Quadratic roots, full range

Percentage Accurate: 51.9% → 85.5%

Time: 9.8s

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: 51.9% 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: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+154)
   (/ (- b) a)
   (if (<= b 4.8e-99)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+154) {
		tmp = -b / a;
	} else if (b <= 4.8e-99) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - 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 <= (-8.5d+154)) then
        tmp = -b / a
    else if (b <= 4.8d-99) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - 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 <= -8.5e+154) {
		tmp = -b / a;
	} else if (b <= 4.8e-99) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e+154:
		tmp = -b / a
	elif b <= 4.8e-99:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+154)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.8e-99)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - 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 <= -8.5e+154)
		tmp = -b / a;
	elseif (b <= 4.8e-99)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+154], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.8e-99], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $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 < -8.5000000000000002e154

   1. Initial program 31.1%

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

   2. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   if -8.5000000000000002e154 < b < 4.8000000000000001e-99

   1. Initial program 82.8%

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

   if 4.8000000000000001e-99 < b

   1. Initial program 15.2%

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

   2. Taylor expanded in b around inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. neg-mul-1 86.6%

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

   4. Simplified 86.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e-87)
   (- (/ c b) (/ b a))
   (if (<= b 3.1e-99)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.1e-99) {
		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 <= (-2.5d-87)) then
        tmp = (c / b) - (b / a)
    else if (b <= 3.1d-99) 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 <= -2.5e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.1e-99) {
		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 <= -2.5e-87:
		tmp = (c / b) - (b / a)
	elif b <= 3.1e-99:
		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 <= -2.5e-87)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 3.1e-99)
		tmp = Float64(Float64(sqrt(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 <= -2.5e-87)
		tmp = (c / b) - (b / a);
	elseif (b <= 3.1e-99)
		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, -2.5e-87], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.1e-99], 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 < -2.50000000000000021e-87

   1. Initial program 64.7%

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

   2. Taylor expanded in b around -inf 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

   4. Simplified 83.5%

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

   if -2.50000000000000021e-87 < b < 3.0999999999999999e-99

   1. Initial program 77.2%

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

      1. prod-diff 76.9%

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

      2. *-commutative 76.9%

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

      3. fma-def 76.9%

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

      4. associate-+l+ 76.9%

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

      5. *-commutative 76.9%

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

      6. distribute-rgt-neg-in 76.9%

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

      7. fma-def 76.9%

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

      8. *-commutative 76.9%

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

      9. distribute-rgt-neg-in 76.9%

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

      10. metadata-eval 76.9%

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

      11. *-commutative 76.9%

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

      12. fma-udef 76.9%

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

      13. distribute-lft-neg-in 76.9%

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

      14. distribute-rgt-neg-in 76.9%

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

      15. fma-def 76.9%

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

   3. Applied egg-rr 76.9%

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

   4. Taylor expanded in b around 0 72.2%

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

      1. distribute-rgt-out 72.6%

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

      2. metadata-eval 72.6%

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

      3. associate-*r* 72.6%

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

      4. neg-mul-1 72.6%

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

      5. unsub-neg 72.6%

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

      6. associate-*r* 72.6%

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

      7. metadata-eval 72.6%

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

      8. distribute-rgt-out 72.2%

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

      9. associate-*r* 72.2%

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

      10. associate-*r* 72.2%

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

      11. distribute-rgt-in 72.6%

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

      12. distribute-rgt-out 72.6%

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

      13. metadata-eval 72.6%

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

      14. *-commutative 72.6%

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

   6. Simplified 72.6%

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

   if 3.0999999999999999e-99 < b

   1. Initial program 15.2%

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

   2. Taylor expanded in b around inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. neg-mul-1 86.6%

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

   4. Simplified 86.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e-87)
   (- (/ c b) (/ b a))
   (if (<= b 7.8e-100) (* 0.5 (/ (sqrt (* c (* a -4.0))) a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.8e-100) {
		tmp = 0.5 * (sqrt((c * (a * -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 <= (-1.2d-87)) then
        tmp = (c / b) - (b / a)
    else if (b <= 7.8d-100) then
        tmp = 0.5d0 * (sqrt((c * (a * (-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 <= -1.2e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.8e-100) {
		tmp = 0.5 * (Math.sqrt((c * (a * -4.0))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.2e-87:
		tmp = (c / b) - (b / a)
	elif b <= 7.8e-100:
		tmp = 0.5 * (math.sqrt((c * (a * -4.0))) / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e-87)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7.8e-100)
		tmp = Float64(0.5 * Float64(sqrt(Float64(c * Float64(a * -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 <= -1.2e-87)
		tmp = (c / b) - (b / a);
	elseif (b <= 7.8e-100)
		tmp = 0.5 * (sqrt((c * (a * -4.0))) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e-87], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.8e-100], N[(0.5 * N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2e-87

