quad2p (problem 3.2.1, positive)

Percentage Accurate: 53.2% → 87.2%

Time: 8.3s

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: 53.2% 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: 87.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}\\ \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{-145}:\\ \;\;\;\;\frac{t\_0}{a} - \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{c \cdot a}{\left(\left(-b\_2\right) - t\_0\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (fma a (- c) (* b_2 b_2)))))
   (if (<= b_2 -2e+154)
     (* (/ b_2 a) -2.0)
     (if (<= b_2 7e-145)
       (- (/ t_0 a) (/ b_2 a))
       (if (<= b_2 3.8e+28)
         (/ (* c a) (* (- (- b_2) t_0) a))
         (* -0.5 (/ c b_2)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fma(a, -c, (b_2 * b_2)));
	double tmp;
	if (b_2 <= -2e+154) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 7e-145) {
		tmp = (t_0 / a) - (b_2 / a);
	} else if (b_2 <= 3.8e+28) {
		tmp = (c * a) / ((-b_2 - t_0) * a);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = sqrt(fma(a, Float64(-c), Float64(b_2 * b_2)))
	tmp = 0.0
	if (b_2 <= -2e+154)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	elseif (b_2 <= 7e-145)
		tmp = Float64(Float64(t_0 / a) - Float64(b_2 / a));
	elseif (b_2 <= 3.8e+28)
		tmp = Float64(Float64(c * a) / Float64(Float64(Float64(-b_2) - t_0) * a));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * (-c) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -2e+154], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[b$95$2, 7e-145], N[(N[(t$95$0 / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.8e+28], N[(N[(c * a), $MachinePrecision] / N[(N[((-b$95$2) - t$95$0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.00000000000000007e154

   1. Initial program 39.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.00000000000000007e154 < b_2 < 6.99999999999999994e-145

   1. Initial program 90.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 90.7

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

   4. Applied rewrites 90.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. div-sub N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 90.7

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 90.7

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

   6. Applied rewrites 90.7%

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

   if 6.99999999999999994e-145 < b_2 < 3.7999999999999999e28

   1. Initial program 62.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 62.3

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

   4. Applied rewrites 62.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 62.7%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 84.5

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

   9. Applied rewrites 84.5%

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

   if 3.7999999999999999e28 < b_2

   1. Initial program 14.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{-145}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}{a} - \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{c \cdot a}{\left(\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+154)
   (* (/ b_2 a) -2.0)
   (if (<= b_2 3e+22)
     (- (/ (sqrt (fma a (- c) (* b_2 b_2))) a) (/ b_2 a))
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+154) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 3e+22) {
		tmp = (sqrt(fma(a, -c, (b_2 * b_2))) / a) - (b_2 / a);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e+154)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	elseif (b_2 <= 3e+22)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-c), Float64(b_2 * b_2))) / a) - Float64(b_2 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e+154], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[b$95$2, 3e+22], N[(N[(N[Sqrt[N[(a * (-c) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.00000000000000007e154

   1. Initial program 39.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.00000000000000007e154 < b_2 < 3e22

   1. Initial program 85.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 85.4

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

   4. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. div-sub N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 85.4

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 85.4

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

   6. Applied rewrites 85.4%

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

   if 3e22 < b_2

   1. Initial program 14.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+154)
   (* (/ b_2 a) -2.0)
   (if (<= b_2 3e+22)
     (/ (- (sqrt (fma a (- c) (* b_2 b_2))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+154) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 3e+22) {
		tmp = (sqrt(fma(a, -c, (b_2 * b_2))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e+154)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	elseif (b_2 <= 3e+22)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-c), Float64(b_2 * b_2))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e+154], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[b$95$2, 3e+22], N[(N[(N[Sqrt[N[(a * (-c) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.00000000000000007e154

