quadm (p42, negative)

Percentage Accurate: 62.9% → 83.7%

Time: 15.9s

Alternatives: 5

Speedup: 0.9×

• could not determine a ground truth

  1. a = -3.3887297905792814e-221
  2. b = -9.757583861191998e-223
  3. c = 1.6794026275760462e-183

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: 62.9% 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: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+47}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;b \cdot \frac{-0.5}{a} - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+47)
   0.0
   (if (<= b 8.6e+32)
     (- (* b (/ -0.5 a)) (* (sqrt (fma a (* c -4.0) (pow b 2.0))) (/ 0.5 a)))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+47) {
		tmp = 0.0;
	} else if (b <= 8.6e+32) {
		tmp = (b * (-0.5 / a)) - (sqrt(fma(a, (c * -4.0), pow(b, 2.0))) * (0.5 / a));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e+47)
		tmp = 0.0;
	elseif (b <= 8.6e+32)
		tmp = Float64(Float64(b * Float64(-0.5 / a)) - Float64(sqrt(fma(a, Float64(c * -4.0), (b ^ 2.0))) * Float64(0.5 / a)));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e+47], 0.0, If[LessEqual[b, 8.6e+32], N[(N[(b * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.6e47

   1. Initial program 29.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 24.9%

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

      2. sub-neg 24.9%

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

      3. neg-mul-1 24.9%

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

      4. *-commutative 24.9%

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

      5. associate-/l* 18.0%

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

      6. distribute-neg-frac 18.0%

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

      7. neg-mul-1 18.0%

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

      8. *-commutative 18.0%

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

      9. associate-/l* 24.9%

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

      10. distribute-rgt-out 29.4%

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

      11. associate-/r* 29.4%

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

      12. metadata-eval 29.4%

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

      13. sub-neg 29.4%

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

      14. +-commutative 29.4%

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

   3. Simplified 29.5%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 20.0%

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

      2. pow2 20.0%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left(\sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{2}}\right) \]

      3. pow1/2 20.0%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.5}}}\right)}^{2}\right) \]

      4. sqrt-pow1 20.0%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\color{blue}{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}\right) \]

      5. pow2 20.0%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}\right) \]

      6. metadata-eval 20.0%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \]

   6. Applied egg-rr 20.0%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right) \]

   7. Taylor expanded in b around -inf 47.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. distribute-rgt-out 47.1%

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

      2. metadata-eval 47.1%

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

      3. mul0-rgt 70.8%

         \[\leadsto \color{blue}{0} \]

   9. Simplified 70.8%

      \[\leadsto \color{blue}{0} \]

   if -1.6e47 < b < 8.5999999999999994e32

   1. Initial program 83.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 83.7%

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

      2. sub-neg 83.7%

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

      3. neg-mul-1 83.7%

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

      4. *-commutative 83.7%

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

      5. associate-/l* 81.3%

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

      6. distribute-neg-frac 81.3%

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

      7. neg-mul-1 81.3%

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

      8. *-commutative 81.3%

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

      9. associate-/l* 83.7%

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

      10. distribute-rgt-out 83.6%

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

      11. associate-/r* 83.6%

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

      12. metadata-eval 83.6%

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

      13. sub-neg 83.6%

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

      14. +-commutative 83.6%

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

   3. Simplified 83.6%

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

   6. Applied egg-rr 83.7%

      \[\leadsto \color{blue}{b \cdot \frac{-0.5}{a} - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a}} \]

   if 8.5999999999999994e32 < b

   1. Initial program 62.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 62.1%

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

      2. sub-neg 62.1%

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

      3. neg-mul-1 62.1%

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

      4. *-commutative 62.1%

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

      5. associate-/l* 62.1%

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

      6. distribute-neg-frac 62.1%

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

      7. neg-mul-1 62.1%

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

      8. *-commutative 62.1%

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

      9. associate-/l* 62.0%

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

      10. distribute-rgt-out 62.0%

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

      11. associate-/r* 62.0%

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

      12. metadata-eval 62.0%

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

      13. sub-neg 62.0%

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

      14. +-commutative 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in a around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. mul-1-neg 97.1%