   1. Initial program 64.7%

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

   2. Taylor expanded in b around -inf 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

   4. Simplified 83.5%

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

   if -1.2e-87 < b < 7.79999999999999955e-100

   1. Initial program 77.2%

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

      1. prod-diff 76.9%

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

      2. *-commutative 76.9%

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

      3. fma-def 76.9%

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

      4. associate-+l+ 76.9%

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

      5. *-commutative 76.9%

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

      6. distribute-rgt-neg-in 76.9%

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

      7. fma-def 76.9%

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

      8. *-commutative 76.9%

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

      9. distribute-rgt-neg-in 76.9%

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

      10. metadata-eval 76.9%

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

      11. *-commutative 76.9%

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

      12. fma-udef 76.9%

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

      13. distribute-lft-neg-in 76.9%

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

      14. distribute-rgt-neg-in 76.9%

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

      15. fma-def 76.9%

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

   3. Applied egg-rr 76.9%

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

   4. Taylor expanded in b around 0 71.5%

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

      1. associate-*l/ 71.5%

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

      2. distribute-rgt-out 71.8%

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

      3. metadata-eval 71.8%

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

      4. associate-*r* 71.8%

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

      5. *-lft-identity 71.8%

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

      6. associate-*r* 71.8%

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

      7. metadata-eval 71.8%

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

      8. distribute-rgt-out 71.5%

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

      9. associate-*r* 71.5%

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

      10. associate-*r* 71.5%

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

      11. distribute-rgt-in 71.8%

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

      12. distribute-rgt-out 71.8%

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

      13. metadata-eval 71.8%

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

      14. *-commutative 71.8%

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

   6. Simplified 71.8%

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

   if 7.79999999999999955e-100 < b

   1. Initial program 15.2%

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

   2. Taylor expanded in b around inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. neg-mul-1 86.6%

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

   4. Simplified 86.6%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 68.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-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 <= (-1d-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 <= -1e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-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 <= -1e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-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 < -9.999999999999969e-311

   1. Initial program 70.0%

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

   2. Taylor expanded in b around -inf 64.1%

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

      1. +-commutative 64.1%

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

      2. mul-1-neg 64.1%

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

      3. unsub-neg 64.1%

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

   4. Simplified 64.1%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.3%

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

   2. Taylor expanded in b around inf 66.4%

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

      1. associate-*r/ 66.4%

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

      2. neg-mul-1 66.4%

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

   4. Simplified 66.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 44.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-20}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.08e-20) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.08e-20) {
		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 <= 1.08d-20) 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 <= 1.08e-20) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.08e-20:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.08e-20)
		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 <= 1.08e-20)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.08e-20], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.08e-20

   1. Initial program 67.4%

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

   2. Taylor expanded in b around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. mul-1-neg 47.2%

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

   4. Simplified 47.2%

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

   if 1.08e-20 < b

   1. Initial program 11.8%

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

   2. Taylor expanded in b around inf 68.2%

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

      1. associate-/l* 70.2%

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

      2. associate-/r/ 64.7%

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

   4. Simplified 64.7%

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

      1. times-frac 64.7%

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

      2. metadata-eval 64.7%

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

      3. associate-/r/ 70.2%

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

      4. add-sqr-sqrt 46.9%

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

      5. sqrt-unprod 43.9%

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

      6. mul-1-neg 43.9%

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

      7. mul-1-neg 43.9%

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

      8. sqr-neg 43.9%

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

      9. sqrt-unprod 28.3%

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

      10. add-sqr-sqrt 29.0%

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

      11. div-inv 29.0%

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

      12. associate-/r/ 29.0%

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

      13. associate-*l/ 28.8%

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

      14. *-commutative 28.8%

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

   6. Applied egg-rr 28.8%

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

   7. Taylor expanded in c around 0 28.6%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-20}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6: 68.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-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 <= (-1d-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 <= -1e-310) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-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 <= -1e-310)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.0%

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

   2. Taylor expanded in b around -inf 63.6%

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

      1. associate-*r/ 63.6%

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

      2. mul-1-neg 63.6%

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

   4. Simplified 63.6%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.3%

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

   2. Taylor expanded in b around inf 66.4%

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

      1. associate-*r/ 66.4%

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

      2. neg-mul-1 66.4%

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

   4. Simplified 66.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 50.4%

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

   1. add-sqr-sqrt 50.2%

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

   2. pow2 50.2%

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

3. Applied egg-rr 28.3%

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

4. Taylor expanded in b around inf 2.7%

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

5. Final simplification 2.7%

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

Alternative 8: 11.2% 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 50.4%

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

2. Taylor expanded in b around inf 24.2%

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

   1. associate-/l* 25.5%

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

   2. associate-/r/ 23.7%

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

4. Simplified 23.7%

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

   1. times-frac 23.7%

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

   2. metadata-eval 23.7%

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

   3. associate-/r/ 25.5%

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

   4. add-sqr-sqrt 16.2%

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

   5. sqrt-unprod 16.2%

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

   6. mul-1-neg 16.2%

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

   7. mul-1-neg 16.2%

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

   8. sqr-neg 16.2%

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

   9. sqrt-unprod 10.0%

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

   10. add-sqr-sqrt 11.1%

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

   11. div-inv 11.1%

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

   12. associate-/r/ 11.0%

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

   13. associate-*l/ 11.0%

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

   14. *-commutative 11.0%

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

6. Applied egg-rr 11.0%

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

7. Taylor expanded in c around 0 11.0%

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

8. Final simplification 11.0%

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

Reproduce

herbie shell --seed 2023271 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))