quadp (p42, positive)

Percentage Accurate: 52.2% → 85.7%

Time: 14.4s

Alternatives: 8

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e+71)
   (/ (fma -2.0 b (* 2.0 (/ a (/ b c)))) (* 2.0 a))
   (if (<= b 2.9e-39)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e+71) {
		tmp = fma(-2.0, b, (2.0 * (a / (b / c)))) / (2.0 * a);
	} else if (b <= 2.9e-39) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e+71)
		tmp = Float64(fma(-2.0, b, Float64(2.0 * Float64(a / Float64(b / c)))) / Float64(2.0 * a));
	elseif (b <= 2.9e-39)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e+71], N[(N[(-2.0 * b + N[(2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e-39], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.09999999999999997e71

   1. Initial program 50.4%

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

      1. *-commutative 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in b around -inf 89.5%

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

      1. fma-def 89.5%

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

      2. associate-/l* 97.8%

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

   6. Simplified 97.8%

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

   if -1.09999999999999997e71 < b < 2.89999999999999988e-39

   1. Initial program 80.5%

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

      1. +-commutative 80.5%

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

      2. unsub-neg 80.5%

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

      3. fma-neg 80.5%

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

      4. distribute-lft-neg-in 80.5%

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

      5. *-commutative 80.5%

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

      6. *-commutative 80.5%

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

      7. associate-*l* 80.6%

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

      8. metadata-eval 80.6%

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

      9. *-commutative 80.6%

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

   3. Simplified 80.6%

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

   if 2.89999999999999988e-39 < b

   1. Initial program 13.5%

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

      1. *-commutative 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in b around inf 88.5%

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

      1. associate-*r/ 88.5%

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

      2. neg-mul-1 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e+70)
   (/ (fma -2.0 b (* 2.0 (/ a (/ b c)))) (* 2.0 a))
   (if (<= b 5.6e-39)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e+70) {
		tmp = fma(-2.0, b, (2.0 * (a / (b / c)))) / (2.0 * a);
	} else if (b <= 5.6e-39) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.8e+70)
		tmp = Float64(fma(-2.0, b, Float64(2.0 * Float64(a / Float64(b / c)))) / Float64(2.0 * a));
	elseif (b <= 5.6e-39)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.8e+70], N[(N[(-2.0 * b + N[(2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.6e-39], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.79999999999999949e70

   1. Initial program 50.4%

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

      1. *-commutative 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in b around -inf 89.5%

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

      1. fma-def 89.5%

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

      2. associate-/l* 97.8%

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

   6. Simplified 97.8%

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

   if -7.79999999999999949e70 < b < 5.6000000000000003e-39

   1. Initial program 80.5%

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

   if 5.6000000000000003e-39 < b

   1. Initial program 13.5%

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

      1. *-commutative 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in b around inf 88.5%

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

      1. associate-*r/ 88.5%

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

      2. neg-mul-1 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-48)
   (- (/ c b) (/ b a))
   (if (<= b 2e-38) (* (/ (- b (sqrt (* c (* a -4.0)))) a) -0.5) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-48) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2e-38) {
		tmp = ((b - sqrt((c * (a * -4.0)))) / a) * -0.5;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.4d-48)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2d-38) then
        tmp = ((b - sqrt((c * (a * (-4.0d0))))) / a) * (-0.5d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-48) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2e-38) {
		tmp = ((b - Math.sqrt((c * (a * -4.0)))) / a) * -0.5;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-48:
		tmp = (c / b) - (b / a)
	elif b <= 2e-38:
		tmp = ((b - math.sqrt((c * (a * -4.0)))) / a) * -0.5
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-48)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2e-38)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) / a) * -0.5);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e-48)
		tmp = (c / b) - (b / a);
	elseif (b <= 2e-38)
		tmp = ((b - sqrt((c * (a * -4.0)))) / a) * -0.5;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-48], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-38], N[(N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4e-48

