quadp (p42, positive)

Percentage Accurate: 51.9% → 85.5%

Time: 12.5s

Alternatives: 8

Speedup: 2.5×

• could not determine a ground truth

  1. a = 5.947568138347641e-268
  2. b = 3.9065475810662966e+138
  3. c = -3.136680613544536e-109

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: 51.9% 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: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+94)
   (/ b (- a))
   (if (<= b 5e-85)
     (- (/ (sqrt (fma a (* c -4.0) (* b b))) (* a 2.0)) (/ b (* a 2.0)))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+94) {
		tmp = b / -a;
	} else if (b <= 5e-85) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) / (a * 2.0)) - (b / (a * 2.0));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+94)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 5e-85)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) / Float64(a * 2.0)) - Float64(b / Float64(a * 2.0)));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+94], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 5e-85], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] - N[(b / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -2e94

   1. Initial program 47.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 94.3

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

   5. Applied rewrites 94.3%

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

   if -2e94 < b < 5.0000000000000002e-85

   1. Initial program 81.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. div-sub N/A

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

      12. lower--.f64 N/A

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

   4. Applied rewrites 81.1%

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

   if 5.0000000000000002e-85 < b

   1. Initial program 15.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-neg.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+94)
   (/ b (- a))
   (if (<= b 5e-85)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+94) {
		tmp = b / -a;
	} else if (b <= 5e-85) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d+94)) then
        tmp = b / -a
    else if (b <= 5d-85) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+94) {
		tmp = b / -a;
	} else if (b <= 5e-85) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+94:
		tmp = b / -a
	elif b <= 5e-85:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+94)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 5e-85)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+94)
		tmp = b / -a;
	elseif (b <= 5e-85)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+94], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 5e-85], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -2e94

   1. Initial program 47.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 94.3

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

   5. Applied rewrites 94.3%

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

   if -2e94 < b < 5.0000000000000002e-85

   1. Initial program 81.1%

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

   if 5.0000000000000002e-85 < b

   1. Initial program 15.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-neg.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+59)
   (/ b (- a))
   (if (<= b 5e-85)
     (* (/ -0.5 a) (- b (sqrt (fma a (* c -4.0) (* b b)))))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+59) {
		tmp = b / -a;
	} else if (b <= 5e-85) {
		tmp = (-0.5 / a) * (b - sqrt(fma(a, (c * -4.0), (b * b))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+59)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 5e-85)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+59], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 5e-85], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -9.99999999999999972e58

   1. Initial program 52.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if -9.99999999999999972e58 < b < 5.0000000000000002e-85

   1. Initial program 80.2%

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

   4. Applied rewrites 80.1%

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

   if 5.0000000000000002e-85 < b

   1. Initial program 15.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-neg.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-18)
   (/ b (- a))
   (if (<= b 2e-136) (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0)) (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-18) {
		tmp = b / -a;
	} else if (b <= 2e-136) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-18)) then
        tmp = b / -a
    else if (b <= 2d-136) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-18) {
		tmp = b / -a;
	} else if (b <= 2e-136) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-18:
		tmp = b / -a
	elif b <= 2e-136:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-18)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2e-136)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-18)
		tmp = b / -a;
	elseif (b <= 2e-136)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-18], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2e-136], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000036e-18

   1. Initial program 59.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if -5.00000000000000036e-18 < b < 2e-136

   1. Initial program 81.1%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-approx N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 66.1

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

   7. Applied rewrites 66.1%

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

   if 2e-136 < b

   1. Initial program 18.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-neg.f64 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-136}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-18)
   (/ b (- a))
   (if (<= b 2e-136) (* (- (sqrt (* a (* c -4.0))) b) (/ 0.5 a)) (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-18) {
		tmp = b / -a;
	} else if (b <= 2e-136) {
		tmp = (sqrt((a * (c * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-18)) then
        tmp = b / -a
    else if (b <= 2d-136) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) * (0.5d0 / a)
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-18) {
		tmp = b / -a;
	} else if (b <= 2e-136) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-18:
		tmp = b / -a
	elif b <= 2e-136:
		tmp = (math.sqrt((a * (c * -4.0))) - b) * (0.5 / a)
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-18)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2e-136)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-18)
		tmp = b / -a;
	elseif (b <= 2e-136)
		tmp = (sqrt((a * (c * -4.0))) - b) * (0.5 / a);
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-18], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2e-136], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000036e-18

   1. Initial program 59.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if -5.00000000000000036e-18 < b < 2e-136

   1. Initial program 81.1%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 66.0%

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

   if 2e-136 < b

   1. Initial program 18.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. lower-/.f64 N/A

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

      4. lower-neg.f64 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-18}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-136}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-300) (/ b (- a)) (- (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-300) {
		tmp = b / -a;
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-300)) then
        tmp = b / -a
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-300) {
		tmp = b / -a;
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-300:
		tmp = b / -a
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-300)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-300)
		tmp = b / -a;
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-300], N[(b / (-a)), $MachinePrecision], (-N[(c / b), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < -1.00000000000000003e-300

   1. Initial program 70.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 59.9

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

   5. Applied rewrites 59.9%

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

   if -1.00000000000000003e-300 < b

   1. Initial program 28.6%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-neg.f64 74.4

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

   5. Applied rewrites 74.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-275}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -5e-275) (/ b (- a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-275) {
		tmp = b / -a;
	} else {
		tmp = 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 <= (-5d-275)) then
        tmp = b / -a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-275) {
		tmp = b / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-275:
		tmp = b / -a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-275)
		tmp = Float64(b / Float64(-a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-275)
		tmp = b / -a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-275], N[(b / (-a)), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -4.99999999999999983e-275

   1. Initial program 70.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 61.6

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

   5. Applied rewrites 61.6%

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

   if -4.99999999999999983e-275 < b

   1. Initial program 30.0%

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

   4. Applied rewrites 30.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-commutative N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. metadata-eval N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

   6. Applied rewrites 24.4%

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

   7. Taylor expanded in c around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

      3. mul0-rgt 16.5

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

   9. Applied rewrites 16.5%

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

Alternative 8: 10.8% 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 50.5%

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

4. Applied rewrites 50.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. *-commutative N/A

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

   10. sub-neg N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. sub-neg N/A

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

   14. *-commutative N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. metadata-eval N/A

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

   17. associate-*r* N/A

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

   18. lift-*.f64 N/A

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

6. Applied rewrites 47.6%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

   3. mul0-rgt 9.6

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

9. Applied rewrites 9.6%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024212 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

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

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