quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.4% → 84.4%

Time: 7.5s

Alternatives: 7

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: 52.4% 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: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 4 \cdot 10^{-130}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+31)
   (fma 0.5 (/ c b_2) (* (/ b_2 a) -2.0))
   (if (<= b_2 4e-130)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+31) {
		tmp = fma(0.5, (c / b_2), ((b_2 / a) * -2.0));
	} else if (b_2 <= 4e-130) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e+31)
		tmp = fma(0.5, Float64(c / b_2), Float64(Float64(b_2 / a) * -2.0));
	elseif (b_2 <= 4e-130)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e+31], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4e-130], 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[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.9999999999999999e31

   1. Initial program 59.6%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 96.4

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 96.7%

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

   if -1.9999999999999999e31 < b_2 < 4.0000000000000003e-130

   1. Initial program 83.2%

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

   if 4.0000000000000003e-130 < b_2

   1. Initial program 19.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 4 \cdot 10^{-130}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.35 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 4 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{\left|c \cdot a\right|} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.35e-17)
   (fma 0.5 (/ c b_2) (* (/ b_2 a) -2.0))
   (if (<= b_2 4e-130)
     (/ (- (sqrt (fabs (* c a))) b_2) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.35e-17) {
		tmp = fma(0.5, (c / b_2), ((b_2 / a) * -2.0));
	} else if (b_2 <= 4e-130) {
		tmp = (sqrt(fabs((c * a))) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.35e-17)
		tmp = fma(0.5, Float64(c / b_2), Float64(Float64(b_2 / a) * -2.0));
	elseif (b_2 <= 4e-130)
		tmp = Float64(Float64(sqrt(abs(Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.35e-17], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4e-130], N[(N[(N[Sqrt[N[Abs[N[(c * a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.3500000000000001e-17

   1. Initial program 65.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 91.9%

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

   if -1.3500000000000001e-17 < b_2 < 4.0000000000000003e-130

   1. Initial program 82.5%

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

   4. Applied rewrites 47.5%

      \[\leadsto \frac{\left(-b\_2\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)\right)}^{2}\right)}^{0.25}}}{a} \]

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 3.5

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

   7. Applied rewrites 3.5%

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

   8. Taylor expanded in b_2 around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 1.9

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

   10. Applied rewrites 1.9%

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

   12. Applied rewrites 71.8%

       \[\leadsto \frac{\sqrt{\left|c \cdot a\right|} - b\_2}{a} \]

   if 4.0000000000000003e-130 < b_2

   1. Initial program 19.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.35 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 4 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{\left|c \cdot a\right|} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.35 \cdot 10^{-17}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 4 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{\left|c \cdot a\right|} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.35e-17)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 4e-130)
     (/ (- (sqrt (fabs (* c a))) b_2) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.35e-17) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 4e-130) {
		tmp = (sqrt(fabs((c * a))) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= (-1.35d-17)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 4d-130) then
        tmp = (sqrt(abs((c * a))) - b_2) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.35e-17) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 4e-130) {
		tmp = (Math.sqrt(Math.abs((c * a))) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.35e-17:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 4e-130:
		tmp = (math.sqrt(math.fabs((c * a))) - b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.35e-17)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 4e-130)
		tmp = Float64(Float64(sqrt(abs(Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.35e-17)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 4e-130)
		tmp = (sqrt(abs((c * a))) - b_2) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.35e-17], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 4e-130], N[(N[(N[Sqrt[N[Abs[N[(c * a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.3500000000000001e-17

   1. Initial program 65.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

   if -1.3500000000000001e-17 < b_2 < 4.0000000000000003e-130

   1. Initial program 82.5%

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

   4. Applied rewrites 47.5%

      \[\leadsto \frac{\left(-b\_2\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)\right)}^{2}\right)}^{0.25}}}{a} \]

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 3.5

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

   7. Applied rewrites 3.5%

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

   8. Taylor expanded in b_2 around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 1.9

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

   10. Applied rewrites 1.9%

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

   12. Applied rewrites 71.8%

       \[\leadsto \frac{\sqrt{\left|c \cdot a\right|} - b\_2}{a} \]

   if 4.0000000000000003e-130 < b_2

   1. Initial program 19.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.35 \cdot 10^{-17}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 4 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{\left|c \cdot a\right|} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310) (/ (* -2.0 b_2) a) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (-2.0 * b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= (-2d-310)) then
        tmp = ((-2.0d0) * b_2) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (-2.0 * b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = (-2.0 * b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-310)
		tmp = (-2.0 * b_2) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 72.2%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 33.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.8

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

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.8% 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 52.8%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 34.5

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

5. Applied rewrites 34.5%

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

6. Final simplification 34.5%

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

Alternative 6: 34.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	return c * (-0.5 / 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 = c * ((-0.5d0) / b_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 34.5

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

5. Applied rewrites 34.5%

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

7. Applied rewrites 34.4%

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

8. Final simplification 34.4%

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

Alternative 7: 10.8% accurate, 40.0× speedup

Math

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

FPCore

(FPCore (a b_2 c) :precision binary64 0.0)

C

double code(double a, double b_2, double c) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0;
}

Python

def code(a, b_2, c):
	return 0.0

Julia

function code(a, b_2, c)
	return 0.0
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0;
end

Wolfram

code[a_, b$95$2_, c_] := 0.0

Derivation

1. Initial program 52.8%

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

3. Taylor expanded in b_2 around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower--.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 37.9

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

5. Applied rewrites 37.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. frac-add N/A

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

   6. lower-/.f64 N/A

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

7. Applied rewrites 27.2%

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

8. Taylor expanded in a around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} \cdot b\_2}{a} \]

   3. mul0-lft N/A

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

   4. div0 9.4

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

10. Applied rewrites 9.4%

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

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

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

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 2024342 
(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))