quadp (p42, positive)

Percentage Accurate: 52.0% → 85.0%

Time: 11.9s

Alternatives: 9

Speedup: 12.9×

• Could not converge on a ground truth

  1. a = 7.094104730402104e-192
  2. b = 6.022756046961695e+278
  3. c = 1.9624601878036035e+162

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: 52.0% 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: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+85)
   (- (/ c b) (/ b a))
   (if (<= b 1.15e-87)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+85) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.15e-87) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	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 <= (-4.5d+85)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.15d-87) then
        tmp = (sqrt(((b * b) - (4.0d0 * (c * a)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+85) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.15e-87) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e+85:
		tmp = (c / b) - (b / a)
	elif b <= 1.15e-87:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+85)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.15e-87)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e+85)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.15e-87)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+85], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.15e-87], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.50000000000000007e85

   1. Initial program 50.3%

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

      1. *-commutative 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in b around -inf 91.7%

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

      1. mul-1-neg 91.7%

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

      2. *-commutative 91.7%

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

      3. distribute-rgt-neg-in 91.7%

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

      4. +-commutative 91.7%

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

      5. mul-1-neg 91.7%

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

      6. unsub-neg 91.7%

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

   7. Simplified 91.7%

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

      1. expm1-log1p-u 91.7%

         \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{{b}^{2}}\right)\right)}\right) \cdot \left(-b\right) \]

      2. expm1-undefine 91.7%

         \[\leadsto \left(\frac{1}{a} - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c}{{b}^{2}}\right)} - 1\right)}\right) \cdot \left(-b\right) \]

      3. div-inv 91.7%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(\color{blue}{c \cdot \frac{1}{{b}^{2}}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

      4. pow-flip 91.7%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{{b}^{\left(-2\right)}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

      5. metadata-eval 91.7%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(c \cdot {b}^{\color{blue}{-2}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

   9. Applied egg-rr 91.7%

      \[\leadsto \left(\frac{1}{a} - \color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot {b}^{-2}\right)} - 1\right)}\right) \cdot \left(-b\right) \]
   10. Step-by-step derivation

       1. expm1-define 91.7%

          \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)}\right) \cdot \left(-b\right) \]

   11. Simplified 91.7%

       \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)}\right) \cdot \left(-b\right) \]

   12. Taylor expanded in a around inf 91.8%

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

       1. +-commutative 91.8%

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

       2. mul-1-neg 91.8%

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

       3. unsub-neg 91.8%

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

   14. Simplified 91.8%

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

   if -4.50000000000000007e85 < b < 1.1500000000000001e-87

   1. Initial program 81.5%

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

   if 1.1500000000000001e-87 < b

   1. Initial program 15.3%

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

      1. *-commutative 15.3%

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

   3. Simplified 15.3%

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

   5. Taylor expanded in b around inf 83.9%

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

      1. associate-*r/ 83.9%

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

      2. neg-mul-1 83.9%

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

   7. Simplified 83.9%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+85}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e+85)
   (- (/ c b) (/ b a))
   (if (<= b 3.8e-86)
     (/ -0.5 (/ a (- b (sqrt (+ (* b b) (* (* c a) -4.0))))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e+85) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.8e-86) {
		tmp = -0.5 / (a / (b - sqrt(((b * b) + ((c * a) * -4.0)))));
	} else {
		tmp = c / -b;
	}
	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 <= (-5.2d+85)) then
        tmp = (c / b) - (b / a)
    else if (b <= 3.8d-86) then
        tmp = (-0.5d0) / (a / (b - sqrt(((b * b) + ((c * a) * (-4.0d0))))))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e+85) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.8e-86) {
		tmp = -0.5 / (a / (b - Math.sqrt(((b * b) + ((c * a) * -4.0)))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.2e+85:
		tmp = (c / b) - (b / a)
	elif b <= 3.8e-86:
		tmp = -0.5 / (a / (b - math.sqrt(((b * b) + ((c * a) * -4.0)))))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e+85)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 3.8e-86)
		tmp = Float64(-0.5 / Float64(a / Float64(b - sqrt(Float64(Float64(b * b) + Float64(Float64(c * a) * -4.0))))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.2e+85)
		tmp = (c / b) - (b / a);
	elseif (b <= 3.8e-86)
		tmp = -0.5 / (a / (b - sqrt(((b * b) + ((c * a) * -4.0)))));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e+85], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e-86], N[(-0.5 / N[(a / N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.20000000000000021e85

