The quadratic formula (r2)

Percentage Accurate: 51.7% → 85.7%

Time: 6.8s

Alternatives: 6

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.7% 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: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{-73}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+107}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.26e-73)
   (/ c (- b))
   (if (<= b 4.4e+107)
     (/ (- (- b) (sqrt (fma a (* c -4.0) (* b b)))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.26e-73) {
		tmp = c / -b;
	} else if (b <= 4.4e+107) {
		tmp = (-b - sqrt(fma(a, (c * -4.0), (b * b)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.26e-73)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 4.4e+107)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.26e-73], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 4.4e+107], N[(N[((-b) - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.26000000000000001e-73

   1. Initial program 21.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.8

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

   5. Simplified 84.8%

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

   if -1.26000000000000001e-73 < b < 4.4e107

   1. Initial program 82.5%

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

   3. Taylor expanded in b around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 82.5

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

   5. Simplified 82.5%

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

   if 4.4e107 < b

   1. Initial program 57.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 94.6

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

   5. Simplified 94.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{-73}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+107}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{-73}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.26e-73)
   (/ c (- b))
   (if (<= b 1.1e-37)
     (/ (- (- b) (sqrt (* a (* c -4.0)))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.26e-73) {
		tmp = c / -b;
	} else if (b <= 1.1e-37) {
		tmp = (-b - sqrt((a * (c * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.26d-73)) then
        tmp = c / -b
    else if (b <= 1.1d-37) then
        tmp = (-b - sqrt((a * (c * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.26e-73) {
		tmp = c / -b;
	} else if (b <= 1.1e-37) {
		tmp = (-b - Math.sqrt((a * (c * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.26e-73:
		tmp = c / -b
	elif b <= 1.1e-37:
		tmp = (-b - math.sqrt((a * (c * -4.0)))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.26e-73)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1.1e-37)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(a * Float64(c * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.26e-73)
		tmp = c / -b;
	elseif (b <= 1.1e-37)
		tmp = (-b - sqrt((a * (c * -4.0)))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.26e-73], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1.1e-37], N[(N[((-b) - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.26000000000000001e-73

   1. Initial program 21.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.8

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

   5. Simplified 84.8%

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

   if -1.26000000000000001e-73 < b < 1.10000000000000001e-37

   1. Initial program 79.7%

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

   3. Taylor expanded in b around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 79.7

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

   5. Simplified 79.7%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.7

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

   8. Simplified 74.7%

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

   if 1.10000000000000001e-37 < b

   1. Initial program 71.7%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 86.4

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

   5. Simplified 86.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{-73}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-311) (/ c (- b)) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-311) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-311)) then
        tmp = c / -b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-311) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-311:
		tmp = c / -b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-311)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-311)
		tmp = c / -b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-311], N[(c / (-b)), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9999999999999e-311

   1. Initial program 40.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 58.2

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

   5. Simplified 58.2%

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

   if -1.9999999999999e-311 < b

   1. Initial program 75.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 68.7

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

   5. Simplified 68.7%

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

Alternative 4: 67.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-311) (/ c (- b)) (/ b (- a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-311) {
		tmp = c / -b;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-311)) then
        tmp = c / -b
    else
        tmp = b / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-311) {
		tmp = c / -b;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-311:
		tmp = c / -b
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-311)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-311)
		tmp = c / -b;
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-311], N[(c / (-b)), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9999999999999e-311

   1. Initial program 40.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 58.2

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

   5. Simplified 58.2%

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

   if -1.9999999999999e-311 < b

   1. Initial program 75.4%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 68.3

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

   5. Simplified 68.3%

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

Alternative 5: 42.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-299}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 3.7e-299) 0.0 (/ b (- a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7e-299) {
		tmp = 0.0;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.7d-299) then
        tmp = 0.0d0
    else
        tmp = b / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7e-299) {
		tmp = 0.0;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.7e-299:
		tmp = 0.0
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.7e-299)
		tmp = 0.0;
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.7e-299)
		tmp = 0.0;
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.7e-299], 0.0, N[(b / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.70000000000000014e-299

   1. Initial program 40.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 13.2

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

   5. Simplified 13.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{0} \]
   7. Step-by-step derivation

   8. Simplified 13.2%

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

   if 3.70000000000000014e-299 < b

   1. Initial program 75.2%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 68.8

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

   5. Simplified 68.8%

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

Alternative 6: 11.5% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 7.8

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

5. Simplified 7.8%

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

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{0} \]
7. Step-by-step derivation

8. Simplified 7.8%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Developer Target 1: 70.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t\_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
     (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp = 0.0;
	if (b < 0.0)
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-b - t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024215 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (sqrt (- (* b b) (* 4 (* a c)))))) (let ((r1 (/ (+ (- b) d) (* 2 a)))) (let ((r2 (/ (- (- b) d) (* 2 a)))) (if (< b 0) (/ c (* a r1)) r2)))))

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