quadm (p42, negative)

Percentage Accurate: 51.8% → 85.4%

Time: 11.7s

Alternatives: 10

Speedup: 2.4×

• could not determine a ground truth

  1. a = 7.137704138478491e-254
  2. b = -8.161671618334053e+83
  3. c = 1.8071085470419346e-172

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{0 - b}\\ \mathbf{elif}\;b \leq 10^{+72}:\\ \;\;\;\;\frac{\left(0 - b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.9e-98)
   (/ (* c (fma a (/ c (* b b)) 1.0)) (- 0.0 b))
   (if (<= b 1e+72)
     (/ (- (- 0.0 b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.9e-98) {
		tmp = (c * fma(a, (c / (b * b)), 1.0)) / (0.0 - b);
	} else if (b <= 1e+72) {
		tmp = ((0.0 - b) - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.9e-98)
		tmp = Float64(Float64(c * fma(a, Float64(c / Float64(b * b)), 1.0)) / Float64(0.0 - b));
	elseif (b <= 1e+72)
		tmp = Float64(Float64(Float64(0.0 - b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.9e-98], N[(N[(c * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+72], N[(N[(N[(0.0 - b), $MachinePrecision] - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.89999999999999971e-98

   1. Initial program 14.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

      5. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{0 - \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

      6. +-commutative N/A

         \[\leadsto \frac{0 - \color{blue}{\left(\frac{a \cdot {c}^{2}}{{b}^{2}} + c\right)}}{b} \]

      7. *-commutative N/A

         \[\leadsto \frac{0 - \left(\frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{2}} + c\right)}{b} \]

      8. associate-/l* N/A

         \[\leadsto \frac{0 - \left(\color{blue}{{c}^{2} \cdot \frac{a}{{b}^{2}}} + c\right)}{b} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{0 - \color{blue}{\mathsf{fma}\left({c}^{2}, \frac{a}{{b}^{2}}, c\right)}}{b} \]

      10. unpow2 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(\color{blue}{c \cdot c}, \frac{a}{{b}^{2}}, c\right)}{b} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(\color{blue}{c \cdot c}, \frac{a}{{b}^{2}}, c\right)}{b} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(c \cdot c, \color{blue}{\frac{a}{{b}^{2}}}, c\right)}{b} \]

      13. unpow2 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(c \cdot c, \frac{a}{\color{blue}{b \cdot b}}, c\right)}{b} \]

      14. *-lowering-*.f64 78.1

          \[\leadsto \frac{0 - \mathsf{fma}\left(c \cdot c, \frac{a}{\color{blue}{b \cdot b}}, c\right)}{b} \]

   5. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{0 - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}} \]

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

      4. +-commutative N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. *-lft-identity N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l/ N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 93.9

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

   8. Simplified 93.9%

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

   if -3.89999999999999971e-98 < b < 9.99999999999999944e71

   1. Initial program 86.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   if 9.99999999999999944e71 < b

   1. Initial program 66.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 96.8

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

   5. Simplified 96.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{0 - b}\\ \mathbf{elif}\;b \leq 10^{+72}:\\ \;\;\;\;\frac{\left(0 - b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{0 - b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e-99)
   (/ (* c (fma a (/ c (* b b)) 1.0)) (- 0.0 b))
   (if (<= b 4e+71)
     (* (/ -0.5 a) (+ b (sqrt (fma b b (* c (* a -4.0))))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e-99) {
		tmp = (c * fma(a, (c / (b * b)), 1.0)) / (0.0 - b);
	} else if (b <= 4e+71) {
		tmp = (-0.5 / a) * (b + sqrt(fma(b, b, (c * (a * -4.0)))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e-99)
		tmp = Float64(Float64(c * fma(a, Float64(c / Float64(b * b)), 1.0)) / Float64(0.0 - b));
	elseif (b <= 4e+71)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e-99], N[(N[(c * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+71], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2e-99

   1. Initial program 14.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

      5. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{0 - \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

      6. +-commutative N/A

         \[\leadsto \frac{0 - \color{blue}{\left(\frac{a \cdot {c}^{2}}{{b}^{2}} + c\right)}}{b} \]

      7. *-commutative N/A

         \[\leadsto \frac{0 - \left(\frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{2}} + c\right)}{b} \]

      8. associate-/l* N/A

         \[\leadsto \frac{0 - \left(\color{blue}{{c}^{2} \cdot \frac{a}{{b}^{2}}} + c\right)}{b} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{0 - \color{blue}{\mathsf{fma}\left({c}^{2}, \frac{a}{{b}^{2}}, c\right)}}{b} \]

