quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.9% → 72.0%

Time: 11.0s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 72.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.2 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\left|\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.2e-294)
   (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
   (/ c (- (- b_2) (sqrt (fabs (fma c a (* b_2 b_2))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.2e-294) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = c / (-b_2 - sqrt(fabs(fma(c, a, (b_2 * b_2)))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.2e-294)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(c / Float64(Float64(-b_2) - sqrt(abs(fma(c, a, Float64(b_2 * b_2))))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.2e-294], 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[(c / N[((-b$95$2) - N[Sqrt[N[Abs[N[(c * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -6.20000000000000007e-294

   1. Initial program 71.8%

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

   if -6.20000000000000007e-294 < b_2

   1. Initial program 30.2%

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

   4. Applied egg-rr 1.2%

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

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 54.2

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

   7. Simplified 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 54.2

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square N/A

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

      6. lower-fabs.f64 77.7

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 77.7

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

   9. Applied egg-rr 77.7%

      \[\leadsto \frac{-c}{b\_2 + \sqrt{\color{blue}{\left|\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)\right|}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.2 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\left|\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)\right|}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\\ \mathbf{if}\;b\_2 \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{t\_0 - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 9.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\left|a \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b_2 b_2 (* a c)))))
   (if (<= b_2 -7.5e-67)
     (/ (- t_0 b_2) a)
     (if (<= b_2 9.2e-42)
       (/ c (- (- b_2) (sqrt (fabs (* a c)))))
       (/ c (- (- b_2) t_0))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fma(b_2, b_2, (a * c)));
	double tmp;
	if (b_2 <= -7.5e-67) {
		tmp = (t_0 - b_2) / a;
	} else if (b_2 <= 9.2e-42) {
		tmp = c / (-b_2 - sqrt(fabs((a * c))));
	} else {
		tmp = c / (-b_2 - t_0);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = sqrt(fma(b_2, b_2, Float64(a * c)))
	tmp = 0.0
	if (b_2 <= -7.5e-67)
		tmp = Float64(Float64(t_0 - b_2) / a);
	elseif (b_2 <= 9.2e-42)
		tmp = Float64(c / Float64(Float64(-b_2) - sqrt(abs(Float64(a * c)))));
	else
		tmp = Float64(c / Float64(Float64(-b_2) - t_0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -7.5e-67], N[(N[(t$95$0 - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 9.2e-42], N[(c / N[((-b$95$2) - N[Sqrt[N[Abs[N[(a * c), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / N[((-b$95$2) - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.5000000000000005e-67

   1. Initial program 69.6%

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

   4. Applied egg-rr 64.0%

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

   if -7.5000000000000005e-67 < b_2 < 9.20000000000000015e-42

   1. Initial program 76.4%

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

   4. Applied egg-rr 0.9%

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

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.5

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

   7. Simplified 6.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 6.5

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square N/A

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

      6. lower-fabs.f64 69.8

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 69.8

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

   9. Applied egg-rr 69.8%

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

   10. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 65.5

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

   12. Simplified 65.5%

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

   if 9.20000000000000015e-42 < b_2

   1. Initial program 10.7%

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

   4. Applied egg-rr 1.5%

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

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 72.3

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

   7. Simplified 72.3%

      \[\leadsto \frac{\color{blue}{-c}}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 9.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\left|a \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)\right|}\\ \mathbf{if}\;b\_2 \leq -6.2 \cdot 10^{-294}:\\ \;\;\;\;\frac{t\_0 - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (fabs (fma c a (* b_2 b_2))))))
   (if (<= b_2 -6.2e-294) (/ (- t_0 b_2) a) (/ c (- (- b_2) t_0)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(fma(c, a, (b_2 * b_2))));
	double tmp;
	if (b_2 <= -6.2e-294) {
		tmp = (t_0 - b_2) / a;
	} else {
		tmp = c / (-b_2 - t_0);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = sqrt(abs(fma(c, a, Float64(b_2 * b_2))))
	tmp = 0.0
	if (b_2 <= -6.2e-294)
		tmp = Float64(Float64(t_0 - b_2) / a);
	else
		tmp = Float64(c / Float64(Float64(-b_2) - t_0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[Abs[N[(c * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -6.2e-294], N[(N[(t$95$0 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c / N[((-b$95$2) - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -6.20000000000000007e-294

