Quadratic roots, full range

Percentage Accurate: 51.7% → 84.9%

Time: 13.9s

Alternatives: 9

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: 51.7% 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: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.2e+108)
   (- (/ c b) (/ b a))
   (if (<= b 1.4e-48)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ 1.0 (- (/ a b) (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e+108) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.4e-48) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.2d+108)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.4d-48) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e+108) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.4e-48) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.2e+108:
		tmp = (c / b) - (b / a)
	elif b <= 1.4e-48:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.2e+108)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.4e-48)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.2e+108)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.4e-48)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.2e+108], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4e-48], 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[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.2000000000000003e108

   1. Initial program 37.8%

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

      1. *-commutative 37.8%

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

   3. Simplified 37.8%

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

   4. Taylor expanded in b around -inf 93.4%

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

      1. +-commutative 93.4%

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

      2. mul-1-neg 93.4%

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

      3. unsub-neg 93.4%

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

   6. Simplified 93.4%

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

   if -6.2000000000000003e108 < b < 1.40000000000000002e-48

   1. Initial program 81.2%

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

   if 1.40000000000000002e-48 < b

   1. Initial program 14.9%

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

      1. *-commutative 14.9%

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

   3. Simplified 14.9%

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

      1. frac-2neg 14.9%

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

      2. div-inv 14.9%

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

   5. Applied egg-rr 14.9%

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

      1. un-div-inv 14.9%

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

      2. clear-num 14.9%

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

      3. fma-udef 14.9%

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

      4. add-sqr-sqrt 12.1%

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

      5. hypot-def 23.1%

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

   7. Applied egg-rr 23.1%

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

      1. *-un-lft-identity 23.1%

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

      2. times-frac 23.1%

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

   9. Applied egg-rr 23.1%

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

   10. Taylor expanded in b around inf 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. times-frac 0.0%

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

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\frac{4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} + \frac{a}{b}} \]

       5. rem-square-sqrt 87.9%

          \[\leadsto \frac{1}{\frac{4}{\color{blue}{-4}} \cdot \frac{b}{c} + \frac{a}{b}} \]

       6. metadata-eval 87.9%

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

   12. Simplified 87.9%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 2: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-80)
   (- (/ c b) (/ b a))
   (if (<= b 9e-47)
     (* (/ 0.5 a) (+ b (sqrt (* c (* a -4.0)))))
     (/ 1.0 (- (/ a b) (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-80) {
		tmp = (c / b) - (b / a);
	} else if (b <= 9e-47) {
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.5d-80)) then
        tmp = (c / b) - (b / a)
    else if (b <= 9d-47) then
        tmp = (0.5d0 / a) * (b + sqrt((c * (a * (-4.0d0)))))
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-80) {
		tmp = (c / b) - (b / a);
	} else if (b <= 9e-47) {
		tmp = (0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.5e-80:
		tmp = (c / b) - (b / a)
	elif b <= 9e-47:
		tmp = (0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e-80)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 9e-47)
		tmp = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.5e-80)
		tmp = (c / b) - (b / a);
	elseif (b <= 9e-47)
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e-80], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9e-47], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.4999999999999997e-80

   1. Initial program 59.7%

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

      1. *-commutative 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in b around -inf 84.5%

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

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

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

   6. Simplified 84.5%

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

   if -5.4999999999999997e-80 < b < 9e-47

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

   3. Simplified 77.3%

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

      1. clear-num 77.2%

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

      2. associate-/r/ 77.2%

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

      3. *-commutative 77.2%

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

      4. associate-/r* 77.2%

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

      5. metadata-eval 77.2%

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

      6. add-sqr-sqrt 44.7%

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

      7. sqrt-unprod 76.2%

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

      8. sqr-neg 76.2%

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

      9. sqrt-prod 31.6%

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

      10. add-sqr-sqrt 67.9%

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

      11. fma-neg 67.9%

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

      12. *-commutative 67.9%

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

      13. distribute-rgt-neg-in 67.9%

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

      14. *-commutative 67.9%

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

      15. distribute-rgt-neg-in 67.9%

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

      16. metadata-eval 67.9%

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

   5. Applied egg-rr 67.9%

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

   6. Taylor expanded in b around 0 67.6%

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

      1. associate-*r* 67.6%

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

      2. *-commutative 67.6%

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

      3. *-commutative 67.6%

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

   8. Simplified 67.6%

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

   if 9e-47 < b

   1. Initial program 14.9%

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

      1. *-commutative 14.9%

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

   3. Simplified 14.9%

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

      1. frac-2neg 14.9%

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

      2. div-inv 14.9%

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

   5. Applied egg-rr 14.9%

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

      1. un-div-inv 14.9%

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

      2. clear-num 14.9%

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

      3. fma-udef 14.9%

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

      4. add-sqr-sqrt 12.1%

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

      5. hypot-def 23.1%

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

   7. Applied egg-rr 23.1%

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

      1. *-un-lft-identity 23.1%

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

      2. times-frac 23.1%

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

   9. Applied egg-rr 23.1%

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

   10. Taylor expanded in b around inf 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. times-frac 0.0%

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

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\frac{4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} + \frac{a}{b}} \]