      10. unpow2 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(\color{blue}{c \cdot c}, \frac{a}{{b}^{2}}, c\right)}{b} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(\color{blue}{c \cdot c}, \frac{a}{{b}^{2}}, c\right)}{b} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(c \cdot c, \color{blue}{\frac{a}{{b}^{2}}}, c\right)}{b} \]

      13. unpow2 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(c \cdot c, \frac{a}{\color{blue}{b \cdot b}}, c\right)}{b} \]

      14. *-lowering-*.f64 78.1

          \[\leadsto \frac{0 - \mathsf{fma}\left(c \cdot c, \frac{a}{\color{blue}{b \cdot b}}, c\right)}{b} \]

   5. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{0 - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}} \]

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

      4. +-commutative N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. *-lft-identity N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l/ N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 93.9

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

   8. Simplified 93.9%

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

   if -1.2e-99 < b < 4.0000000000000002e71

   1. Initial program 86.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied egg-rr 86.4%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]

   if 4.0000000000000002e71 < b

   1. Initial program 66.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 96.8

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

   5. Simplified 96.8%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{0 - b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-98}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{0 - b}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-65}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-98)
   (/ (* c (fma a (/ c (* b b)) 1.0)) (- 0.0 b))
   (if (<= b 1.35e-65)
     (/ (+ b (sqrt (* c (* a -4.0)))) (* a -2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-98) {
		tmp = (c * fma(a, (c / (b * b)), 1.0)) / (0.0 - b);
	} else if (b <= 1.35e-65) {
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * -2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-98)
		tmp = Float64(Float64(c * fma(a, Float64(c / Float64(b * b)), 1.0)) / Float64(0.0 - b));
	elseif (b <= 1.35e-65)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * -2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-98], N[(N[(c * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-65], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7e-98

   1. Initial program 14.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

      5. --lowering--.f64 N/A

         \[\leadsto \frac{\color{blue}{0 - \left(c + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

      6. +-commutative N/A

         \[\leadsto \frac{0 - \color{blue}{\left(\frac{a \cdot {c}^{2}}{{b}^{2}} + c\right)}}{b} \]

      7. *-commutative N/A

         \[\leadsto \frac{0 - \left(\frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{2}} + c\right)}{b} \]

      8. associate-/l* N/A

         \[\leadsto \frac{0 - \left(\color{blue}{{c}^{2} \cdot \frac{a}{{b}^{2}}} + c\right)}{b} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{0 - \color{blue}{\mathsf{fma}\left({c}^{2}, \frac{a}{{b}^{2}}, c\right)}}{b} \]

      10. unpow2 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(\color{blue}{c \cdot c}, \frac{a}{{b}^{2}}, c\right)}{b} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(\color{blue}{c \cdot c}, \frac{a}{{b}^{2}}, c\right)}{b} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(c \cdot c, \color{blue}{\frac{a}{{b}^{2}}}, c\right)}{b} \]

      13. unpow2 N/A

          \[\leadsto \frac{0 - \mathsf{fma}\left(c \cdot c, \frac{a}{\color{blue}{b \cdot b}}, c\right)}{b} \]

      14. *-lowering-*.f64 78.1

          \[\leadsto \frac{0 - \mathsf{fma}\left(c \cdot c, \frac{a}{\color{blue}{b \cdot b}}, c\right)}{b} \]

   5. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{0 - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}} \]

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

      4. +-commutative N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. *-lft-identity N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l/ N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-/l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 93.9

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

   8. Simplified 93.9%

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

   if -3.7e-98 < b < 1.3499999999999999e-65

   1. Initial program 85.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied egg-rr 85.4%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(-4 \cdot c\right)}}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right) \]

      6. *-lowering-*.f64 78.7

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right) \]

   7. Simplified 78.7%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{\frac{-1}{2}}{a}} \]

      2. clear-num N/A

         \[\leadsto \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right) \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-1}{2}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{\frac{a}{\frac{-1}{2}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{\frac{a}{\frac{-1}{2}}}} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{\color{blue}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}{\frac{a}{\frac{-1}{2}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{b + \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}}{\frac{a}{\frac{-1}{2}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{b + \color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{b + \sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

      12. div-inv N/A

          \[\leadsto \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{\color{blue}{a \cdot \frac{1}{\frac{-1}{2}}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot \color{blue}{-2}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot \color{blue}{\left(-1 + -1\right)}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{\color{blue}{a \cdot \left(-1 + -1\right)}} \]

      16. metadata-eval 78.9

          \[\leadsto \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot \color{blue}{-2}} \]

