quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.7% → 86.4%

Time: 7.2s

Alternatives: 9

Speedup: 1.7×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 52.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.5e-71)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.5e+129)
     (/ (- (- b_2) (sqrt (fma (- c) a (* b_2 b_2)))) a)
     (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.5e-71) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.5e+129) {
		tmp = (-b_2 - sqrt(fma(-c, a, (b_2 * b_2)))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.5e-71)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.5e+129)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(fma(Float64(-c), a, Float64(b_2 * b_2)))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.5e-71], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.5e+129], N[(N[((-b$95$2) - N[Sqrt[N[((-c) * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.49999999999999988e-71

   1. Initial program 13.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]

      2. lower-/.f64 93.3

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b\_2}} \]

   5. Applied rewrites 93.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.3%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -8.49999999999999988e-71 < b_2 < 2.5000000000000001e129

   1. Initial program 83.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(\mathsf{neg}\left(a \cdot c\right)\right)}}}{a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(a \cdot c\right)\right) + b\_2 \cdot b\_2}}}{a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{neg}\left(\color{blue}{a \cdot c}\right)\right) + b\_2 \cdot b\_2}}{a} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{neg}\left(\color{blue}{c \cdot a}\right)\right) + b\_2 \cdot b\_2}}{a} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot a} + b\_2 \cdot b\_2}}{a} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(c\right), a, b\_2 \cdot b\_2\right)}}}{a} \]

      8. lower-neg.f64 83.3

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(\color{blue}{-c}, a, b\_2 \cdot b\_2\right)}}{a} \]

   4. Applied rewrites 83.3%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}}{a} \]

   if 2.5000000000000001e129 < b_2

   1. Initial program 41.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

      3. lower-/.f64 97.1

         \[\leadsto \color{blue}{\frac{b\_2}{a}} \cdot -2 \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.5e-71)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.5e+129)
     (/ (+ b_2 (sqrt (- (* b_2 b_2) (* a c)))) (- a))
     (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.5e-71) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.5e+129) {
		tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-8.5d-71)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.5d+129) then
        tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -a
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.5e-71) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.5e+129) {
		tmp = (b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / -a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8.5e-71:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.5e+129:
		tmp = (b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / -a
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.5e-71)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.5e+129)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / Float64(-a));
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8.5e-71)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.5e+129)
		tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -a;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.5e-71], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.5e+129], N[(N[(b$95$2 + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.49999999999999988e-71

   1. Initial program 13.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]

      2. lower-/.f64 93.3

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b\_2}} \]

   5. Applied rewrites 93.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.3%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -8.49999999999999988e-71 < b_2 < 2.5000000000000001e129

   1. Initial program 83.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 2.5000000000000001e129 < b_2

   1. Initial program 41.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

      3. lower-/.f64 97.1

         \[\leadsto \color{blue}{\frac{b\_2}{a}} \cdot -2 \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{b\_2 + \sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.5e-71)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.6e-20)
     (/ (+ b_2 (sqrt (* (- a) c))) (- a))
     (fma (/ 0.5 b_2) c (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.5e-71) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.6e-20) {
		tmp = (b_2 + sqrt((-a * c))) / -a;
	} else {
		tmp = fma((0.5 / b_2), c, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.5e-71)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.6e-20)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(-a) * c))) / Float64(-a));
	else
		tmp = fma(Float64(0.5 / b_2), c, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.5e-71], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.6e-20], N[(N[(b$95$2 + N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(0.5 / b$95$2), $MachinePrecision] * c + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.49999999999999988e-71

   1. Initial program 13.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]

      2. lower-/.f64 93.3

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b\_2}} \]

   5. Applied rewrites 93.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.3%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -8.49999999999999988e-71 < b_2 < 1.59999999999999985e-20

   1. Initial program 78.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot c}}{a} \]

      4. lower-neg.f64 70.7

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

   5. Applied rewrites 70.7%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]

   if 1.59999999999999985e-20 < b_2

   1. Initial program 58.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2} + -2 \cdot \frac{b\_2}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot c}{b\_2}} + -2 \cdot \frac{b\_2}{a} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{b\_2} \cdot c} + -2 \cdot \frac{b\_2}{a} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot 1}}{b\_2} \cdot c + -2 \cdot \frac{b\_2}{a} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)} \cdot c + -2 \cdot \frac{b\_2}{a} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{1}{b\_2}, c, -2 \cdot \frac{b\_2}{a}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{b\_2}}, c, -2 \cdot \frac{b\_2}{a}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{2}}}{b\_2}, c, -2 \cdot \frac{b\_2}{a}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2}}{b\_2}}, c, -2 \cdot \frac{b\_2}{a}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{b\_2}, c, \color{blue}{\frac{b\_2}{a} \cdot -2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{b\_2}, c, \color{blue}{\frac{b\_2}{a} \cdot -2}\right) \]

      12. lower-/.f64 94.4

          \[\leadsto \mathsf{fma}\left(\frac{0.5}{b\_2}, c, \color{blue}{\frac{b\_2}{a}} \cdot -2\right) \]

   5. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{b\_2 + \sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (/ (* -0.5 c) b_2)
   (fma (/ 0.5 b_2) c (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = fma((0.5 / b_2), c, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = fma(Float64(0.5 / b_2), c, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(0.5 / b$95$2), $MachinePrecision] * c + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 27.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]

