Quadratic roots, full range

Percentage Accurate: 51.8% → 85.4%

Time: 13.2s

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: 51.8% 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: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+93)
   (- (/ c b) (/ b a))
   (if (<= b 1.95e-57)
     (/ (- (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 <= -5e+93) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-57) {
		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 <= (-5d+93)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.95d-57) 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 <= -5e+93) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-57) {
		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 <= -5e+93:
		tmp = (c / b) - (b / a)
	elif b <= 1.95e-57:
		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 <= -5e+93)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.95e-57)
		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 <= -5e+93)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.95e-57)
		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, -5e+93], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-57], 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 < -5.0000000000000001e93

   1. Initial program 52.9%

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

   2. Taylor expanded in b around -inf 93.5%

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

      1. +-commutative 93.5%

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

      2. mul-1-neg 93.5%

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

      3. unsub-neg 93.5%

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

   4. Simplified 93.5%

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

   if -5.0000000000000001e93 < b < 1.95000000000000003e-57

   1. Initial program 77.5%

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

   if 1.95000000000000003e-57 < b

   1. Initial program 14.4%

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

   2. Taylor expanded in b around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

   4. Simplified 86.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 2}{\frac{b}{c}} - b\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-57}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-69)
   (/ (- (- (/ (* a 2.0) (/ b c)) b) b) (* a 2.0))
   (if (<= b 4.9e-57)
     (/ (+ b (sqrt (* c (* a -4.0)))) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-69) {
		tmp = ((((a * 2.0) / (b / c)) - b) - b) / (a * 2.0);
	} else if (b <= 4.9e-57) {
		tmp = (b + sqrt((c * (a * -4.0)))) / (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-69)) then
        tmp = ((((a * 2.0d0) / (b / c)) - b) - b) / (a * 2.0d0)
    else if (b <= 4.9d-57) then
        tmp = (b + sqrt((c * (a * (-4.0d0))))) / (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-69) {
		tmp = ((((a * 2.0) / (b / c)) - b) - b) / (a * 2.0);
	} else if (b <= 4.9e-57) {
		tmp = (b + Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e-69:
		tmp = ((((a * 2.0) / (b / c)) - b) - b) / (a * 2.0)
	elif b <= 4.9e-57:
		tmp = (b + math.sqrt((c * (a * -4.0)))) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-69)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 2.0) / Float64(b / c)) - b) - b) / Float64(a * 2.0));
	elseif (b <= 4.9e-57)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / 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-69)
		tmp = ((((a * 2.0) / (b / c)) - b) - b) / (a * 2.0);
	elseif (b <= 4.9e-57)
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-69], N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.9e-57], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.50000000000000009e-69

   1. Initial program 67.4%

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

      1. add-cube-cbrt 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \]

      2. pow3 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}^{3}}}}{2 \cdot a} \]

      3. add-sqr-sqrt 67.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\right)}^{3}}}{2 \cdot a} \]

      4. cbrt-prod 66.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}}^{3}}}{2 \cdot a} \]

      5. cbrt-prod 67.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}}^{3}}}{2 \cdot a} \]

      6. add-sqr-sqrt 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{3}}}{2 \cdot a} \]

      7. fma-neg 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}}\right)}^{3}}}{2 \cdot a} \]

      8. *-commutative 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(4 \cdot a\right)}\right)}\right)}^{3}}}{2 \cdot a} \]

      9. distribute-rgt-neg-in 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)}\right)}^{3}}}{2 \cdot a} \]

      10. *-commutative 67.2%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)}\right)}^{3}}}{2 \cdot a} \]

      11. distribute-rgt-neg-in 67.2%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)}\right)}^{3}}}{2 \cdot a} \]

      12. metadata-eval 67.2%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)}\right)}^{3}}}{2 \cdot a} \]

