quad2p (problem 3.2.1, positive)

Percentage Accurate: 50.9% → 84.8%

Time: 10.3s

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: 50.9% 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: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.1 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, \frac{c \cdot 0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.1e+123)
   (fma (/ b_2 a) -2.0 (/ (* c 0.5) b_2))
   (if (<= b_2 6.5e-21)
     (/ (- (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 <= -4.1e+123) {
		tmp = fma((b_2 / a), -2.0, ((c * 0.5) / b_2));
	} else if (b_2 <= 6.5e-21) {
		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 <= -4.1e+123)
		tmp = fma(Float64(b_2 / a), -2.0, Float64(Float64(c * 0.5) / b_2));
	elseif (b_2 <= 6.5e-21)
		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, -4.1e+123], 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, 6.5e-21], 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 < -4.09999999999999989e123

   1. Initial program 41.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 93.5

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

   5. Applied rewrites 93.5%

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

   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 2.2%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 94.6%

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

   if -4.09999999999999989e123 < b_2 < 6.49999999999999987e-21

   1. Initial program 78.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 78.4

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

   4. Applied rewrites 78.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 78.4

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.4

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

   6. Applied rewrites 78.4%

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

   if 6.49999999999999987e-21 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 2.3

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

   5. Applied rewrites 2.3%

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

   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 89.7

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

   8. Applied rewrites 89.7%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.1 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, \frac{c \cdot 0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.7 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, \frac{c \cdot 0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{-a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.7e-39)
   (fma (/ b_2 a) -2.0 (/ (* c 0.5) b_2))
   (if (<= b_2 6.5e-21) (/ (- (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 <= -5.7e-39) {
		tmp = fma((b_2 / a), -2.0, ((c * 0.5) / b_2));
	} else if (b_2 <= 6.5e-21) {
		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 <= -5.7e-39)
		tmp = fma(Float64(b_2 / a), -2.0, Float64(Float64(c * 0.5) / b_2));
	elseif (b_2 <= 6.5e-21)
		tmp = Float64(Float64(sqrt(Float64(-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, -5.7e-39], 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, 6.5e-21], 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 < -5.6999999999999997e-39

   1. Initial program 65.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 86.8

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

   5. Applied rewrites 86.8%

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

   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 3.3%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 87.6%

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

   if -5.6999999999999997e-39 < b_2 < 6.49999999999999987e-21

   1. Initial program 71.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 71.1

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

   4. Applied rewrites 71.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 71.1

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.1

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

   6. Applied rewrites 71.1%

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

   7. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 63.8

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

   9. Applied rewrites 63.8%

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

   if 6.49999999999999987e-21 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 2.3

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

   5. Applied rewrites 2.3%

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

   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 89.7

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

   8. Applied rewrites 89.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.7 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, \frac{c \cdot 0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{-a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{-a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.7e-39)
   (* (/ b_2 a) -2.0)
   (if (<= b_2 6.5e-21) (/ (- (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 <= -5.7e-39) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 6.5e-21) {
		tmp = (sqrt(-(a * c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	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 <= (-5.7d-39)) then
        tmp = (b_2 / a) * (-2.0d0)
    else if (b_2 <= 6.5d-21) then
        tmp = (sqrt(-(a * c)) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.7e-39) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 6.5e-21) {
		tmp = (Math.sqrt(-(a * c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.7e-39:
		tmp = (b_2 / a) * -2.0
	elif b_2 <= 6.5e-21:
		tmp = (math.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 <= -5.7e-39)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	elseif (b_2 <= 6.5e-21)
		tmp = Float64(Float64(sqrt(Float64(-Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.7e-39)
		tmp = (b_2 / a) * -2.0;
	elseif (b_2 <= 6.5e-21)
		tmp = (sqrt(-(a * c)) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.7e-39], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[b$95$2, 6.5e-21], 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 < -5.6999999999999997e-39

   1. Initial program 65.4%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if -5.6999999999999997e-39 < b_2 < 6.49999999999999987e-21

   1. Initial program 71.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 71.1

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

   4. Applied rewrites 71.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 71.1

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.1

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

   6. Applied rewrites 71.1%

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

   7. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 63.8

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

   9. Applied rewrites 63.8%

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

   if 6.49999999999999987e-21 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 2.3

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

   5. Applied rewrites 2.3%

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

   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 89.7

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

   8. Applied rewrites 89.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{-a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.8e-308) (* (/ b_2 a) -2.0) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.8e-308) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 2.8d-308) then
        tmp = (b_2 / a) * (-2.0d0)
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.8e-308) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.8e-308:
		tmp = (b_2 / a) * -2.0
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.8e-308)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.8e-308)
		tmp = (b_2 / a) * -2.0;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.8e-308], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.79999999999999984e-308

   1. Initial program 68.2%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if 2.79999999999999984e-308 < b_2

   1. Initial program 39.7%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 2.8

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

   5. Applied rewrites 2.8%

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

   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 62.7

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

   8. Applied rewrites 62.7%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.8e-308) (* (/ b_2 a) -2.0) (* c (/ -0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.8e-308) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 2.8d-308) then
        tmp = (b_2 / a) * (-2.0d0)
    else
        tmp = c * ((-0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.8e-308) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.8e-308:
		tmp = (b_2 / a) * -2.0
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.8e-308)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.8e-308)
		tmp = (b_2 / a) * -2.0;
	else
		tmp = c * (-0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.8e-308], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.79999999999999984e-308

   1. Initial program 68.2%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if 2.79999999999999984e-308 < b_2

   1. Initial program 39.7%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 62.6

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

   5. Applied rewrites 62.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.5e-19) (* (/ b_2 a) -2.0) (* c (/ 0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.5e-19) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = c * (0.5 / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 2.5d-19) then
        tmp = (b_2 / a) * (-2.0d0)
    else
        tmp = c * (0.5d0 / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.5e-19) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = c * (0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.5e-19:
		tmp = (b_2 / a) * -2.0
	else:
		tmp = c * (0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.5e-19)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	else
		tmp = Float64(c * Float64(0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.5e-19)
		tmp = (b_2 / a) * -2.0;
	else
		tmp = c * (0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.5e-19], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.5000000000000002e-19

   1. Initial program 68.2%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 52.1

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

   5. Applied rewrites 52.1%

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

   if 2.5000000000000002e-19 < 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

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   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 2.3

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

   5. Applied rewrites 2.3%

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

   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 33.1%

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

   10. Applied rewrites 33.1%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 11.2% 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 55.6%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
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 36.4

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

5. Applied rewrites 36.4%

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

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 11.3%

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

10. Applied rewrites 11.3%

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

11. Final simplification 11.3%

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

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

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

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 2024232 
(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))