   9. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}} \]

   if 1.3499999999999999e-65 < b

   1. Initial program 73.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 85.9

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

   5. Simplified 85.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-98}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{0 - b}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-65}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.2e-106)
   (- 0.0 (/ c b))
   (if (<= b 1.9e-70)
     (/ (+ b (sqrt (* c (* a -4.0)))) (* a -2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e-106) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.9e-70) {
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * -2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.2d-106)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.9d-70) then
        tmp = (b + sqrt((c * (a * (-4.0d0))))) / (a * (-2.0d0))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e-106) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.9e-70) {
		tmp = (b + Math.sqrt((c * (a * -4.0)))) / (a * -2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.2e-106:
		tmp = 0.0 - (c / b)
	elif b <= 1.9e-70:
		tmp = (b + math.sqrt((c * (a * -4.0)))) / (a * -2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.2e-106)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.9e-70)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * -2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.2e-106)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.9e-70)
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * -2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.2e-106], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e-70], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.19999999999999971e-106

   1. Initial program 15.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 91.5

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -6.19999999999999971e-106 < b < 1.8999999999999999e-70

   1. Initial program 87.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied egg-rr 87.4%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(-4 \cdot c\right)}}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right) \]

      6. *-lowering-*.f64 80.5

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right) \]

   7. Simplified 80.5%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{\frac{-1}{2}}{a}} \]

      2. clear-num N/A

         \[\leadsto \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right) \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-1}{2}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{\frac{a}{\frac{-1}{2}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{\frac{a}{\frac{-1}{2}}}} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{\color{blue}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}{\frac{a}{\frac{-1}{2}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{b + \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}}{\frac{a}{\frac{-1}{2}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{b + \color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{b + \sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)}}}{\frac{a}{\frac{-1}{2}}} \]

      12. div-inv N/A

          \[\leadsto \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{\color{blue}{a \cdot \frac{1}{\frac{-1}{2}}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot \color{blue}{-2}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot \color{blue}{\left(-1 + -1\right)}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{\color{blue}{a \cdot \left(-1 + -1\right)}} \]

      16. metadata-eval 80.8

          \[\leadsto \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot \color{blue}{-2}} \]

   9. Applied egg-rr 80.8%

      \[\leadsto \color{blue}{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}} \]

   if 1.8999999999999999e-70 < b

   1. Initial program 73.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 85.9

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

   5. Simplified 85.9%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.45 \cdot 10^{-105}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.45e-105)
   (- 0.0 (/ c b))
   (if (<= b 2.8e-70)
     (* (/ -0.5 a) (+ b (sqrt (* a (* c -4.0)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.45e-105) {
		tmp = 0.0 - (c / b);
	} else if (b <= 2.8e-70) {
		tmp = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.45d-105)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 2.8d-70) then
        tmp = ((-0.5d0) / a) * (b + sqrt((a * (c * (-4.0d0)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.45e-105) {
		tmp = 0.0 - (c / b);
	} else if (b <= 2.8e-70) {
		tmp = (-0.5 / a) * (b + Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.45e-105:
		tmp = 0.0 - (c / b)
	elif b <= 2.8e-70:
		tmp = (-0.5 / a) * (b + math.sqrt((a * (c * -4.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.45e-105)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 2.8e-70)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.45e-105)
		tmp = 0.0 - (c / b);
	elseif (b <= 2.8e-70)
		tmp = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.45e-105], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e-70], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.45000000000000014e-105

   1. Initial program 15.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 91.5

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 91.5%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -3.45000000000000014e-105 < b < 2.7999999999999999e-70

   1. Initial program 87.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied egg-rr 87.4%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(-4 \cdot c\right)}}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right) \]

      6. *-lowering-*.f64 80.5

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right) \]

   7. Simplified 80.5%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \]

   if 2.7999999999999999e-70 < b

   1. Initial program 73.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 85.9

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

   5. Simplified 85.9%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (- 0.0 (/ c b)) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = 0.0d0 - (c / b)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = 0.0 - (c / b)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(0.0 - Float64(c / b));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = 0.0 - (c / b);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 31.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 69.8

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 69.8%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 79.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 64.4

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

   5. Simplified 64.4%

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

Alternative 7: 67.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-290}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.45e-290) (- 0.0 (/ c b)) (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e-290) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.45d-290)) then
        tmp = 0.0d0 - (c / b)
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.45e-290) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.45e-290:
		tmp = 0.0 - (c / b)
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.45e-290)
		tmp = Float64(0.0 - Float64(c / b));
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.45e-290)
		tmp = 0.0 - (c / b);
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.45e-290], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.45e-290