       5. rem-square-sqrt 87.9%

          \[\leadsto \frac{1}{\frac{4}{\color{blue}{-4}} \cdot \frac{b}{c} + \frac{a}{b}} \]

       6. metadata-eval 87.9%

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

   12. Simplified 87.9%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 3: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.4e-29)
   (- (/ c b) (/ b a))
   (if (<= b 7.6e-48)
     (/ (- (sqrt (* -4.0 (* c a))) b) (* a 2.0))
     (/ 1.0 (- (/ a b) (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e-29) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.6e-48) {
		tmp = (sqrt((-4.0 * (c * a))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	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 <= (-1.4d-29)) then
        tmp = (c / b) - (b / a)
    else if (b <= 7.6d-48) then
        tmp = (sqrt(((-4.0d0) * (c * a))) - b) / (a * 2.0d0)
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e-29) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.6e-48) {
		tmp = (Math.sqrt((-4.0 * (c * a))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.4e-29:
		tmp = (c / b) - (b / a)
	elif b <= 7.6e-48:
		tmp = (math.sqrt((-4.0 * (c * a))) - b) / (a * 2.0)
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.4e-29)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7.6e-48)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.4e-29)
		tmp = (c / b) - (b / a);
	elseif (b <= 7.6e-48)
		tmp = (sqrt((-4.0 * (c * a))) - b) / (a * 2.0);
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.4e-29], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.6e-48], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.4000000000000001e-29

   1. Initial program 57.3%

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

      1. *-commutative 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in b around -inf 89.3%

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

      1. +-commutative 89.3%

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

      2. mul-1-neg 89.3%

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

      3. unsub-neg 89.3%

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

   6. Simplified 89.3%

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

   if -1.4000000000000001e-29 < b < 7.60000000000000005e-48

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

   3. Simplified 77.7%

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

      1. add-sqr-sqrt 77.4%

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

      2. pow2 77.4%

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

      3. pow1/2 77.4%

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

      4. sqrt-pow1 77.5%

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

      5. fma-neg 77.5%

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

      6. *-commutative 77.5%

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

      7. distribute-rgt-neg-in 77.5%

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

      8. *-commutative 77.5%

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

      9. distribute-rgt-neg-in 77.5%

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

      10. metadata-eval 77.5%

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

      11. metadata-eval 77.5%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a \cdot 2} \]

   5. Applied egg-rr 77.5%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}\right)}^{2}}}{a \cdot 2} \]

   6. Taylor expanded in c around inf 37.3%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25 \cdot \left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)}\right)}^{2} - b}}{a \cdot 2} \]
   7. Step-by-step derivation

   8. Simplified 66.9%

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

   if 7.60000000000000005e-48 < b

   1. Initial program 14.9%

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

      1. *-commutative 14.9%

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

   3. Simplified 14.9%

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

      1. frac-2neg 14.9%

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

      2. div-inv 14.9%

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

   5. Applied egg-rr 14.9%

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

      1. un-div-inv 14.9%

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

      2. clear-num 14.9%

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

      3. fma-udef 14.9%

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

      4. add-sqr-sqrt 12.1%

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

      5. hypot-def 23.1%

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

   7. Applied egg-rr 23.1%

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

      1. *-un-lft-identity 23.1%

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

      2. times-frac 23.1%

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

   9. Applied egg-rr 23.1%

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

   10. Taylor expanded in b around inf 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. times-frac 0.0%

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

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\frac{4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} + \frac{a}{b}} \]

       5. rem-square-sqrt 87.9%

          \[\leadsto \frac{1}{\frac{4}{\color{blue}{-4}} \cdot \frac{b}{c} + \frac{a}{b}} \]

       6. metadata-eval 87.9%

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

   12. Simplified 87.9%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 4: 67.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e-305) (- (/ c b) (/ b a)) (/ 1.0 (- (/ a b) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-305) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	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 <= (-7.5d-305)) then
        tmp = (c / b) - (b / a)
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-305) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e-305:
		tmp = (c / b) - (b / a)
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e-305)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e-305)
		tmp = (c / b) - (b / a);
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e-305], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.5000000000000003e-305