   1. Initial program 71.8%

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

   4. Applied egg-rr 51.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 51.6

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square N/A

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

      6. lower-fabs.f64 70.1

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 70.1

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

   6. Applied egg-rr 70.1%

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

   if -6.20000000000000007e-294 < b_2

   1. Initial program 30.2%

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

   4. Applied egg-rr 1.2%

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

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 54.2

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

   7. Simplified 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 54.2

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square N/A

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

      6. lower-fabs.f64 77.7

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 77.7

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

   9. Applied egg-rr 77.7%

      \[\leadsto \frac{-c}{b\_2 + \sqrt{\color{blue}{\left|\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)\right|}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.2 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{\left|\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)\right|} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\left|\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)\right|}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 8.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{\left|\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)\right|} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 8.5e-42)
   (/ (- (sqrt (fabs (fma c a (* b_2 b_2)))) b_2) a)
   (/ c (- (- b_2) (sqrt (fma b_2 b_2 (* a c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 8.5e-42) {
		tmp = (sqrt(fabs(fma(c, a, (b_2 * b_2)))) - b_2) / a;
	} else {
		tmp = c / (-b_2 - sqrt(fma(b_2, b_2, (a * c))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 8.5e-42)
		tmp = Float64(Float64(sqrt(abs(fma(c, a, Float64(b_2 * b_2)))) - b_2) / a);
	else
		tmp = Float64(c / Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(a * c)))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 8.5e-42], N[(N[(N[Sqrt[N[Abs[N[(c * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c / N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 8.4999999999999996e-42

   1. Initial program 72.8%

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

   4. Applied egg-rr 39.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 39.0

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square N/A

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

      6. lower-fabs.f64 71.6

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 71.6

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

   6. Applied egg-rr 71.6%

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

   if 8.4999999999999996e-42 < b_2

   1. Initial program 10.7%

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

   4. Applied egg-rr 1.5%

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

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 72.3

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

   7. Simplified 72.3%

      \[\leadsto \frac{\color{blue}{-c}}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 8.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{\left|\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)\right|} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\left|a \cdot c\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.5e-67)
   (/ (- (sqrt (fma b_2 b_2 (* a c))) b_2) a)
   (/ c (- (- b_2) (sqrt (fabs (* a c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.5e-67) {
		tmp = (sqrt(fma(b_2, b_2, (a * c))) - b_2) / a;
	} else {
		tmp = c / (-b_2 - sqrt(fabs((a * c))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.5e-67)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(c / Float64(Float64(-b_2) - sqrt(abs(Float64(a * c)))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.5e-67], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c / N[((-b$95$2) - N[Sqrt[N[Abs[N[(a * c), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -7.5000000000000005e-67

   1. Initial program 69.6%

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

   4. Applied egg-rr 64.0%

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

   if -7.5000000000000005e-67 < b_2

   1. Initial program 40.3%

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

   4. Applied egg-rr 1.2%

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

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 42.7

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

   7. Simplified 42.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 42.7

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square N/A

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

      6. lower-fabs.f64 71.8

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 71.8

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

   9. Applied egg-rr 71.8%

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

   10. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 54.8

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

   12. Simplified 54.8%

       \[\leadsto \frac{-c}{b\_2 + \sqrt{\left|\color{blue}{c \cdot a}\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\left|a \cdot c\right|}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\left|a \cdot c\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.3e+31) (/ (- b_2) a) (/ c (- (- b_2) (sqrt (fabs (* a c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.3e+31) {
		tmp = -b_2 / a;
	} else {
		tmp = c / (-b_2 - sqrt(fabs((a * 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 <= (-5.3d+31)) then
        tmp = -b_2 / a
    else
        tmp = c / (-b_2 - sqrt(abs((a * c))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.3e+31) {
		tmp = -b_2 / a;
	} else {
		tmp = c / (-b_2 - Math.sqrt(Math.abs((a * c))));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.3e+31:
		tmp = -b_2 / a
	else:
		tmp = c / (-b_2 - math.sqrt(math.fabs((a * c))))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.3e+31)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(c / Float64(Float64(-b_2) - sqrt(abs(Float64(a * c)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.3e+31)
		tmp = -b_2 / a;
	else
		tmp = c / (-b_2 - sqrt(abs((a * c))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.3e+31], N[((-b$95$2) / a), $MachinePrecision], N[(c / N[((-b$95$2) - N[Sqrt[N[Abs[N[(a * c), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.3000000000000003e31