   1. Initial program 30.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 71.4

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 71.4%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -2.45e-290 < b

   1. Initial program 80.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

      4. /-lowering-/.f64 62.4

         \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

   5. Simplified 62.4%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 62.4

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

   7. Applied egg-rr 62.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-290}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.85e+17) (/ c b) (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e+17) {
		tmp = c / b;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.85d+17)) then
        tmp = c / b
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e+17) {
		tmp = c / b;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.85e+17:
		tmp = c / b
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.85e+17)
		tmp = Float64(c / b);
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.85e+17)
		tmp = c / b;
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.85e+17], N[(c / b), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.85e17

   1. Initial program 13.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 2.7

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

   5. Simplified 2.7%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{c}{b}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 33.6

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

   8. Simplified 33.6%

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

   if -1.85e17 < b

   1. Initial program 73.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

      4. /-lowering-/.f64 45.9

         \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

   5. Simplified 45.9%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 45.9

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

   7. Applied egg-rr 45.9%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+17}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 10.9% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

double code(double a, double b, double c) {
	return c / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / b
end function

Java

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

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 55.7%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. /-lowering-/.f64 N/A

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

   5. --lowering--.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. /-lowering-/.f64 33.3

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

5. Simplified 33.3%

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

6. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{c}{b}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 12.0

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

8. Simplified 12.0%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
9. Add Preprocessing

Alternative 10: 2.5% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (a b c) :precision binary64 (/ b a))

C

double code(double a, double b, double c) {
	return b / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b / a
end function

Java

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

Python

def code(a, b, c):
	return b / a

Julia

function code(a, b, c)
	return Float64(b / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(b / a), $MachinePrecision]

Derivation

1. Initial program 55.7%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

   4. /-lowering-/.f64 33.4

      \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

5. Simplified 33.4%

   \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
6. Step-by-step derivation

   1. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {\left(\frac{b}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - {\left(\frac{b}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   3. sub0-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({\left(\frac{b}{a}\right)}^{3}\right)}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   4. cube-neg N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(\frac{b}{a}\right)\right)}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   5. distribute-neg-frac N/A

      \[\leadsto \frac{{\color{blue}{\left(\frac{\mathsf{neg}\left(b\right)}{a}\right)}}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   6. cube-div N/A

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{neg}\left(b\right)\right)}^{3}}{{a}^{3}}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{{\left(\mathsf{neg}\left(b\right)\right)}^{\color{blue}{\left(2 \cdot \frac{3}{2}\right)}}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{{\left(\mathsf{neg}\left(b\right)\right)}^{\left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot 3\right)}\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   9. pow-pow N/A

      \[\leadsto \frac{\frac{\color{blue}{{\left({\left(\mathsf{neg}\left(b\right)\right)}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   10. pow2 N/A

       \[\leadsto \frac{\frac{{\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   11. sqr-neg N/A

       \[\leadsto \frac{\frac{{\color{blue}{\left(b \cdot b\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{{\color{blue}{\left({b}^{2}\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   13. pow-pow N/A

       \[\leadsto \frac{\frac{\color{blue}{{b}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\frac{{b}^{\left(2 \cdot \color{blue}{\frac{3}{2}}\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{{b}^{\color{blue}{3}}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   16. cube-div N/A

       \[\leadsto \frac{\color{blue}{{\left(\frac{b}{a}\right)}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   17. metadata-eval N/A

       \[\leadsto \frac{{\left(\frac{b}{a}\right)}^{3}}{\color{blue}{0} + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   18. +-lft-identity N/A

       \[\leadsto \frac{{\left(\frac{b}{a}\right)}^{3}}{\color{blue}{\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}}} \]

   19. distribute-rgt-out N/A

       \[\leadsto \frac{{\left(\frac{b}{a}\right)}^{3}}{\color{blue}{\frac{b}{a} \cdot \left(\frac{b}{a} + 0\right)}} \]

   20. +-commutative N/A

       \[\leadsto \frac{{\left(\frac{b}{a}\right)}^{3}}{\frac{b}{a} \cdot \color{blue}{\left(0 + \frac{b}{a}\right)}} \]

   21. +-lft-identity N/A

       \[\leadsto \frac{{\left(\frac{b}{a}\right)}^{3}}{\frac{b}{a} \cdot \color{blue}{\frac{b}{a}}} \]

   22. pow2 N/A

       \[\leadsto \frac{{\left(\frac{b}{a}\right)}^{3}}{\color{blue}{{\left(\frac{b}{a}\right)}^{2}}} \]

7. Applied egg-rr 2.5%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024197 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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 0) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))