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in b around -inf 66.6%

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

      1. +-commutative 66.6%

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

      2. mul-1-neg 66.6%

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

      3. unsub-neg 66.6%

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

   6. Simplified 66.6%

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

   if -7.5000000000000003e-305 < b

   1. Initial program 32.1%

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

      1. *-commutative 32.1%

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

   3. Simplified 32.1%

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

      1. frac-2neg 32.1%

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

      2. div-inv 32.1%

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

   5. Applied egg-rr 32.1%

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

      1. un-div-inv 32.1%

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

      2. clear-num 32.0%

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

      3. fma-udef 32.0%

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

      4. add-sqr-sqrt 30.1%

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

      5. hypot-def 37.5%

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

   7. Applied egg-rr 37.5%

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

      1. *-un-lft-identity 37.5%

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

      2. times-frac 37.5%

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

   9. Applied egg-rr 37.5%

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

   10. Taylor expanded in b around inf 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. times-frac 0.0%

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

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\frac{4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} + \frac{a}{b}} \]

       5. rem-square-sqrt 65.6%

          \[\leadsto \frac{1}{\frac{4}{\color{blue}{-4}} \cdot \frac{b}{c} + \frac{a}{b}} \]

       6. metadata-eval 65.6%

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

   12. Simplified 65.6%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 5: 68.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-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 <= (-5d-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 <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-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 <= -5e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-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 < -4.999999999999985e-310

   1. Initial program 67.8%

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

      1. *-commutative 67.8%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in b around -inf 66.2%

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

      1. +-commutative 66.2%

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

      2. mul-1-neg 66.2%

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

      3. unsub-neg 66.2%

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

   6. Simplified 66.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.5%

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

      1. *-commutative 31.5%

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

   3. Simplified 31.5%

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

   4. Taylor expanded in b around inf 65.5%

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

      1. mul-1-neg 65.5%

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

      2. distribute-neg-frac 65.5%

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

   6. Simplified 65.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 43.1% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{+26}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.85e+26) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.85e+26) {
		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 <= 1.85d+26) 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 <= 1.85e+26) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.85e+26:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.85e+26)
		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 <= 1.85e+26)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.85e+26], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.84999999999999994e26

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in b around -inf 50.1%

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

      1. associate-*r/ 50.1%

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

      2. mul-1-neg 50.1%

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

   6. Simplified 50.1%

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

   if 1.84999999999999994e26 < b

   1. Initial program 12.6%

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

      1. *-commutative 12.6%

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

   3. Simplified 12.6%

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

   4. Taylor expanded in b around -inf 2.4%

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

   5. Taylor expanded in b around 0 28.2%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{+26}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 7: 67.9% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{-270}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.6e-270) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.6e-270) {
		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 <= 3.6d-270) 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 <= 3.6e-270) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.6e-270:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.6e-270)
		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 <= 3.6e-270)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.6e-270], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5999999999999998e-270

   1. Initial program 68.4%

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

      1. *-commutative 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in b around -inf 63.2%

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

      1. associate-*r/ 63.2%

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

      2. mul-1-neg 63.2%

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

   6. Simplified 63.2%

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

   if 3.5999999999999998e-270 < b

   1. Initial program 28.8%

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

      1. *-commutative 28.8%

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

   3. Simplified 28.8%

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

   4. Taylor expanded in b around inf 68.8%

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

      1. mul-1-neg 68.8%

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

      2. distribute-neg-frac 68.8%

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

   6. Simplified 68.8%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{-270}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 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 50.9%

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

   1. *-commutative 50.9%

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

3. Simplified 50.9%

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

   1. clear-num 50.8%

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

   2. associate-/r/ 50.8%

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

   3. *-commutative 50.8%

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

   4. associate-/r* 50.8%

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

   5. metadata-eval 50.8%

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

   6. add-sqr-sqrt 36.2%

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

   7. sqrt-unprod 48.1%

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

   8. sqr-neg 48.1%

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

   9. sqrt-prod 11.9%

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

   10. add-sqr-sqrt 31.8%

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

   11. fma-neg 31.8%

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

   12. *-commutative 31.8%

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

   13. distribute-rgt-neg-in 31.8%

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

   14. *-commutative 31.8%

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

   15. distribute-rgt-neg-in 31.8%

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

   16. metadata-eval 31.8%

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

5. Applied egg-rr 31.8%

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

6. Taylor expanded in a around 0 2.6%

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

7. Final simplification 2.6%

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

Alternative 9: 11.0% 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 50.9%

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

   1. *-commutative 50.9%

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

3. Simplified 50.9%

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

4. Taylor expanded in b around -inf 33.9%

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

5. Taylor expanded in b around 0 10.4%

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

6. Final simplification 10.4%

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

Reproduce

herbie shell --seed 2023309 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))