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

      3. unsub-neg 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+47}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;b \cdot \frac{-0.5}{a} - \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+136}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+136)
   0.0
   (if (<= b 8.6e+32)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e+136) {
		tmp = 0.0;
	} else if (b <= 8.6e+32) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (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 <= (-3d+136)) then
        tmp = 0.0d0
    else if (b <= 8.6d+32) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (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 <= -3e+136) {
		tmp = 0.0;
	} else if (b <= 8.6e+32) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e+136:
		tmp = 0.0
	elif b <= 8.6e+32:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e+136)
		tmp = 0.0;
	elseif (b <= 8.6e+32)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / 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 <= -3e+136)
		tmp = 0.0;
	elseif (b <= 8.6e+32)
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e+136], 0.0, If[LessEqual[b, 8.6e+32], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $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 < -2.99999999999999979e136

   1. Initial program 6.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 5.6%

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

      2. sub-neg 5.6%

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

      3. neg-mul-1 5.6%

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

      4. *-commutative 5.6%

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

      5. associate-/l* 5.6%

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

      6. distribute-neg-frac 5.6%

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

      7. neg-mul-1 5.6%

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

      8. *-commutative 5.6%

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

      9. associate-/l* 5.6%

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

      10. distribute-rgt-out 6.2%

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

      11. associate-/r* 6.2%

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

      12. metadata-eval 6.2%

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

      13. sub-neg 6.2%

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

      14. +-commutative 6.2%

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

   3. Simplified 6.2%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 4.0%

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

      2. pow2 4.0%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left(\sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{2}}\right) \]

      3. pow1/2 4.0%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.5}}}\right)}^{2}\right) \]

      4. sqrt-pow1 4.0%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\color{blue}{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}\right) \]

      5. pow2 4.0%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}\right) \]

      6. metadata-eval 4.0%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \]

   6. Applied egg-rr 4.0%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right) \]

   7. Taylor expanded in b around -inf 42.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. distribute-rgt-out 42.8%

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

      2. metadata-eval 42.8%

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

      3. mul0-rgt 75.4%

         \[\leadsto \color{blue}{0} \]

   9. Simplified 75.4%

      \[\leadsto \color{blue}{0} \]

   if -2.99999999999999979e136 < b < 8.5999999999999994e32

   1. Initial program 79.9%

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

   if 8.5999999999999994e32 < b

   1. Initial program 62.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 62.1%

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

      2. sub-neg 62.1%

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

      3. neg-mul-1 62.1%

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

      4. *-commutative 62.1%

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

      5. associate-/l* 62.1%

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

      6. distribute-neg-frac 62.1%

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

      7. neg-mul-1 62.1%

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

      8. *-commutative 62.1%

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

      9. associate-/l* 62.0%

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

      10. distribute-rgt-out 62.0%

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

      11. associate-/r* 62.0%

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

      12. metadata-eval 62.0%

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

      13. sub-neg 62.0%

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

      14. +-commutative 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in a around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. mul-1-neg 97.1%

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

      3. unsub-neg 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+136}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) 0.0 (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = 0.0;
	} 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 <= (-5d-310)) then
        tmp = 0.0d0
    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 <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = 0.0
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], 0.0, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 54.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 52.6%

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

      2. sub-neg 52.6%

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

      3. neg-mul-1 52.6%

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

      4. *-commutative 52.6%

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

      5. associate-/l* 46.6%

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

      6. distribute-neg-frac 46.6%

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

      7. neg-mul-1 46.6%

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

      8. *-commutative 46.6%

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

      9. associate-/l* 52.5%

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

      10. distribute-rgt-out 54.9%

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

      11. associate-/r* 54.9%

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

      12. metadata-eval 54.9%

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

      13. sub-neg 54.9%

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

      14. +-commutative 54.9%

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

   3. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 43.2%

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

      2. pow2 43.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left(\sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{2}}\right) \]

      3. pow1/2 43.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.5}}}\right)}^{2}\right) \]

      4. sqrt-pow1 43.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\color{blue}{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}\right) \]

      5. pow2 43.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}\right) \]

      6. metadata-eval 43.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \]

   6. Applied egg-rr 43.2%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right) \]

   7. Taylor expanded in b around -inf 38.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. distribute-rgt-out 38.7%

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

      2. metadata-eval 38.7%

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

      3. mul0-rgt 51.5%

         \[\leadsto \color{blue}{0} \]

   9. Simplified 51.5%

      \[\leadsto \color{blue}{0} \]

   if -4.999999999999985e-310 < b

   1. Initial program 71.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 71.9%

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

      2. sub-neg 71.9%

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

      3. neg-mul-1 71.9%

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

      4. *-commutative 71.9%

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

      5. associate-/l* 71.9%

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

      6. distribute-neg-frac 71.9%

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

      7. neg-mul-1 71.9%

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

      8. *-commutative 71.9%

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

      9. associate-/l* 71.9%

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

      10. distribute-rgt-out 71.9%

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

      11. associate-/r* 71.9%

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

      12. metadata-eval 71.9%

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

      13. sub-neg 71.9%

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

      14. +-commutative 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in a around 0 66.8%