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in b around -inf 88.8%

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

      1. +-commutative 88.8%

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

      2. mul-1-neg 88.8%

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

      3. unsub-neg 88.8%

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

   6. Simplified 88.8%

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

   if -2.4e-48 < b < 1.9999999999999999e-38

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in b around 0 72.0%

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

      1. *-commutative 72.0%

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

   6. Simplified 72.0%

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

      1. expm1-log1p-u 56.7%

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

      2. expm1-udef 26.9%

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

      3. add-sqr-sqrt 15.9%

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

      4. sqrt-unprod 27.0%

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

      5. sqr-neg 27.0%

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

      6. sqrt-prod 11.0%

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

      7. add-sqr-sqrt 26.9%

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

      8. associate-*l* 27.0%

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

   8. Applied egg-rr 27.0%

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

      1. expm1-def 56.7%

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

      2. expm1-log1p 70.5%

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

   10. Simplified 70.5%

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

       1. frac-2neg 70.5%

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

       2. div-inv 70.4%

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

       3. distribute-neg-in 70.4%

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

       4. add-sqr-sqrt 38.8%

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

       5. sqrt-unprod 70.2%

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

       6. sqr-neg 70.2%

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

       7. sqrt-unprod 31.3%

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

       8. add-sqr-sqrt 70.6%

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

       9. sub-neg 70.6%

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

       10. distribute-rgt-neg-in 70.6%

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

       11. metadata-eval 70.6%

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

   12. Applied egg-rr 70.6%

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

       1. associate-*r/ 70.8%

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

       2. times-frac 70.8%

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

       3. associate-*r* 72.0%

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

       4. *-commutative 72.0%

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

       5. associate-*r* 72.2%

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

       6. metadata-eval 72.2%

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

   14. Simplified 72.2%

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

   if 1.9999999999999999e-38 < b

   1. Initial program 13.5%

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

      1. *-commutative 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in b around inf 88.5%

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

      1. associate-*r/ 88.5%

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

      2. neg-mul-1 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-39}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-48)
   (/ (fma -2.0 b (* 2.0 (/ a (/ b c)))) (* 2.0 a))
   (if (<= b 3e-39) (* (/ (- b (sqrt (* c (* a -4.0)))) a) -0.5) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-48) {
		tmp = fma(-2.0, b, (2.0 * (a / (b / c)))) / (2.0 * a);
	} else if (b <= 3e-39) {
		tmp = ((b - sqrt((c * (a * -4.0)))) / a) * -0.5;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-48)
		tmp = Float64(fma(-2.0, b, Float64(2.0 * Float64(a / Float64(b / c)))) / Float64(2.0 * a));
	elseif (b <= 3e-39)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) / a) * -0.5);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-48], N[(N[(-2.0 * b + N[(2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3e-39], N[(N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.90000000000000001e-48

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in b around -inf 83.1%

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

      1. fma-def 83.1%

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

      2. associate-/l* 88.8%

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

   6. Simplified 88.8%

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

   if -1.90000000000000001e-48 < b < 3.00000000000000028e-39

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in b around 0 72.0%

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

      1. *-commutative 72.0%

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

   6. Simplified 72.0%

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

      1. expm1-log1p-u 56.7%

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

      2. expm1-udef 26.9%

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

      3. add-sqr-sqrt 15.9%

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

      4. sqrt-unprod 27.0%

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

      5. sqr-neg 27.0%

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

      6. sqrt-prod 11.0%

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

      7. add-sqr-sqrt 26.9%

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

      8. associate-*l* 27.0%

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

   8. Applied egg-rr 27.0%

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

      1. expm1-def 56.7%

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

      2. expm1-log1p 70.5%

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

   10. Simplified 70.5%

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

       1. frac-2neg 70.5%

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

       2. div-inv 70.4%

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

       3. distribute-neg-in 70.4%

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

       4. add-sqr-sqrt 38.8%

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

       5. sqrt-unprod 70.2%

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

       6. sqr-neg 70.2%

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

       7. sqrt-unprod 31.3%

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

       8. add-sqr-sqrt 70.6%

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

       9. sub-neg 70.6%

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

       10. distribute-rgt-neg-in 70.6%

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

       11. metadata-eval 70.6%

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

   12. Applied egg-rr 70.6%

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

       1. associate-*r/ 70.8%

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

       2. times-frac 70.8%

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

       3. associate-*r* 72.0%

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

       4. *-commutative 72.0%

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

       5. associate-*r* 72.2%

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

       6. metadata-eval 72.2%

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

   14. Simplified 72.2%

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

   if 3.00000000000000028e-39 < b

   1. Initial program 13.5%

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

      1. *-commutative 13.5%

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

   3. Simplified 13.5%

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

   4. Taylor expanded in b around inf 88.5%

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

      1. associate-*r/ 88.5%

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

      2. neg-mul-1 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-39}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 68.2% 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 70.9%

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

      1. *-commutative 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in b around -inf 69.7%

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

      1. +-commutative 69.7%

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

      2. mul-1-neg 69.7%

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

      3. unsub-neg 69.7%

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

   6. Simplified 69.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

   3. Simplified 26.9%

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

   4. Taylor expanded in b around inf 70.6%

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

      1. associate-*r/ 70.6%

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

      2. neg-mul-1 70.6%

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

   6. Simplified 70.6%

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

4. Final simplification 70.1%

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

Alternative 6: 43.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{+44}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.9e+44) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.9e+44) {
		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 <= 1.9d+44) 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 <= 1.9e+44) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.9e+44:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.9e+44)
		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 <= 1.9e+44)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.9e+44], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9000000000000001e44