      2. lower-/.f64 71.6

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b\_2}} \]

   5. Applied rewrites 71.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.6%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 68.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2} + -2 \cdot \frac{b\_2}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot c}{b\_2}} + -2 \cdot \frac{b\_2}{a} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{b\_2} \cdot c} + -2 \cdot \frac{b\_2}{a} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot 1}}{b\_2} \cdot c + -2 \cdot \frac{b\_2}{a} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)} \cdot c + -2 \cdot \frac{b\_2}{a} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{1}{b\_2}, c, -2 \cdot \frac{b\_2}{a}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{b\_2}}, c, -2 \cdot \frac{b\_2}{a}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{2}}}{b\_2}, c, -2 \cdot \frac{b\_2}{a}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{2}}{b\_2}}, c, -2 \cdot \frac{b\_2}{a}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{b\_2}, c, \color{blue}{\frac{b\_2}{a} \cdot -2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{b\_2}, c, \color{blue}{\frac{b\_2}{a} \cdot -2}\right) \]

      12. lower-/.f64 67.7

          \[\leadsto \mathsf{fma}\left(\frac{0.5}{b\_2}, c, \color{blue}{\frac{b\_2}{a}} \cdot -2\right) \]

   5. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.25 \cdot 10^{-293}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.25e-293) (/ (* -0.5 c) b_2) (* (/ b_2 a) -2.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e-293) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.25d-293)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e-293) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.25e-293:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.25e-293)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.25e-293)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.25e-293], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.2500000000000001e-293

   1. Initial program 27.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]

      2. lower-/.f64 72.2

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b\_2}} \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.2%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -2.2500000000000001e-293 < b_2

   1. Initial program 67.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

      3. lower-/.f64 67.1

         \[\leadsto \color{blue}{\frac{b\_2}{a}} \cdot -2 \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.25 \cdot 10^{-293}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.25e-293) (* -0.5 (/ c b_2)) (* (/ b_2 a) -2.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e-293) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.25d-293)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e-293) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.25e-293:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.25e-293)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.25e-293)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.25e-293], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.2500000000000001e-293

   1. Initial program 27.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]

      2. lower-/.f64 72.2

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b\_2}} \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -2.2500000000000001e-293 < b_2

   1. Initial program 67.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

      3. lower-/.f64 67.1

         \[\leadsto \color{blue}{\frac{b\_2}{a}} \cdot -2 \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.25 \cdot 10^{-293}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.25e-293) (* -0.5 (/ c b_2)) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e-293) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.25d-293)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e-293) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.25e-293:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.25e-293)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.25e-293)
		tmp = -0.5 * (c / b_2);
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.25e-293], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.2500000000000001e-293

   1. Initial program 27.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]

      2. lower-/.f64 72.2

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b\_2}} \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -2.2500000000000001e-293 < b_2

   1. Initial program 67.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

      3. lower-/.f64 67.1

         \[\leadsto \color{blue}{\frac{b\_2}{a}} \cdot -2 \]

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

      \[\leadsto b\_2 \cdot \color{blue}{\frac{-2}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 34.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -0.5 (/ c b_2)))

C

double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

Python

def code(a, b_2, c):
	return -0.5 * (c / b_2)

Julia

function code(a, b_2, c)
	return Float64(-0.5 * Float64(c / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -0.5 * (c / b_2);
end

Wolfram

code[a_, b$95$2_, c_] := N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.2%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in b_2 around -inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]

   2. lower-/.f64 33.1

      \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b\_2}} \]

5. Applied rewrites 33.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
6. Add Preprocessing

Alternative 9: 11.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b\_2} \cdot 0.5 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* (/ c b_2) 0.5))

C

double code(double a, double b_2, double c) {
	return (c / b_2) * 0.5;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (c / b_2) * 0.5d0
end function

Java

public static double code(double a, double b_2, double c) {
	return (c / b_2) * 0.5;
}

Python

def code(a, b_2, c):
	return (c / b_2) * 0.5

Julia

function code(a, b_2, c)
	return Float64(Float64(c / b_2) * 0.5)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (c / b_2) * 0.5;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 50.2%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

   3. lower-/.f64 38.6

      \[\leadsto \color{blue}{\frac{b\_2}{a}} \cdot -2 \]

5. Applied rewrites 38.6%

   \[\leadsto \color{blue}{\frac{b\_2}{a} \cdot -2} \]

6. Taylor expanded in c around 0

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2} + -2 \cdot \frac{b\_2}{a}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{1}{2}} + -2 \cdot \frac{b\_2}{a} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, -2 \cdot \frac{b\_2}{a}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{c}{b\_2}}, \frac{1}{2}, -2 \cdot \frac{b\_2}{a}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \color{blue}{\frac{b\_2}{a} \cdot -2}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \color{blue}{\frac{b\_2}{a} \cdot -2}\right) \]

   7. lower-/.f64 38.5

      \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \color{blue}{\frac{b\_2}{a}} \cdot -2\right) \]

8. Applied rewrites 38.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)} \]

9. Taylor expanded in a around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]
10. Step-by-step derivation

11. Applied rewrites 11.3%

    \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{0.5} \]
12. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

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

Reproduce

herbie shell --seed 2024313 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, 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_2 0) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))