quadm (p42, negative)

Percentage Accurate: 51.4% → 85.3%

Time: 11.0s

Alternatives: 10

Speedup: 2.5×

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: 51.4% 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: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{a}, \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \frac{b}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.8e-66)
   (/ (- c) b)
   (if (<= b 1.1e+45)
     (fma (/ -0.5 a) (sqrt (fma a (* c -4.0) (* b b))) (/ b (* a -2.0)))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-66) {
		tmp = -c / b;
	} else if (b <= 1.1e+45) {
		tmp = fma((-0.5 / a), sqrt(fma(a, (c * -4.0), (b * b))), (b / (a * -2.0)));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.8e-66)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.1e+45)
		tmp = fma(Float64(-0.5 / a), sqrt(fma(a, Float64(c * -4.0), Float64(b * b))), Float64(b / Float64(a * -2.0)));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.8e-66], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.1e+45], N[(N[(-0.5 / a), $MachinePrecision] * N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(b / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.80000000000000023e-66

   1. Initial program 13.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -5.80000000000000023e-66 < b < 1.1e45

   1. Initial program 87.1%

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

   4. Applied rewrites 87.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 87.1%

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

   if 1.1e45 < b

   1. Initial program 60.8%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 96.2

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

   5. Applied rewrites 96.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{a}, \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \frac{b}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.8e-66)
   (/ (- c) b)
   (if (<= b 1.1e+45)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-66) {
		tmp = -c / b;
	} else if (b <= 1.1e+45) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (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 <= (-5.8d-66)) then
        tmp = -c / b
    else if (b <= 1.1d+45) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (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 <= -5.8e-66) {
		tmp = -c / b;
	} else if (b <= 1.1e+45) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.8e-66:
		tmp = -c / b
	elif b <= 1.1e+45:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.8e-66)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.1e+45)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.8e-66)
		tmp = -c / b;
	elseif (b <= 1.1e+45)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.8e-66], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.1e+45], 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[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.80000000000000023e-66

   1. Initial program 13.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -5.80000000000000023e-66 < b < 1.1e45

   1. Initial program 87.1%

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

   if 1.1e45 < b

   1. Initial program 60.8%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 96.2

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

   5. Applied rewrites 96.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.8e-66)
   (/ (- c) b)
   (if (<= b 1.1e+45)
     (* (/ -0.5 a) (+ b (sqrt (fma b b (* c (* a -4.0))))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-66) {
		tmp = -c / b;
	} else if (b <= 1.1e+45) {
		tmp = (-0.5 / a) * (b + sqrt(fma(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 <= -5.8e-66)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.1e+45)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.8e-66], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.1e+45], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $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 < -5.80000000000000023e-66

   1. Initial program 13.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -5.80000000000000023e-66 < b < 1.1e45

   1. Initial program 87.1%

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

   4. Applied rewrites 87.0%

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

   if 1.1e45 < b

   1. Initial program 60.8%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 96.2

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

   5. Applied rewrites 96.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.8e-66)
   (/ (- c) b)
   (if (<= b 1.25e-123)
     (/ (+ b (sqrt (* a (* c -4.0)))) (* a -2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-66) {
		tmp = -c / b;
	} else if (b <= 1.25e-123) {
		tmp = (b + sqrt((a * (c * -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 <= (-5.8d-66)) then
        tmp = -c / b
    else if (b <= 1.25d-123) then
        tmp = (b + sqrt((a * (c * (-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 <= -5.8e-66) {
		tmp = -c / b;
	} else if (b <= 1.25e-123) {
		tmp = (b + Math.sqrt((a * (c * -4.0)))) / (a * -2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.8e-66:
		tmp = -c / b
	elif b <= 1.25e-123:
		tmp = (b + math.sqrt((a * (c * -4.0)))) / (a * -2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.8e-66)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.25e-123)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(c * -4.0)))) / Float64(a * -2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.8e-66)
		tmp = -c / b;
	elseif (b <= 1.25e-123)
		tmp = (b + sqrt((a * (c * -4.0)))) / (a * -2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.8e-66], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.25e-123], N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.80000000000000023e-66

   1. Initial program 13.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -5.80000000000000023e-66 < b < 1.25000000000000007e-123

   1. Initial program 81.7%

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. rem-square-sqrt N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 79.1

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

   7. Applied rewrites 79.1%

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

      1. metadata-eval N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 79.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 79.2

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

   9. Applied rewrites 79.2%

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

   if 1.25000000000000007e-123 < b

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e-67)
   (/ (- c) b)
   (if (<= b 1.25e-123)
     (* (/ -0.5 a) (+ b (sqrt (* a (* c -4.0)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-67) {
		tmp = -c / b;
	} else if (b <= 1.25e-123) {
		tmp = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.5d-67)) then
        tmp = -c / b
    else if (b <= 1.25d-123) then
        tmp = ((-0.5d0) / a) * (b + sqrt((a * (c * (-4.0d0)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-67) {
		tmp = -c / b;
	} else if (b <= 1.25e-123) {
		tmp = (-0.5 / a) * (b + Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.5e-67:
		tmp = -c / b
	elif b <= 1.25e-123:
		tmp = (-0.5 / a) * (b + math.sqrt((a * (c * -4.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e-67)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.25e-123)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.5e-67)
		tmp = -c / b;
	elseif (b <= 1.25e-123)
		tmp = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e-67], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.25e-123], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5e-67