   1. Initial program 50.3%

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

      1. *-commutative 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in b around -inf 91.7%

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

      1. mul-1-neg 91.7%

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

      2. *-commutative 91.7%

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

      3. distribute-rgt-neg-in 91.7%

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

      4. +-commutative 91.7%

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

      5. mul-1-neg 91.7%

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

      6. unsub-neg 91.7%

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

   7. Simplified 91.7%

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

      1. expm1-log1p-u 91.7%

         \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{{b}^{2}}\right)\right)}\right) \cdot \left(-b\right) \]

      2. expm1-undefine 91.7%

         \[\leadsto \left(\frac{1}{a} - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c}{{b}^{2}}\right)} - 1\right)}\right) \cdot \left(-b\right) \]

      3. div-inv 91.7%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(\color{blue}{c \cdot \frac{1}{{b}^{2}}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

      4. pow-flip 91.7%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{{b}^{\left(-2\right)}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

      5. metadata-eval 91.7%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(c \cdot {b}^{\color{blue}{-2}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

   9. Applied egg-rr 91.7%

      \[\leadsto \left(\frac{1}{a} - \color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot {b}^{-2}\right)} - 1\right)}\right) \cdot \left(-b\right) \]
   10. Step-by-step derivation

       1. expm1-define 91.7%

          \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)}\right) \cdot \left(-b\right) \]

   11. Simplified 91.7%

       \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)}\right) \cdot \left(-b\right) \]

   12. Taylor expanded in a around inf 91.8%

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

       1. +-commutative 91.8%

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

       2. mul-1-neg 91.8%

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

       3. unsub-neg 91.8%

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

   14. Simplified 91.8%

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

   if -5.20000000000000021e85 < b < 3.8e-86

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

   3. Simplified 81.5%

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

      1. add-sqr-sqrt 44.8%

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

      2. pow2 44.8%

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

   6. Applied egg-rr 44.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 44.8%

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

      2. add-sqr-sqrt 81.5%

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

      3. clear-num 81.3%

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

      4. *-commutative 81.3%

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

      5. *-un-lft-identity 81.3%

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

      6. times-frac 81.3%

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

      7. metadata-eval 81.3%

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

   8. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\frac{1}{-2 \cdot \frac{a}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]
   9. Step-by-step derivation

      1. associate-/r* 81.3%

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

      2. metadata-eval 81.3%

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

   10. Simplified 81.3%

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

       1. fma-undefine 81.3%

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

   12. Applied egg-rr 81.3%

       \[\leadsto \frac{-0.5}{\frac{a}{b - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + {b}^{2}}}}} \]
   13. Step-by-step derivation

       1. unpow2 81.3%

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

   14. Applied egg-rr 81.3%

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

   if 3.8e-86 < b

   1. Initial program 15.3%

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

      1. *-commutative 15.3%

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

   3. Simplified 15.3%

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

   5. Taylor expanded in b around inf 83.9%

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

      1. associate-*r/ 83.9%

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

      2. neg-mul-1 83.9%

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

   7. Simplified 83.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+85}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e-91)
   (- (/ c b) (/ b a))
   (if (<= b 1.05e-85)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-91) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.05e-85) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	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 <= (-5.2d-91)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.05d-85) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-91) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.05e-85) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.2e-91:
		tmp = (c / b) - (b / a)
	elif b <= 1.05e-85:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e-91)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.05e-85)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.2e-91)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.05e-85)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e-91], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-85], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.20000000000000028e-91

