quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.8% → 85.5%

Time: 7.9s

Alternatives: 7

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+139}:\\ \;\;\;\;\frac{\left(-b\_2\right) \cdot \left(2 + -0.5 \cdot \left(a \cdot \frac{c}{{b\_2}^{2}}\right)\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 1.15 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+139)
   (/ (* (- b_2) (+ 2.0 (* -0.5 (* a (/ c (pow b_2 2.0)))))) a)
   (if (<= b_2 1.15e-75)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ 1.0 (fma 0.5 (/ a b_2) (* -2.0 (/ b_2 c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+139) {
		tmp = (-b_2 * (2.0 + (-0.5 * (a * (c / pow(b_2, 2.0)))))) / a;
	} else if (b_2 <= 1.15e-75) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = 1.0 / fma(0.5, (a / b_2), (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+139)
		tmp = Float64(Float64(Float64(-b_2) * Float64(2.0 + Float64(-0.5 * Float64(a * Float64(c / (b_2 ^ 2.0)))))) / a);
	elseif (b_2 <= 1.15e-75)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(1.0 / fma(0.5, Float64(a / b_2), Float64(-2.0 * Float64(b_2 / c))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+139], N[(N[((-b$95$2) * N[(2.0 + N[(-0.5 * N[(a * N[(c / N[Power[b$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.15e-75], 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[(1.0 / N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.0000000000000003e139

   1. Initial program 36.9%

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

      1. +-commutative 36.9%

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

      2. unsub-neg 36.9%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in b_2 around -inf 86.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 \cdot \left(2 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)}}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 86.2%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(2 + -0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}}{a} \]

      2. neg-mul-1 86.2%

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

      3. associate-/l* 94.8%

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

   7. Simplified 94.8%

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

   if -5.0000000000000003e139 < b_2 < 1.15e-75

   1. Initial program 86.5%

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

      1. +-commutative 86.5%

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

      2. unsub-neg 86.5%

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

   3. Simplified 86.5%

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

   if 1.15e-75 < b_2

   1. Initial program 19.0%

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

      1. +-commutative 19.0%

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

      2. unsub-neg 19.0%

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

   3. Simplified 19.0%

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

      1. clear-num 18.9%

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

      2. inv-pow 18.9%

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

      3. sub-neg 18.9%

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

      4. add-sqr-sqrt 15.0%

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

      5. hypot-define 27.8%

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

      6. *-commutative 27.8%

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

      7. distribute-rgt-neg-in 27.8%

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

   6. Applied egg-rr 27.8%

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

      1. unpow-1 27.8%

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

   8. Simplified 27.8%

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

   9. Taylor expanded in a around 0 0.0%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b\_2} + 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)}} \]

       2. associate-*r/ 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2 \cdot b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}\right)} \]

       3. *-commutative 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot c}}\right)} \]

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c}\right)} \]

       5. rem-square-sqrt 84.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{-1} \cdot c}\right)} \]

       6. times-frac 84.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2}{-1} \cdot \frac{b\_2}{c}}\right)} \]

       7. metadata-eval 84.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{-2} \cdot \frac{b\_2}{c}\right)} \]

   11. Simplified 84.8%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.6 \cdot 10^{+139}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.15 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.6e+139)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 1.15e-74)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ 1.0 (fma 0.5 (/ a b_2) (* -2.0 (/ b_2 c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.6e+139) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 1.15e-74) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = 1.0 / fma(0.5, (a / b_2), (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.6e+139)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 1.15e-74)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(1.0 / fma(0.5, Float64(a / b_2), Float64(-2.0 * Float64(b_2 / c))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.6e+139], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.15e-74], 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[(1.0 / N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.6e139

   1. Initial program 36.9%

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

      1. +-commutative 36.9%

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

      2. unsub-neg 36.9%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in b_2 around -inf 94.8%

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

   if -4.6e139 < b_2 < 1.1499999999999999e-74

   1. Initial program 86.5%

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

      1. +-commutative 86.5%

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

      2. unsub-neg 86.5%

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

   3. Simplified 86.5%

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

   if 1.1499999999999999e-74 < b_2

   1. Initial program 19.0%

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

      1. +-commutative 19.0%

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

      2. unsub-neg 19.0%

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

   3. Simplified 19.0%

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

      1. clear-num 18.9%

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

      2. inv-pow 18.9%

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

      3. sub-neg 18.9%

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

      4. add-sqr-sqrt 15.0%

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

      5. hypot-define 27.8%

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

      6. *-commutative 27.8%

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

      7. distribute-rgt-neg-in 27.8%

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

   6. Applied egg-rr 27.8%

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

      1. unpow-1 27.8%

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

   8. Simplified 27.8%

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

   9. Taylor expanded in a around 0 0.0%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b\_2} + 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)}} \]

       2. associate-*r/ 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2 \cdot b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}\right)} \]

       3. *-commutative 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot c}}\right)} \]

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c}\right)} \]

       5. rem-square-sqrt 84.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{-1} \cdot c}\right)} \]

       6. times-frac 84.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2}{-1} \cdot \frac{b\_2}{c}}\right)} \]

       7. metadata-eval 84.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{-2} \cdot \frac{b\_2}{c}\right)} \]

   11. Simplified 84.8%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-62}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.25 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.9e-62)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 1.25e-77)
     (/ (- (sqrt (* a (- c))) b_2) a)
     (/ 1.0 (fma 0.5 (/ a b_2) (* -2.0 (/ b_2 c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.9e-62) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 1.25e-77) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = 1.0 / fma(0.5, (a / b_2), (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.9e-62)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 1.25e-77)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(1.0 / fma(0.5, Float64(a / b_2), Float64(-2.0 * Float64(b_2 / c))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.9e-62], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.25e-77], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.90000000000000003e-62