   1. Initial program 39.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.00000000000000007e154 < b_2 < 3e22

   1. Initial program 85.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 85.4

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

   4. Applied rewrites 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 85.4

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 85.4

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

   6. Applied rewrites 85.4%

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

   if 3e22 < b_2

   1. Initial program 14.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 3.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.3e-22)
   (* (/ b_2 a) -2.0)
   (if (<= b_2 3.9e-31) (/ (- (sqrt (* (- a) c)) b_2) a) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.3e-22) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 3.9e-31) {
		tmp = (sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / 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 <= (-1.3d-22)) then
        tmp = (b_2 / a) * (-2.0d0)
    else if (b_2 <= 3.9d-31) then
        tmp = (sqrt((-a * c)) - b_2) / a
    else
        tmp = (-0.5d0) * (c / 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 <= -1.3e-22) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 3.9e-31) {
		tmp = (Math.sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.3e-22:
		tmp = (b_2 / a) * -2.0
	elif b_2 <= 3.9e-31:
		tmp = (math.sqrt((-a * c)) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.3e-22)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	elseif (b_2 <= 3.9e-31)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.3e-22)
		tmp = (b_2 / a) * -2.0;
	elseif (b_2 <= 3.9e-31)
		tmp = (sqrt((-a * c)) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.3e-22], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[b$95$2, 3.9e-31], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.3e-22

   1. Initial program 67.9%

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

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

   5. Applied rewrites 93.5%

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

   if -1.3e-22 < b_2 < 3.9000000000000001e-31

   1. Initial program 80.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 80.9

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

   4. Applied rewrites 80.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 80.9

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 80.9

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

   6. Applied rewrites 80.9%

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

   7. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 73.3

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

   9. Applied rewrites 73.3%

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

   if 3.9000000000000001e-31 < b_2

   1. Initial program 18.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 3.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.15 \cdot 10^{-257}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.15e-257) (* (/ b_2 a) -2.0) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.15e-257) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = -0.5 * (c / 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 <= 1.15d-257) then
        tmp = (b_2 / a) * (-2.0d0)
    else
        tmp = (-0.5d0) * (c / 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 <= 1.15e-257) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.15e-257:
		tmp = (b_2 / a) * -2.0
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.15e-257)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.15e-257)
		tmp = (b_2 / a) * -2.0;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.15e-257], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.15e-257

   1. Initial program 74.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 1.15e-257 < b_2

   1. Initial program 35.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.1

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

   5. Applied rewrites 66.1%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.15 \cdot 10^{-257}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.15 \cdot 10^{-257}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b\_2} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.15e-257) (* (/ b_2 a) -2.0) (* (/ -0.5 b_2) c)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.15e-257) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = (-0.5 / b_2) * c;
	}
	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 <= 1.15d-257) then
        tmp = (b_2 / a) * (-2.0d0)
    else
        tmp = ((-0.5d0) / b_2) * c
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.15e-257) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = (-0.5 / b_2) * c;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.15e-257:
		tmp = (b_2 / a) * -2.0
	else:
		tmp = (-0.5 / b_2) * c
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.15e-257)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	else
		tmp = Float64(Float64(-0.5 / b_2) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.15e-257)
		tmp = (b_2 / a) * -2.0;
	else
		tmp = (-0.5 / b_2) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.15e-257], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 / b$95$2), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.15e-257

   1. Initial program 74.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 1.15e-257 < b_2

   1. Initial program 35.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.1

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 65.8%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.15 \cdot 10^{-257}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b\_2} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{b\_2}{a} \cdot -2 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* (/ b_2 a) -2.0))

C

double code(double a, double b_2, double c) {
	return (b_2 / a) * -2.0;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (b_2 / a) * (-2.0d0)
end function

Java

public static double code(double a, double b_2, double c) {
	return (b_2 / a) * -2.0;
}

Python

def code(a, b_2, c):
	return (b_2 / a) * -2.0

Julia

function code(a, b_2, c)
	return Float64(Float64(b_2 / a) * -2.0)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (b_2 / a) * -2.0;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in b_2 around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 38.9

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

5. Applied rewrites 38.9%

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

6. Final simplification 38.9%

   \[\leadsto \frac{b\_2}{a} \cdot -2 \]
7. 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 2024278 
(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))