Quadratic roots, full range

Percentage Accurate: 51.1% → 86.1%

Time: 11.8s

Alternatives: 8

Speedup: 19.1×

Specification

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.1% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+82)
   (- (/ c b) (/ b a))
   (if (<= b 4.2e-59)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+82) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.2e-59) {
		tmp = (sqrt(((b * b) - (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 <= (-4.5d+82)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4.2d-59) then
        tmp = (sqrt(((b * b) - (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 <= -4.5e+82) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.2e-59) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e+82:
		tmp = (c / b) - (b / a)
	elif b <= 4.2e-59:
		tmp = (math.sqrt(((b * b) - (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 <= -4.5e+82)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4.2e-59)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e+82)
		tmp = (c / b) - (b / a);
	elseif (b <= 4.2e-59)
		tmp = (sqrt(((b * b) - (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, -4.5e+82], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.2e-59], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $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 < -4.4999999999999997e82

   1. Initial program 44.8%

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

      1. *-commutative 44.8%

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

   3. Simplified 44.8%

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

   4. Taylor expanded in b around -inf 95.3%

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

      1. +-commutative 95.3%

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

      2. mul-1-neg 95.3%

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

      3. unsub-neg 95.3%

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

   6. Simplified 95.3%

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

   if -4.4999999999999997e82 < b < 4.19999999999999993e-59

   1. Initial program 78.0%

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

   if 4.19999999999999993e-59 < b

   1. Initial program 16.8%

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

      1. *-commutative 16.8%

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

   3. Simplified 16.8%

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

   4. Taylor expanded in b around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. distribute-neg-frac 89.9%

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

   6. Simplified 89.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e-63)
   (- (/ c b) (/ b a))
   (if (<= b 4.7e-58)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e-63) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.7e-58) {
		tmp = (sqrt((a * (c * -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 <= (-6.5d-63)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4.7d-58) then
        tmp = (sqrt((a * (c * (-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 <= -6.5e-63) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.7e-58) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e-63:
		tmp = (c / b) - (b / a)
	elif b <= 4.7e-58:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e-63)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4.7e-58)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.5e-63)
		tmp = (c / b) - (b / a);
	elseif (b <= 4.7e-58)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-63], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.7e-58], N[(N[(N[Sqrt[N[(a * N[(c * -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 < -6.4999999999999998e-63

   1. Initial program 63.0%

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

      1. *-commutative 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in b around -inf 88.6%

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

      1. +-commutative 88.6%

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

      2. mul-1-neg 88.6%

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

      3. unsub-neg 88.6%

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

   6. Simplified 88.6%

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

   if -6.4999999999999998e-63 < b < 4.69999999999999994e-58

   1. Initial program 71.2%

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

      1. *-commutative 71.2%

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

   3. Simplified 71.2%

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

      1. prod-diff 70.9%

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

      2. *-commutative 70.9%

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

      3. fma-def 70.9%

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

      4. associate-+l+ 70.9%

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

      5. pow2 70.9%

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

      6. distribute-lft-neg-in 70.9%

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

      7. *-commutative 70.9%

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

      8. distribute-rgt-neg-in 70.9%

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

      9. metadata-eval 70.9%

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

      10. associate-*r* 70.9%

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

      11. *-commutative 70.9%

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

      12. *-commutative 70.9%

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

      13. fma-udef 70.9%

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

   5. Applied egg-rr 70.9%

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

      1. fma-def 70.9%

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

      2. fma-def 70.7%

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

      3. associate-*l* 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in b around 0 66.3%

      \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. mul-1-neg 66.3%

         \[\leadsto \frac{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + \color{blue}{\left(-b\right)}}{a \cdot 2} \]

      2. unsub-neg 66.3%

         \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} - b}}{a \cdot 2} \]

      3. distribute-rgt-out 66.6%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 + 4\right)}} - b}{a \cdot 2} \]