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

      1. +-commutative 66.8%

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

      2. mul-1-neg 66.8%

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

      3. unsub-neg 66.8%

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

   7. Simplified 66.8%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -5e-310) 0.0 (/ b (- a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = 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 <= (-5d-310)) then
        tmp = 0.0d0
    else
        tmp = b / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = 0.0
	else:
		tmp = b / -a
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], 0.0, N[(b / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 54.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 52.6%

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

      2. sub-neg 52.6%

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

      3. neg-mul-1 52.6%

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

      4. *-commutative 52.6%

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

      5. associate-/l* 46.6%

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

      6. distribute-neg-frac 46.6%

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

      7. neg-mul-1 46.6%

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

      8. *-commutative 46.6%

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

      9. associate-/l* 52.5%

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

      10. distribute-rgt-out 54.9%

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

      11. associate-/r* 54.9%

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

      12. metadata-eval 54.9%

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

      13. sub-neg 54.9%

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

      14. +-commutative 54.9%

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

   3. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 43.2%

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

      2. pow2 43.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left(\sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{2}}\right) \]

      3. pow1/2 43.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.5}}}\right)}^{2}\right) \]

      4. sqrt-pow1 43.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\color{blue}{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}\right) \]

      5. pow2 43.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}\right) \]

      6. metadata-eval 43.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \]

   6. Applied egg-rr 43.2%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right) \]

   7. Taylor expanded in b around -inf 38.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. distribute-rgt-out 38.7%

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

      2. metadata-eval 38.7%

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

      3. mul0-rgt 51.5%

         \[\leadsto \color{blue}{0} \]

   9. Simplified 51.5%

      \[\leadsto \color{blue}{0} \]

   if -4.999999999999985e-310 < b

   1. Initial program 71.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 71.9%

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

      2. sub-neg 71.9%

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

      3. neg-mul-1 71.9%

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

      4. *-commutative 71.9%

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

      5. associate-/l* 71.9%

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

      6. distribute-neg-frac 71.9%

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

      7. neg-mul-1 71.9%

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

      8. *-commutative 71.9%

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

      9. associate-/l* 71.9%

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

      10. distribute-rgt-out 71.9%

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

      11. associate-/r* 71.9%

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

      12. metadata-eval 71.9%

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

      13. sub-neg 71.9%

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

      14. +-commutative 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in a around 0 66.4%

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

      1. associate-*r/ 66.4%

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

      2. mul-1-neg 66.4%

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

   7. Simplified 66.4%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.6% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 63.1%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Step-by-step derivation

   1. div-sub 61.9%

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

   2. sub-neg 61.9%

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

   3. neg-mul-1 61.9%

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

   4. *-commutative 61.9%

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

   5. associate-/l* 58.8%

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

   6. distribute-neg-frac 58.8%

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

   7. neg-mul-1 58.8%

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

   8. *-commutative 58.8%

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

   9. associate-/l* 61.8%

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

   10. distribute-rgt-out 63.1%

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

   11. associate-/r* 63.1%

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

   12. metadata-eval 63.1%

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

   13. sub-neg 63.1%

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

   14. +-commutative 63.1%

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

3. Simplified 63.1%

   \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 56.9%

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

   2. pow2 56.9%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left(\sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{2}}\right) \]

   3. pow1/2 56.9%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.5}}}\right)}^{2}\right) \]

   4. sqrt-pow1 56.9%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\color{blue}{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}\right) \]

   5. pow2 56.9%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}\right) \]

   6. metadata-eval 56.9%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \]

6. Applied egg-rr 56.9%

   \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right) \]

7. Taylor expanded in b around -inf 21.3%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{b}{a}} \]
8. Step-by-step derivation

   1. distribute-rgt-out 21.3%

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

   2. metadata-eval 21.3%

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

   3. mul0-rgt 28.1%

      \[\leadsto \color{blue}{0} \]

9. Simplified 28.1%

   \[\leadsto \color{blue}{0} \]

10. Final simplification 28.1%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer target: 81.6% accurate, 0.1× 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 2024046 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b 0.0) (/ c (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0))) (/ (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (- a)))

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