = ((2.0 * ((c * a) / b)) + (b * -2.0)) / (a * 2.0);
	}
	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-310)) then
        tmp = -c / b
    else
        tmp = ((2.0d0 * ((c * a) / b)) + (b * (-2.0d0))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -c / b;
	} else {
		tmp = ((2.0 * ((c * a) / b)) + (b * -2.0)) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = -c / b
	else:
		tmp = ((2.0 * ((c * a) / b)) + (b * -2.0)) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(Float64(c * a) / b)) + Float64(b * -2.0)) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = -c / b;
	else
		tmp = ((2.0 * ((c * a) / b)) + (b * -2.0)) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[((-c) / b), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 27.3%

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

   2. Taylor expanded in b around -inf 79.5%

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

      1. mul-1-neg 79.5%

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

      2. distribute-neg-frac 79.5%

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

   4. Simplified 79.5%

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

   if -3.999999999999988e-310 < b

   1. Initial program 90.6%

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

   2. Taylor expanded in b around inf 72.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}{a \cdot 2}\\ \end{array} \]

Alternative 6: 77.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-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 <= (-4d-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 <= -4e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-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 <= -4e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-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 < -3.999999999999988e-310

   1. Initial program 27.3%

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

   2. Taylor expanded in b around -inf 79.5%

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

      1. mul-1-neg 79.5%

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

      2. distribute-neg-frac 79.5%

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

   4. Simplified 79.5%

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

   if -3.999999999999988e-310 < b

   1. Initial program 90.6%

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

   2. Taylor expanded in b around inf 72.2%

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

      1. mul-1-neg 72.2%

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

      2. unsub-neg 72.2%

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

   4. Simplified 72.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 7: 77.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (- c) b) (- (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		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-310)) 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-310) {
		tmp = -c / b;
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = -c / b
	else:
		tmp = -(b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		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-310)
		tmp = -c / b;
	else
		tmp = -(b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[((-c) / b), $MachinePrecision], (-N[(b / a), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 27.3%

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

   2. Taylor expanded in b around -inf 79.5%

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

      1. mul-1-neg 79.5%

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

      2. distribute-neg-frac 79.5%

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

   4. Simplified 79.5%

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

   if -3.999999999999988e-310 < b

   1. Initial program 90.6%

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

   2. Taylor expanded in b around inf 72.0%

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

      1. associate-*r/ 72.0%

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

      2. mul-1-neg 72.0%

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

   4. Simplified 72.0%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 8: 40.0% accurate, 29.0× 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(-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 58.9%

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

2. Taylor expanded in b around inf 37.1%

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

   1. associate-*r/ 37.1%

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

   2. mul-1-neg 37.1%

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

4. Simplified 37.1%

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

5. Final simplification 37.1%

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

Developer target: 79.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 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 = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-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 = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-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 = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-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 = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = 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 = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023278 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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