   1. Initial program 68.7%

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

      1. +-commutative 68.7%

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

      2. unsub-neg 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in b_2 around -inf 87.8%

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

   if -1.90000000000000003e-62 < b_2 < 1.24999999999999991e-77

   1. Initial program 82.3%

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

      1. +-commutative 82.3%

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

      2. unsub-neg 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in b_2 around 0 78.6%

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

      1. associate-*r* 78.6%

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

      2. neg-mul-1 78.6%

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

      3. *-commutative 78.6%

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

   7. Simplified 78.6%

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

   if 1.24999999999999991e-77 < b_2

   1. Initial program 19.0%

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

      1. +-commutative 19.0%

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

      2. unsub-neg 19.0%

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

   3. Simplified 19.0%

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

      1. clear-num 18.9%

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

      2. inv-pow 18.9%

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

      3. sub-neg 18.9%

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

      4. add-sqr-sqrt 15.0%

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

      5. hypot-define 27.8%

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

      6. *-commutative 27.8%

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

      7. distribute-rgt-neg-in 27.8%

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

   6. Applied egg-rr 27.8%

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

      1. unpow-1 27.8%

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

   8. Simplified 27.8%

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

   9. Taylor expanded in a around 0 0.0%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b\_2} + 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)}} \]

       2. associate-*r/ 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2 \cdot b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}\right)} \]

       3. *-commutative 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot c}}\right)} \]

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c}\right)} \]

       5. rem-square-sqrt 84.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{-1} \cdot c}\right)} \]

       6. times-frac 84.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2}{-1} \cdot \frac{b\_2}{c}}\right)} \]

       7. metadata-eval 84.8%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{-2} \cdot \frac{b\_2}{c}\right)} \]

   11. Simplified 84.8%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-62}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.25 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.2e-64)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 1.5e-75) (/ (- (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 <= -6.2e-64) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 1.5e-75) {
		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 <= (-6.2d-64)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 1.5d-75) 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 <= -6.2e-64) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 1.5e-75) {
		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 <= -6.2e-64:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 1.5e-75:
		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 <= -6.2e-64)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 1.5e-75)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.2e-64)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 1.5e-75)
		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, -6.2e-64], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.5e-75], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.20000000000000049e-64

   1. Initial program 68.7%

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

      1. +-commutative 68.7%

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

      2. unsub-neg 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in b_2 around -inf 87.8%

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

   if -6.20000000000000049e-64 < b_2 < 1.4999999999999999e-75

   1. Initial program 82.3%

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

      1. +-commutative 82.3%

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

      2. unsub-neg 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in b_2 around 0 78.6%

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

      1. associate-*r* 78.6%

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

      2. neg-mul-1 78.6%

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

      3. *-commutative 78.6%

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

   7. Simplified 78.6%

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

   if 1.4999999999999999e-75 < b_2

   1. Initial program 19.0%

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

      1. +-commutative 19.0%

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

      2. unsub-neg 19.0%

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

   3. Simplified 19.0%

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

   5. Taylor expanded in b_2 around inf 84.6%

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

      1. associate-*r/ 84.6%

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

      2. *-commutative 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.35 \cdot 10^{-302}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.35e-302) (* -2.0 (/ b_2 a)) (/ (* -0.5 c) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.35e-302) {
		tmp = -2.0 * (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 <= 1.35d-302) then
        tmp = (-2.0d0) * (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 <= 1.35e-302) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.35e-302:
		tmp = -2.0 * (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 <= 1.35e-302)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.35e-302)
		tmp = -2.0 * (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, 1.35e-302], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.35000000000000003e-302

   1. Initial program 74.5%

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

      1. +-commutative 74.5%

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

      2. unsub-neg 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in b_2 around -inf 65.1%

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

   if 1.35000000000000003e-302 < b_2

   1. Initial program 30.8%

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

      1. +-commutative 30.8%

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

      2. unsub-neg 30.8%

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

   3. Simplified 30.8%

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

   5. Taylor expanded in b_2 around inf 70.9%

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

      1. associate-*r/ 70.9%

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

      2. *-commutative 70.9%

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

   7. Simplified 70.9%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.35 \cdot 10^{-302}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.5% accurate, 22.4× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / 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 = (-2.0d0) * (b_2 / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

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

   1. +-commutative 51.8%

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

   2. unsub-neg 51.8%

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

3. Simplified 51.8%

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

5. Taylor expanded in b_2 around -inf 32.7%

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
6. Add Preprocessing

Alternative 7: 15.4% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

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

   1. +-commutative 51.8%

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

   2. unsub-neg 51.8%

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

3. Simplified 51.8%

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

5. Taylor expanded in b_2 around 0 32.5%

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

   1. associate-*r* 32.5%

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

   2. neg-mul-1 32.5%

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

   3. *-commutative 32.5%

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

7. Simplified 32.5%

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

8. Taylor expanded in b_2 around inf 12.8%

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

   1. associate-*r/ 12.8%

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

   2. neg-mul-1 12.8%

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

10. Simplified 12.8%

    \[\leadsto \color{blue}{\frac{-b\_2}{a}} \]
11. Add Preprocessing

Developer target: 99.7% accurate, 0.1× 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 2024119 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b_2 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))) b_2) a) (/ (- c) (+ b_2 (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))))))

  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))