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 84.9%

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

      1. mul-1-neg 84.9%

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

      2. *-commutative 84.9%

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

      3. distribute-rgt-neg-in 84.9%

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

      4. +-commutative 84.9%

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

      5. mul-1-neg 84.9%

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

      6. unsub-neg 84.9%

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

   7. Simplified 84.9%

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

      1. expm1-log1p-u 82.1%

         \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{{b}^{2}}\right)\right)}\right) \cdot \left(-b\right) \]

      2. expm1-undefine 81.9%

         \[\leadsto \left(\frac{1}{a} - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c}{{b}^{2}}\right)} - 1\right)}\right) \cdot \left(-b\right) \]

      3. div-inv 81.9%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(\color{blue}{c \cdot \frac{1}{{b}^{2}}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

      4. pow-flip 81.9%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{{b}^{\left(-2\right)}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

      5. metadata-eval 81.9%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(c \cdot {b}^{\color{blue}{-2}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

   9. Applied egg-rr 81.9%

      \[\leadsto \left(\frac{1}{a} - \color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot {b}^{-2}\right)} - 1\right)}\right) \cdot \left(-b\right) \]
   10. Step-by-step derivation

       1. expm1-define 82.1%

          \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)}\right) \cdot \left(-b\right) \]

   11. Simplified 82.1%

       \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)}\right) \cdot \left(-b\right) \]

   12. Taylor expanded in a around inf 85.1%

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

       1. +-commutative 85.1%

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

       2. mul-1-neg 85.1%

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

       3. unsub-neg 85.1%

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

   14. Simplified 85.1%

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

   if -5.20000000000000028e-91 < b < 1.05e-85

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in b around 0 71.2%

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

      1. associate-*r* 71.2%

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

      2. *-commutative 71.2%

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

      3. *-commutative 71.2%

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

   7. Simplified 71.2%

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

   if 1.05e-85 < b

   1. Initial program 15.3%

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

      1. *-commutative 15.3%

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

   3. Simplified 15.3%

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

   5. Taylor expanded in b around inf 83.9%

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

      1. associate-*r/ 83.9%

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

      2. neg-mul-1 83.9%

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

   7. Simplified 83.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-86}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-86}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b - \sqrt{\left(c \cdot a\right) \cdot -4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-86)
   (- (/ c b) (/ b a))
   (if (<= b 1.85e-86)
     (/ -0.5 (/ a (- b (sqrt (* (* c a) -4.0)))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-86) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.85e-86) {
		tmp = -0.5 / (a / (b - sqrt(((c * a) * -4.0))));
	} else {
		tmp = c / -b;
	}
	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-86)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.85d-86) then
        tmp = (-0.5d0) / (a / (b - sqrt(((c * a) * (-4.0d0)))))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-86) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.85e-86) {
		tmp = -0.5 / (a / (b - Math.sqrt(((c * a) * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-86:
		tmp = (c / b) - (b / a)
	elif b <= 1.85e-86:
		tmp = -0.5 / (a / (b - math.sqrt(((c * a) * -4.0))))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-86)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.85e-86)
		tmp = Float64(-0.5 / Float64(a / Float64(b - sqrt(Float64(Float64(c * a) * -4.0)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-86)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.85e-86)
		tmp = -0.5 / (a / (b - sqrt(((c * a) * -4.0))));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-86], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.85e-86], N[(-0.5 / N[(a / N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999999e-86

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 84.9%

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

      1. mul-1-neg 84.9%

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

      2. *-commutative 84.9%

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

      3. distribute-rgt-neg-in 84.9%

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

      4. +-commutative 84.9%

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

      5. mul-1-neg 84.9%

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

      6. unsub-neg 84.9%

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

   7. Simplified 84.9%

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

      1. expm1-log1p-u 82.1%

         \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{{b}^{2}}\right)\right)}\right) \cdot \left(-b\right) \]