   1. Initial program 65.9%

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

   3. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 36.7

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

   5. Simplified 36.7%

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

   6. Taylor expanded in b_2 around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 45.3

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

   8. Simplified 45.3%

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

   if -5.3000000000000003e31 < b_2

   1. Initial program 44.1%

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

   4. Applied egg-rr 1.2%

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

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.3

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

   7. Simplified 39.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 39.3

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square N/A

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

      6. lower-fabs.f64 68.3

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 68.3

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

   9. Applied egg-rr 68.3%

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

   10. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 52.7

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

   12. Simplified 52.7%

       \[\leadsto \frac{-c}{b\_2 + \sqrt{\left|\color{blue}{c \cdot a}\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) - \sqrt{\left|a \cdot c\right|}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = -(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 <= (-5d-310)) then
        tmp = -b_2 / a
    else
        tmp = -(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 <= -5e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = -(c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -b_2 / a
	else:
		tmp = -(c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(-Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -b_2 / a;
	else
		tmp = -(c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[((-b$95$2) / a), $MachinePrecision], (-N[(c / b$95$2), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 72.2%

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

   3. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 24.1

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

   5. Simplified 24.1%

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

   6. Taylor expanded in b_2 around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 30.2

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

   8. Simplified 30.2%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 29.2%

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

   4. Applied egg-rr 1.2%

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

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 55.0

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

   7. Simplified 55.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 55.0

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square N/A

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

      6. lower-fabs.f64 77.3

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 77.3

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

   9. Applied egg-rr 77.3%

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

   10. Taylor expanded in b_2 around inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 34.2

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

   12. Simplified 34.2%

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

4. Final simplification 32.3%

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

Alternative 8: 14.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	return -(c / 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 / b_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

4. Applied egg-rr 0.9%

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

5. Taylor expanded in b_2 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 31.5

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

7. Simplified 31.5%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 31.5

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

   3. rem-square-sqrt N/A

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

   4. sqrt-unprod N/A

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

   5. rem-sqrt-square N/A

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

   6. lower-fabs.f64 52.6

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 52.6

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

9. Applied egg-rr 52.6%

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

10. Taylor expanded in b_2 around inf

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

    1. mul-1-neg N/A

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

    2. distribute-neg-frac2 N/A

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

    3. mul-1-neg N/A

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

    4. lower-/.f64 N/A

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

    5. mul-1-neg N/A

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

    6. lower-neg.f64 18.9

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

12. Simplified 18.9%

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

13. Final simplification 18.9%

    \[\leadsto -\frac{c}{b\_2} \]
14. Add Preprocessing

Alternative 9: 4.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(b\_2 \cdot b\_2\right) \cdot 0 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (b_2 * b_2) * 0.0d0
end function

Java

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

Python

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

Julia

function code(a, b_2, c)
	return Float64(Float64(b_2 * b_2) * 0.0)
end

MATLAB

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

Wolfram

code[a_, b$95$2_, c_] := N[(N[(b$95$2 * b$95$2), $MachinePrecision] * 0.0), $MachinePrecision]

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in b_2 around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. sub-neg N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 12.8

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

5. Simplified 12.8%

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

6. Taylor expanded in b_2 around inf

   \[\leadsto \left(b\_2 \cdot b\_2\right) \cdot \color{blue}{0} \]
7. Step-by-step derivation

8. Simplified 4.7%

   \[\leadsto \left(b\_2 \cdot b\_2\right) \cdot \color{blue}{0} \]
9. Add Preprocessing

Alternative 10: 3.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ -b\_2 \end{array} \]

FPCore

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

C

double code(double a, double b_2, double c) {
	return -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 = -b_2
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b$95$2_, c_] := (-b$95$2)

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in b_2 around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. sub-neg N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 12.8

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

5. Simplified 12.8%

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

6. Taylor expanded in a around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(b\_2\right)} \]

   2. lower-neg.f64 3.2

      \[\leadsto \color{blue}{-b\_2} \]

8. Simplified 3.2%

   \[\leadsto \color{blue}{-b\_2} \]
9. 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 2024214 
(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))