quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.6% → 85.7%

Time: 8.7s

Alternatives: 6

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: 52.6% 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: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+102)
   (fma (/ b_2 a) -2.0 (* 0.5 (/ c b_2)))
   (if (<= b_2 1.25e-69)
     (/ (- (sqrt (fma (- c) a (* b_2 b_2))) b_2) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+102) {
		tmp = fma((b_2 / a), -2.0, (0.5 * (c / b_2)));
	} else if (b_2 <= 1.25e-69) {
		tmp = (sqrt(fma(-c, a, (b_2 * b_2))) - 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 <= -1e+102)
		tmp = fma(Float64(b_2 / a), -2.0, Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.25e-69)
		tmp = Float64(Float64(sqrt(fma(Float64(-c), a, Float64(b_2 * b_2))) - 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, -1e+102], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0 + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.25e-69], N[(N[(N[Sqrt[N[((-c) * a + N[(b$95$2 * b$95$2), $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 < -9.99999999999999977e101

   1. Initial program 52.0%

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

   4. Applied rewrites 17.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 17.4

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

   6. Applied rewrites 52.2%

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

   7. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   8. 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. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 96.2

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

   9. Applied rewrites 96.2%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 96.6%

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

   if -9.99999999999999977e101 < b_2 < 1.25000000000000008e-69

   1. Initial program 84.3%

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

   4. Applied rewrites 58.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 58.6

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

   6. Applied rewrites 84.3%

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

   if 1.25000000000000008e-69 < b_2

   1. Initial program 13.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.22 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.22e-97)
   (fma (/ b_2 a) -2.0 (* 0.5 (/ c b_2)))
   (if (<= b_2 1.25e-69) (/ (- (sqrt (* (- a) c)) b_2) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.22e-97) {
		tmp = fma((b_2 / a), -2.0, (0.5 * (c / b_2)));
	} else if (b_2 <= 1.25e-69) {
		tmp = (sqrt((-a * c)) - 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 <= -1.22e-97)
		tmp = fma(Float64(b_2 / a), -2.0, Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.25e-69)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - 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, -1.22e-97], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0 + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.25e-69], N[(N[(N[Sqrt[N[((-a) * c), $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 < -1.22e-97

   1. Initial program 69.3%

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

   4. Applied rewrites 43.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 43.5

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

   6. Applied rewrites 69.4%

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

   7. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   8. 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. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 79.6

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

   9. Applied rewrites 79.6%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 79.9%

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

   if -1.22e-97 < b_2 < 1.25000000000000008e-69

   1. Initial program 81.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 81.5

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

   5. Applied rewrites 81.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 81.5

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

   7. Applied rewrites 81.5%

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

   if 1.25000000000000008e-69 < b_2

   1. Initial program 13.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.22 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.22 \cdot 10^{-97}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.22e-97)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.25e-69) (/ (- (sqrt (* (- a) c)) b_2) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.22e-97) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.25e-69) {
		tmp = (sqrt((-a * c)) - 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 <= (-1.22d-97)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.25d-69) then
        tmp = (sqrt((-a * c)) - 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 <= -1.22e-97) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.25e-69) {
		tmp = (Math.sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.22e-97:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.25e-69:
		tmp = (math.sqrt((-a * c)) - 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 <= -1.22e-97)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.25e-69)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - 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 <= -1.22e-97)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.25e-69)
		tmp = (sqrt((-a * c)) - 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, -1.22e-97], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.25e-69], N[(N[(N[Sqrt[N[((-a) * c), $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 < -1.22e-97

   1. Initial program 69.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 79.3

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

   5. Applied rewrites 79.3%

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

   if -1.22e-97 < b_2 < 1.25000000000000008e-69

   1. Initial program 81.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 81.5

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

   5. Applied rewrites 81.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 81.5

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

   7. Applied rewrites 81.5%

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

   if 1.25000000000000008e-69 < b_2

   1. Initial program 13.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.22 \cdot 10^{-97}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.4% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5e-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 <= 5e-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 <= 5d-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 <= 5e-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 <= 5e-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 <= 5e-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 <= 5e-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, 5e-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 < 4.999999999999985e-310

   1. Initial program 75.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 58.7

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

   5. Applied rewrites 58.7%

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

   if 4.999999999999985e-310 < b_2

   1. Initial program 30.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 69.4

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

   5. Applied rewrites 69.4%

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

4. Final simplification 64.2%

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

Alternative 5: 35.1% 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 52.3%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
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 36.5

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

5. Applied rewrites 36.5%

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

6. Final simplification 36.5%

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

Alternative 6: 35.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

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) / b_2) * c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.3%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right) \cdot c} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right) \cdot c} \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)\right)} \cdot c \]

   4. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{8} \cdot \left(a \cdot c\right)}{{b\_2}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)\right) \cdot c \]

   5. associate-*r* N/A

      \[\leadsto \left(\frac{\color{blue}{\left(\frac{-1}{8} \cdot a\right) \cdot c}}{{b\_2}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)\right) \cdot c \]

   6. associate-*l/ N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{-1}{8} \cdot a}{{b\_2}^{3}} \cdot c} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)\right) \cdot c \]

   7. associate-*r/ N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right)} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)\right) \cdot c \]

   8. *-commutative N/A

      \[\leadsto \left(\color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a}{{b\_2}^{3}}\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)\right) \cdot c \]

   9. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(c \cdot \frac{-1}{8}\right) \cdot \frac{a}{{b\_2}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)\right) \cdot c \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot \frac{-1}{8}, \frac{a}{{b\_2}^{3}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)} \cdot c \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{c \cdot \frac{-1}{8}}, \frac{a}{{b\_2}^{3}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \cdot c \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(c \cdot \frac{-1}{8}, \color{blue}{\frac{a}{{b\_2}^{3}}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \cdot c \]

   13. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(c \cdot \frac{-1}{8}, \frac{a}{\color{blue}{{b\_2}^{3}}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \cdot c \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(c \cdot \frac{-1}{8}, \frac{a}{{b\_2}^{3}}, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{b\_2}}\right)\right) \cdot c \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(c \cdot \frac{-1}{8}, \frac{a}{{b\_2}^{3}}, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{b\_2}\right)\right) \cdot c \]

   16. distribute-neg-frac N/A

       \[\leadsto \mathsf{fma}\left(c \cdot \frac{-1}{8}, \frac{a}{{b\_2}^{3}}, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{b\_2}}\right) \cdot c \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(c \cdot \frac{-1}{8}, \frac{a}{{b\_2}^{3}}, \frac{\color{blue}{\frac{-1}{2}}}{b\_2}\right) \cdot c \]

   18. lower-/.f64 36.0

       \[\leadsto \mathsf{fma}\left(c \cdot -0.125, \frac{a}{{b\_2}^{3}}, \color{blue}{\frac{-0.5}{b\_2}}\right) \cdot c \]

5. Applied rewrites 36.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot -0.125, \frac{a}{{b\_2}^{3}}, \frac{-0.5}{b\_2}\right) \cdot c} \]

6. Taylor expanded in a around 0

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

8. Applied rewrites 36.4%

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

9. Final simplification 36.4%

   \[\leadsto \frac{-0.5}{b\_2} \cdot c \]
10. 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 2024323 
(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))