   1. Initial program 64.9%

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

      1. *-commutative 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in b around -inf 48.9%

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

      1. associate-*r/ 48.9%

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

      2. mul-1-neg 48.9%

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

   6. Simplified 48.9%

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

   if 1.9000000000000001e44 < b

   1. Initial program 8.3%

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

      1. *-commutative 8.3%

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

   3. Simplified 8.3%

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

   4. Taylor expanded in b around inf 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-/l* 73.7%

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

   6. Simplified 73.7%

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

      1. clear-num 71.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{-2 \cdot \frac{c}{\frac{b}{a}}}}} \]

      2. inv-pow 71.5%

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

      3. *-commutative 71.5%

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

      4. times-frac 71.5%

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

      5. metadata-eval 71.5%

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

      6. associate-/r/ 82.1%

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

   8. Applied egg-rr 82.1%

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

      1. unpow-1 82.1%

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

      2. mul-1-neg 82.1%

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

      3. *-commutative 82.1%

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

   10. Simplified 82.1%

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

       1. add-sqr-sqrt 52.3%

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

       2. sqrt-unprod 46.5%

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

       3. frac-times 46.5%

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

       4. sqr-neg 46.5%

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

       5. associate-/r* 46.6%

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

       6. *-inverses 46.6%

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

       7. clear-num 46.6%

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

       8. associate-/r* 49.3%

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

       9. *-inverses 49.3%

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

       10. clear-num 49.2%

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

       11. frac-times 49.2%

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

       12. clear-num 49.2%

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

       13. clear-num 49.3%

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

       14. sqrt-unprod 27.7%

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

       15. add-sqr-sqrt 28.7%

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

       16. expm1-log1p-u 28.5%

           \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{b}\right)\right)} \]

       17. expm1-udef 29.4%

           \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c}{b}\right)} - 1} \]

   12. Applied egg-rr 29.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c}{b}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 28.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{b}\right)\right)} \]

       2. expm1-log1p 28.7%

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

   14. Simplified 28.7%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{+44}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 7: 68.1% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 70.9%

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

      1. *-commutative 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in b around -inf 69.4%

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

      1. associate-*r/ 69.4%

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

      2. mul-1-neg 69.4%

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

   6. Simplified 69.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

   3. Simplified 26.9%

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

   4. Taylor expanded in b around inf 70.6%

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

      1. associate-*r/ 70.6%

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

      2. neg-mul-1 70.6%

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

   6. Simplified 70.6%

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

4. Final simplification 70.0%

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

Alternative 8: 10.8% 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 48.5%

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

   1. *-commutative 48.5%

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

3. Simplified 48.5%

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

4. Taylor expanded in b around inf 29.2%

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

   1. *-commutative 29.2%

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

   2. associate-/l* 29.3%

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

6. Simplified 29.3%

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

   1. clear-num 28.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{-2 \cdot \frac{c}{\frac{b}{a}}}}} \]

   2. inv-pow 28.6%

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

   3. *-commutative 28.6%

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

   4. times-frac 28.6%

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

   5. metadata-eval 28.6%

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

   6. associate-/r/ 31.7%

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

8. Applied egg-rr 31.7%

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

   1. unpow-1 31.7%

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

   2. mul-1-neg 31.7%

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

   3. *-commutative 31.7%

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

10. Simplified 31.7%

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

    1. add-sqr-sqrt 19.8%

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

    2. sqrt-unprod 19.2%

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

    3. frac-times 19.2%

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

    4. sqr-neg 19.2%

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

    5. associate-/r* 19.2%

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

    6. *-inverses 19.2%

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

    7. clear-num 19.2%

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

    8. associate-/r* 20.4%

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

    9. *-inverses 20.4%

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

    10. clear-num 20.4%

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

    11. frac-times 20.3%

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

    12. clear-num 20.4%

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

    13. clear-num 20.4%

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

    14. sqrt-unprod 9.3%

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

    15. add-sqr-sqrt 10.5%

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

    16. expm1-log1p-u 10.1%

        \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{b}\right)\right)} \]

    17. expm1-udef 10.3%

        \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c}{b}\right)} - 1} \]

12. Applied egg-rr 10.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c}{b}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 10.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{b}\right)\right)} \]

    2. expm1-log1p 10.5%

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

14. Simplified 10.5%

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

15. Final simplification 10.5%

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

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t_0 - t_1} \cdot \sqrt{t_0 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2023322 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))))))

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