quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.9% → 85.3%

Time: 8.5s

Alternatives: 8

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: 51.9% 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: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.5 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, \frac{c \cdot 0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.5e+100)
   (fma (/ b_2 a) -2.0 (/ (* c 0.5) b_2))
   (if (<= b_2 6e-153)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.5e+100) {
		tmp = fma((b_2 / a), -2.0, ((c * 0.5) / b_2));
	} else if (b_2 <= 6e-153) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.5e+100)
		tmp = fma(Float64(b_2 / a), -2.0, Float64(Float64(c * 0.5) / b_2));
	elseif (b_2 <= 6e-153)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.5e+100], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0 + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6e-153], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.50000000000000043e100

   1. Initial program 43.8%

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

   3. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in b_2 around 0

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

   8. Applied rewrites 2.7%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 96.3%

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

   if -8.50000000000000043e100 < b_2 < 6e-153

   1. Initial program 88.2%

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

   if 6e-153 < b_2

   1. Initial program 22.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 85.3

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

   8. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.5 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, \frac{c \cdot 0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, \frac{c \cdot 0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e+15)
   (fma (/ b_2 a) -2.0 (/ (* c 0.5) b_2))
   (if (<= b_2 6e-153) (/ (- (sqrt (* a (- c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e+15) {
		tmp = fma((b_2 / a), -2.0, ((c * 0.5) / b_2));
	} else if (b_2 <= 6e-153) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e+15)
		tmp = fma(Float64(b_2 / a), -2.0, Float64(Float64(c * 0.5) / b_2));
	elseif (b_2 <= 6e-153)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e+15], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0 + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6e-153], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.15e15

   1. Initial program 65.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 91.9

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in b_2 around 0

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

   8. Applied rewrites 3.0%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 92.3%

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

   if -1.15e15 < b_2 < 6e-153

   1. Initial program 81.1%

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 63.1

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

   7. Applied rewrites 63.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 63.1

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

   9. Applied rewrites 63.1%

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

   if 6e-153 < b_2

   1. Initial program 22.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 80.4

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

   8. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

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) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	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(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	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 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	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[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024227 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, 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_2 0) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

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