      2. expm1-undefine 81.9%

         \[\leadsto \left(\frac{1}{a} - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c}{{b}^{2}}\right)} - 1\right)}\right) \cdot \left(-b\right) \]

      3. div-inv 81.9%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(\color{blue}{c \cdot \frac{1}{{b}^{2}}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

      4. pow-flip 81.9%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{{b}^{\left(-2\right)}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

      5. metadata-eval 81.9%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(c \cdot {b}^{\color{blue}{-2}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

   9. Applied egg-rr 81.9%

      \[\leadsto \left(\frac{1}{a} - \color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot {b}^{-2}\right)} - 1\right)}\right) \cdot \left(-b\right) \]
   10. Step-by-step derivation

       1. expm1-define 82.1%

          \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)}\right) \cdot \left(-b\right) \]

   11. Simplified 82.1%

       \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)}\right) \cdot \left(-b\right) \]

   12. Taylor expanded in a around inf 85.1%

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

       1. +-commutative 85.1%

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

       2. mul-1-neg 85.1%

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

       3. unsub-neg 85.1%

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

   14. Simplified 85.1%

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

   if -4.9999999999999999e-86 < b < 1.8499999999999999e-86

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

   3. Simplified 75.0%

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

      1. add-sqr-sqrt 37.3%

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

      2. pow2 37.3%

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

   6. Applied egg-rr 37.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 37.3%

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

      2. add-sqr-sqrt 75.0%

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

      3. clear-num 74.9%

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

      4. *-commutative 74.9%

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

      5. *-un-lft-identity 74.9%

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

      6. times-frac 74.9%

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

      7. metadata-eval 74.9%

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

   8. Applied egg-rr 74.9%

      \[\leadsto \color{blue}{\frac{1}{-2 \cdot \frac{a}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]
   9. Step-by-step derivation

      1. associate-/r* 74.9%

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

      2. metadata-eval 74.9%

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

   10. Simplified 74.9%

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

   11. Taylor expanded in a around inf 71.1%

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

   if 1.8499999999999999e-86 < b

   1. Initial program 15.3%

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

      1. *-commutative 15.3%

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

   3. Simplified 15.3%

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

   5. Taylor expanded in b around inf 83.9%

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

      1. associate-*r/ 83.9%

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

      2. neg-mul-1 83.9%

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

   7. Simplified 83.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-86}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-86}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b - \sqrt{\left(c \cdot a\right) \cdot -4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-312) (- (/ c b) (/ b a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-312) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = c / -b;
	}
	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 <= (-2d-312)) then
        tmp = (c / b) - (b / a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-312) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-312:
		tmp = (c / b) - (b / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-312)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-312)
		tmp = (c / b) - (b / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-312], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.0000000000019e-312

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in b around -inf 62.3%

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

      1. mul-1-neg 62.3%

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

      2. *-commutative 62.3%

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

      3. distribute-rgt-neg-in 62.3%

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

      4. +-commutative 62.3%

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

      5. mul-1-neg 62.3%

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

      6. unsub-neg 62.3%

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

   7. Simplified 62.3%

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

      1. expm1-log1p-u 59.2%

         \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{{b}^{2}}\right)\right)}\right) \cdot \left(-b\right) \]

      2. expm1-undefine 59.1%

         \[\leadsto \left(\frac{1}{a} - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{c}{{b}^{2}}\right)} - 1\right)}\right) \cdot \left(-b\right) \]

      3. div-inv 59.1%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(\color{blue}{c \cdot \frac{1}{{b}^{2}}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

      4. pow-flip 59.1%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{{b}^{\left(-2\right)}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

      5. metadata-eval 59.1%

         \[\leadsto \left(\frac{1}{a} - \left(e^{\mathsf{log1p}\left(c \cdot {b}^{\color{blue}{-2}}\right)} - 1\right)\right) \cdot \left(-b\right) \]