   3. Applied egg-rr 67.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}^{3}}}}{2 \cdot a} \]

   4. Taylor expanded in b around -inf 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

      4. associate-/l* 89.0%

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

      5. associate-*r/ 89.0%

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

      6. *-commutative 89.0%

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

   6. Simplified 89.0%

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

   if -4.50000000000000009e-69 < b < 4.89999999999999988e-57

   1. Initial program 70.6%

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

   2. Taylor expanded in b around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*r* 65.5%

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

   4. Simplified 65.5%

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

      1. expm1-log1p-u 47.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\right)\right)} \]

      2. expm1-udef 19.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\right)} - 1} \]

      3. add-sqr-sqrt 10.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      4. sqrt-unprod 18.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      5. sqr-neg 18.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{b \cdot b}} + \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      6. sqrt-unprod 8.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      7. add-sqr-sqrt 18.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{b} + \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      8. *-commutative 18.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{\color{blue}{a \cdot 2}}\right)} - 1 \]

   6. Applied egg-rr 18.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 45.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\right)\right)} \]

      2. expm1-log1p 64.1%

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

   8. Simplified 64.1%

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

   if 4.89999999999999988e-57 < b

   1. Initial program 14.4%

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

   2. Taylor expanded in b around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

   4. Simplified 86.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 2}{\frac{b}{c}} - b\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-57}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 2}{\frac{b}{c}} - b\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.3e-69)
   (/ (- (- (/ (* a 2.0) (/ b c)) b) b) (* a 2.0))
   (if (<= b 9.2e-57)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-69) {
		tmp = ((((a * 2.0) / (b / c)) - b) - b) / (a * 2.0);
	} else if (b <= 9.2e-57) {
		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 <= (-4.3d-69)) then
        tmp = ((((a * 2.0d0) / (b / c)) - b) - b) / (a * 2.0d0)
    else if (b <= 9.2d-57) 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 <= -4.3e-69) {
		tmp = ((((a * 2.0) / (b / c)) - b) - b) / (a * 2.0);
	} else if (b <= 9.2e-57) {
		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 <= -4.3e-69:
		tmp = ((((a * 2.0) / (b / c)) - b) - b) / (a * 2.0)
	elif b <= 9.2e-57:
		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 <= -4.3e-69)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 2.0) / Float64(b / c)) - b) - b) / Float64(a * 2.0));
	elseif (b <= 9.2e-57)
		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 <= -4.3e-69)
		tmp = ((((a * 2.0) / (b / c)) - b) - b) / (a * 2.0);
	elseif (b <= 9.2e-57)
		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, -4.3e-69], N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.2e-57], 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 < -4.3e-69

   1. Initial program 67.4%

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

      1. add-cube-cbrt 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \]

      2. pow3 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}^{3}}}}{2 \cdot a} \]

      3. add-sqr-sqrt 67.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\right)}^{3}}}{2 \cdot a} \]

      4. cbrt-prod 66.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}}^{3}}}{2 \cdot a} \]

      5. cbrt-prod 67.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left(\sqrt[3]{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}}^{3}}}{2 \cdot a} \]

      6. add-sqr-sqrt 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{3}}}{2 \cdot a} \]

      7. fma-neg 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}}\right)}^{3}}}{2 \cdot a} \]

      8. *-commutative 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(4 \cdot a\right)}\right)}\right)}^{3}}}{2 \cdot a} \]

      9. distribute-rgt-neg-in 67.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)}\right)}^{3}}}{2 \cdot a} \]

      10. *-commutative 67.2%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 4}\right)\right)}\right)}^{3}}}{2 \cdot a} \]

      11. distribute-rgt-neg-in 67.2%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)}\right)}^{3}}}{2 \cdot a} \]

      12. metadata-eval 67.2%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)}\right)}^{3}}}{2 \cdot a} \]