   1. Initial program 13.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -3.5e-67 < b < 1.25000000000000007e-123

   1. Initial program 81.7%

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. rem-square-sqrt N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if 1.25000000000000007e-123 < b

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 91.8

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

   5. Applied rewrites 91.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-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 <= (-2d-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 <= -2e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-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 <= -2e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-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 < -1.999999999999994e-310

   1. Initial program 32.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 69.0

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

   5. Applied rewrites 69.0%

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

   if -1.999999999999994e-310 < b

   1. Initial program 74.2%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 75.7

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

   5. Applied rewrites 75.7%

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

4. Final simplification 72.5%

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

Alternative 7: 68.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-309) (/ (- c) b) (/ b (- a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-309) {
		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 <= (-2d-309)) 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 <= -2e-309) {
		tmp = -c / b;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-309:
		tmp = -c / b
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-309)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-309)
		tmp = -c / b;
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-309], N[((-c) / b), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9999999999999988e-309

   1. Initial program 32.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 69.0

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

   5. Applied rewrites 69.0%

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

   if -1.9999999999999988e-309 < b

   1. Initial program 74.2%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      3. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-305}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -1.95e-305) 0.0 (/ b (- a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-305) {
		tmp = 0.0;
	} 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 <= (-1.95d-305)) then
        tmp = 0.0d0
    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 <= -1.95e-305) {
		tmp = 0.0;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e-305:
		tmp = 0.0
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e-305)
		tmp = 0.0;
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.95e-305)
		tmp = 0.0;
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e-305], 0.0, N[(b / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.95000000000000013e-305

   1. Initial program 32.8%

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{-1 \cdot b}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      2. lower-neg.f64 3.7

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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 20.1%

      \[\leadsto \color{blue}{0} \]

   if -1.95000000000000013e-305 < b

   1. Initial program 73.6%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)} \]

      3. lower-/.f64 74.6

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

   5. Applied rewrites 74.6%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-305}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 15.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-20}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 6.4e-20) 0.0 (/ 0.5 0.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.4e-20) {
		tmp = 0.0;
	} else {
		tmp = 0.5 / 0.0;
	}
	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 <= 6.4d-20) then
        tmp = 0.0d0
    else
        tmp = 0.5d0 / 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.4e-20) {
		tmp = 0.0;
	} else {
		tmp = 0.5 / 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.4e-20:
		tmp = 0.0
	else:
		tmp = 0.5 / 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.4e-20)
		tmp = 0.0;
	else
		tmp = Float64(0.5 / 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6.4e-20)
		tmp = 0.0;
	else
		tmp = 0.5 / 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.4e-20], 0.0, N[(0.5 / 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.39999999999999941e-20

   1. Initial program 46.8%

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{-1 \cdot b}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      2. lower-neg.f64 3.5

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

   7. Applied rewrites 3.5%

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

   9. Applied rewrites 16.0%

      \[\leadsto \color{blue}{0} \]

   if 6.39999999999999941e-20 < b

   1. Initial program 67.1%

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{-1 \cdot b}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      2. lower-neg.f64 0.7

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

   7. Applied rewrites 0.7%

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

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      2. flip-- N/A

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

      3. sqr-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. flip-+ N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{a}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{a}{\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{a}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. distribute-neg-in N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{a}{\color{blue}{\mathsf{neg}\left(\left(b + b\right)\right)}}} \]

      14. remove-double-neg N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{a}{\mathsf{neg}\left(\left(b + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(b\right)\right)\right)\right)}\right)\right)}} \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{a}{\mathsf{neg}\left(\left(b + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right)\right)\right)}} \]

      16. sub-neg N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{a}{\mathsf{neg}\left(\color{blue}{\left(b - \left(\mathsf{neg}\left(b\right)\right)\right)}\right)}} \]

      17. lift--.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\frac{a}{\mathsf{neg}\left(\color{blue}{\left(b - \left(\mathsf{neg}\left(b\right)\right)\right)}\right)}} \]

      18. frac-2neg N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\mathsf{neg}\left(a\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(b - \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right)}}} \]

      19. clear-num N/A

          \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(b - \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\right)}{\mathsf{neg}\left(a\right)}}}} \]

   9. Applied rewrites 16.0%

      \[\leadsto \frac{0.5}{\color{blue}{0}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 11.4% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 54.2%

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

4. Applied rewrites 31.6%

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

5. Taylor expanded in b around -inf

   \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{-1 \cdot b}}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{a}{b - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

   2. lower-neg.f64 2.5

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

7. Applied rewrites 2.5%

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

9. Applied rewrites 11.0%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024214 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

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