   9. Applied egg-rr 59.1%

      \[\leadsto \left(\frac{1}{a} - \color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot {b}^{-2}\right)} - 1\right)}\right) \cdot \left(-b\right) \]
   10. Step-by-step derivation

       1. expm1-define 59.2%

          \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)}\right) \cdot \left(-b\right) \]

   11. Simplified 59.2%

       \[\leadsto \left(\frac{1}{a} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)}\right) \cdot \left(-b\right) \]

   12. Taylor expanded in a around inf 63.4%

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

       1. +-commutative 63.4%

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

       2. mul-1-neg 63.4%

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

       3. unsub-neg 63.4%

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

   14. Simplified 63.4%

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

   if -2.0000000000019e-312 < b

   1. Initial program 28.8%

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

      1. *-commutative 28.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in b around inf 68.2%

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

   7. Simplified 68.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-312) (/ (- b) a) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-312) {
		tmp = -b / a;
	} else {
		tmp = c / -b;
	}
	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 <= (-2d-312)) then
        tmp = -b / a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-312) {
		tmp = -b / a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-312:
		tmp = -b / a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-312)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-312)
		tmp = -b / a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-312], N[((-b) / a), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.0000000000019e-312

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in b around -inf 62.9%

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

      1. associate-*r/ 62.9%

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

      2. mul-1-neg 62.9%

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

   7. Simplified 62.9%

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

   if -2.0000000000019e-312 < b

   1. Initial program 28.8%

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

      1. *-commutative 28.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in b around inf 68.2%

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

   7. Simplified 68.2%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-312}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 5.1e+14) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e+14) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	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 <= 5.1d+14) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e+14) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e+14:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e+14)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e+14)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e+14], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.1e14

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in b around -inf 43.3%

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

      1. associate-*r/ 43.3%

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

      2. mul-1-neg 43.3%

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

   7. Simplified 43.3%

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

   if 5.1e14 < b

   1. Initial program 11.3%

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

      1. *-commutative 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in b around -inf 2.5%

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

      1. mul-1-neg 2.5%

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

      2. *-commutative 2.5%

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

      3. distribute-rgt-neg-in 2.5%

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

      4. +-commutative 2.5%

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

      5. mul-1-neg 2.5%

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

      6. unsub-neg 2.5%

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

   7. Simplified 2.5%

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

   8. Taylor expanded in a around inf 18.7%

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

Alternative 8: 11.1% accurate, 38.7× 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 48.7%

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

   1. *-commutative 48.7%

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

3. Simplified 48.7%

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

5. Taylor expanded in b around -inf 29.5%

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

   1. mul-1-neg 29.5%

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

   2. *-commutative 29.5%

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

   3. distribute-rgt-neg-in 29.5%

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

   4. +-commutative 29.5%

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

   5. mul-1-neg 29.5%

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

   6. unsub-neg 29.5%

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

7. Simplified 29.5%

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

8. Taylor expanded in a around inf 8.2%

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

Alternative 9: 2.5% accurate, 38.7× 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 48.7%

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

   1. *-commutative 48.7%

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

3. Simplified 48.7%

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

   1. clear-num 48.6%

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

   2. inv-pow 48.6%

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

   3. add-sqr-sqrt 32.8%

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

   4. sqrt-unprod 46.8%

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

   5. sqr-neg 46.8%

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

   6. sqrt-prod 14.0%

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

   7. add-sqr-sqrt 31.1%

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

   8. sub-neg 31.1%

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

   9. +-commutative 31.1%

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

   10. *-commutative 31.1%

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

   11. distribute-rgt-neg-in 31.1%

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

   12. fma-define 31.1%

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

   13. metadata-eval 31.1%

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

   14. pow2 31.1%

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

6. Applied egg-rr 31.1%

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

   1. unpow-1 31.1%

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

   2. *-commutative 31.1%

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

8. Simplified 31.1%

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

9. Taylor expanded in a around 0 2.8%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

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) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

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 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	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 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	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 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	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(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	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 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	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[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024184 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

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