   3. Applied egg-rr 67.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}^{3}}}}{2 \cdot a} \]

   4. Taylor expanded in b around -inf 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

      4. associate-/l* 89.0%

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

      5. associate-*r/ 89.0%

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

      6. *-commutative 89.0%

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

   6. Simplified 89.0%

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

   if -4.3e-69 < b < 9.2000000000000001e-57

   1. Initial program 70.6%

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

   2. Taylor expanded in b around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*r* 65.5%

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

   4. Simplified 65.5%

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

   if 9.2000000000000001e-57 < b

   1. Initial program 14.4%

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

   2. Taylor expanded in b around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

   4. Simplified 86.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 2}{\frac{b}{c}} - b\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 67.4% 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 68.1%

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

   2. Taylor expanded in b around -inf 64.8%

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

      1. +-commutative 64.8%

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

      2. mul-1-neg 64.8%

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

      3. unsub-neg 64.8%

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

   4. Simplified 64.8%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.0%

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

   2. Taylor expanded in b around inf 67.9%

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

      1. associate-*r/ 67.9%

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

      2. neg-mul-1 67.9%

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

   4. Simplified 67.9%

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

4. Final simplification 66.4%

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

Alternative 5: 42.9% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 9.5e+33) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.5e+33) {
		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 <= 9.5d+33) 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 <= 9.5e+33) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9.5e+33:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.5e+33)
		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 <= 9.5e+33)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.5e+33], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.5000000000000003e33

   1. Initial program 65.6%

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

   2. Taylor expanded in b around -inf 46.0%

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

      1. associate-*r/ 46.0%

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

      2. mul-1-neg 46.0%

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

   4. Simplified 46.0%

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

   if 9.5000000000000003e33 < b

   1. Initial program 9.3%

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

      1. clear-num 9.3%

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

      2. inv-pow 9.3%

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

   3. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{{\left(\frac{a}{\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot 0.5}\right)}^{-1}} \]

   4. Taylor expanded in b around -inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. rem-square-sqrt 0.0%

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

      3. unpow2 0.0%

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

      4. *-commutative 0.0%

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

      5. times-frac 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 0.0%

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

      8. unpow2 0.0%

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

      9. rem-square-sqrt 31.6%

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

      10. metadata-eval 31.6%

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

   6. Simplified 31.6%

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

   7. Taylor expanded in b around 0 31.6%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6: 67.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-261}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1e-261) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1e-261) {
		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-261) 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-261) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1e-261:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1e-261)
		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-261)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1e-261], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999984e-262

   1. Initial program 68.6%

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

   2. Taylor expanded in b around -inf 61.8%

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

      1. associate-*r/ 61.8%

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

      2. mul-1-neg 61.8%

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

   4. Simplified 61.8%

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

   if 9.99999999999999984e-262 < b

   1. Initial program 28.0%

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

   2. Taylor expanded in b around inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

   4. Simplified 70.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-261}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 2.6% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(b / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(b / a), $MachinePrecision]

Derivation

1. Initial program 48.9%

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

   1. clear-num 48.8%

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

   2. inv-pow 48.8%

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

3. Applied egg-rr 25.0%

   \[\leadsto \color{blue}{{\left(\frac{a}{\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot 0.5}\right)}^{-1}} \]

4. Taylor expanded in a around 0 2.5%

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

5. Final simplification 2.5%

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

Alternative 8: 10.9% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 48.9%

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

   1. clear-num 48.8%

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

   2. inv-pow 48.8%

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

3. Applied egg-rr 25.0%

   \[\leadsto \color{blue}{{\left(\frac{a}{\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot 0.5}\right)}^{-1}} \]

4. Taylor expanded in b around -inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. rem-square-sqrt 0.0%

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

   3. unpow2 0.0%

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

   4. *-commutative 0.0%

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

   5. times-frac 0.0%

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

   6. unpow2 0.0%

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

   7. rem-square-sqrt 0.0%

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

   8. unpow2 0.0%

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

   9. rem-square-sqrt 11.7%

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

   10. metadata-eval 11.7%

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

6. Simplified 11.7%

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

7. Taylor expanded in b around 0 11.7%

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

8. Final simplification 11.7%

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

Reproduce

herbie shell --seed 2023287 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))