      4. metadata-eval 66.6%

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

      5. associate-*r* 66.7%

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

      6. *-commutative 66.7%

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

   10. Simplified 66.7%

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

   if 4.69999999999999994e-58 < b

   1. Initial program 16.8%

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

      1. *-commutative 16.8%

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

   3. Simplified 16.8%

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

   4. Taylor expanded in b around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. distribute-neg-frac 89.9%

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

   6. Simplified 89.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-84)
   (- (/ c b) (/ b a))
   (if (<= b 9.2e-52) (* 0.5 (/ (sqrt (* a (* c -4.0))) a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-84) {
		tmp = (c / b) - (b / a);
	} else if (b <= 9.2e-52) {
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / 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 <= (-2.9d-84)) then
        tmp = (c / b) - (b / a)
    else if (b <= 9.2d-52) then
        tmp = 0.5d0 * (sqrt((a * (c * (-4.0d0)))) / 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 <= -2.9e-84) {
		tmp = (c / b) - (b / a);
	} else if (b <= 9.2e-52) {
		tmp = 0.5 * (Math.sqrt((a * (c * -4.0))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e-84:
		tmp = (c / b) - (b / a)
	elif b <= 9.2e-52:
		tmp = 0.5 * (math.sqrt((a * (c * -4.0))) / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-84)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 9.2e-52)
		tmp = Float64(0.5 * Float64(sqrt(Float64(a * Float64(c * -4.0))) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.9e-84)
		tmp = (c / b) - (b / a);
	elseif (b <= 9.2e-52)
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-84], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.2e-52], N[(0.5 * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.90000000000000019e-84

   1. Initial program 63.8%

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

      1. *-commutative 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in b around -inf 87.9%

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

      1. +-commutative 87.9%

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

      2. mul-1-neg 87.9%

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

      3. unsub-neg 87.9%

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

   6. Simplified 87.9%

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

   if -2.90000000000000019e-84 < b < 9.19999999999999977e-52

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

   3. Simplified 70.5%

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

      1. prod-diff 70.1%

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

      2. *-commutative 70.1%

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

      3. fma-def 70.1%

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

      4. associate-+l+ 70.1%

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

      5. pow2 70.1%

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

      6. distribute-lft-neg-in 70.1%

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

      7. *-commutative 70.1%

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

      8. distribute-rgt-neg-in 70.1%

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

      9. metadata-eval 70.1%

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

      10. associate-*r* 70.1%

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

      11. *-commutative 70.1%

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

      12. *-commutative 70.1%

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

      13. fma-udef 70.1%

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

   5. Applied egg-rr 70.1%

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

      1. fma-def 70.1%

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

      2. fma-def 69.9%

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

      3. associate-*l* 69.9%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in b around 0 64.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{a} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 65.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}}{a}} \]

      2. *-lft-identity 65.1%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}}}{a} \]

      3. distribute-rgt-out 65.4%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 + 4\right)}}}{a} \]

      4. metadata-eval 65.4%

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

      5. associate-*r* 65.5%

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

      6. *-commutative 65.5%

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

   10. Simplified 65.5%

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

   if 9.19999999999999977e-52 < b

   1. Initial program 16.8%

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

      1. *-commutative 16.8%

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

   3. Simplified 16.8%

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

   4. Taylor expanded in b around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. distribute-neg-frac 89.9%

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

   6. Simplified 89.9%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 68.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		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-310)) 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-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in b around -inf 68.9%

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

      1. +-commutative 68.9%

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

      2. mul-1-neg 68.9%

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

      3. unsub-neg 68.9%

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

   6. Simplified 68.9%

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

   if -1.999999999999994e-310 < b

   1. Initial program 32.8%

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

      1. *-commutative 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in b around inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. distribute-neg-frac 69.3%

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

   6. Simplified 69.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 42.9% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+47}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 3e+47) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e+47) {
		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 <= 3d+47) 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 <= 3e+47) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e+47:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e+47)
		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 <= 3e+47)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e+47], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.0000000000000001e47

   1. Initial program 64.0%

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

      1. *-commutative 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in b around -inf 49.6%

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

      1. associate-*r/ 49.6%

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

      2. mul-1-neg 49.6%

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

   6. Simplified 49.6%

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

   if 3.0000000000000001e47 < b

   1. Initial program 10.2%

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

      1. *-commutative 10.2%

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

   3. Simplified 10.2%

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

   5. Applied egg-rr 3.9%

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

   6. Taylor expanded in b around -inf 33.0%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+47}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6: 68.7% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		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-310)) 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-310) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in b around -inf 68.9%

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

      1. associate-*r/ 68.9%

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

      2. mul-1-neg 68.9%

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

   6. Simplified 68.9%

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

   if -1.999999999999994e-310 < b

   1. Initial program 32.8%

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

      1. *-commutative 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in b around inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. distribute-neg-frac 69.3%

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

   6. Simplified 69.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 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 50.5%

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

   1. *-commutative 50.5%

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

3. Simplified 50.5%

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

5. Applied egg-rr 31.6%

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

6. Taylor expanded in a around 0 2.4%

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

7. Final simplification 2.4%

   \[\leadsto \frac{b}{a} \]

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 50.5%

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

   1. *-commutative 50.5%

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

3. Simplified 50.5%

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

5. Applied egg-rr 31.6%

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

6. Taylor expanded in b around -inf 10.5%

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

7. Final simplification 10.5%

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

Reproduce

herbie shell --seed 2023319 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))