Quadratic roots, full range

Percentage Accurate: 52.1% → 85.9%

Time: 13.3s

Alternatives: 10

Speedup: 12.9×

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: 52.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e+121)
   (- (/ c b) (/ b a))
   (if (<= b 5.1e-92)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e+121) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.1e-92) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e+121)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5.1e-92)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e+121], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.1e-92], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.49999999999999965e121

   1. Initial program 44.6%

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

      1. *-commutative 44.6%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in b around -inf 93.6%

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

      1. +-commutative 93.6%

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

      2. mul-1-neg 93.6%

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

      3. unsub-neg 93.6%

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

   7. Simplified 93.6%

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

   if -7.49999999999999965e121 < b < 5.09999999999999972e-92

   1. Initial program 85.2%

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

      1. sqr-neg 85.2%

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

      2. +-commutative 85.2%

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

      3. unsub-neg 85.2%

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

      4. sqr-neg 85.2%

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

      5. fma-neg 85.2%

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

      6. distribute-lft-neg-in 85.2%

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

      7. *-commutative 85.2%

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

      8. *-commutative 85.2%

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

      9. distribute-rgt-neg-in 85.2%

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

      10. metadata-eval 85.2%

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

      11. *-commutative 85.2%

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

   3. Simplified 85.2%

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

   if 5.09999999999999972e-92 < b

   1. Initial program 13.8%

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

      1. *-commutative 13.8%

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

   3. Simplified 13.8%

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

   5. Taylor expanded in b around inf 85.4%

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

      1. mul-1-neg 85.4%

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

      2. distribute-neg-frac 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+120}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+120)
   (- (/ c b) (/ b a))
   (if (<= b 5.2e-92)
     (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+120) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.2e-92) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e+120)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5.2e-92)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e+120], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-92], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.59999999999999991e120

   1. Initial program 44.6%

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

      1. *-commutative 44.6%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in b around -inf 93.6%

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

      1. +-commutative 93.6%

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

      2. mul-1-neg 93.6%

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

      3. unsub-neg 93.6%

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

   7. Simplified 93.6%

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

   if -1.59999999999999991e120 < b < 5.2e-92

   1. Initial program 85.2%

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

   3. Simplified 85.2%

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

   if 5.2e-92 < b

   1. Initial program 13.8%

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

      1. *-commutative 13.8%

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

   3. Simplified 13.8%

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

   5. Taylor expanded in b around inf 85.4%

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

      1. mul-1-neg 85.4%

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

      2. distribute-neg-frac 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+120}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e+120)
   (- (/ c b) (/ b a))
   (if (<= b 5.8e-93)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e+120) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.8e-93) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.8d+120)) then
        tmp = (c / b) - (b / a)
    else if (b <= 5.8d-93) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e+120) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.8e-93) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.8e+120:
		tmp = (c / b) - (b / a)
	elif b <= 5.8e-93:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.8e+120)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5.8e-93)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.8e+120)
		tmp = (c / b) - (b / a);
	elseif (b <= 5.8e-93)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.8e+120], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e-93], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.7999999999999997e120

   1. Initial program 44.6%

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

      1. *-commutative 44.6%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in b around -inf 93.6%

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

      1. +-commutative 93.6%

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

      2. mul-1-neg 93.6%

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

      3. unsub-neg 93.6%

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

   7. Simplified 93.6%

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

   if -7.7999999999999997e120 < b < 5.7999999999999997e-93

   1. Initial program 85.2%

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

   if 5.7999999999999997e-93 < b

   1. Initial program 13.8%

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

      1. *-commutative 13.8%

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

   3. Simplified 13.8%

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

   5. Taylor expanded in b around inf 85.4%

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

      1. mul-1-neg 85.4%

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

      2. distribute-neg-frac 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-61)
   (- (/ c b) (/ b a))
   (if (<= b 1.7e-92)
     (* (/ 0.5 a) (+ b (sqrt (* a (* c -4.0)))))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-61) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.7e-92) {
		tmp = (0.5 / a) * (b + sqrt((a * (c * -4.0))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.5d-61)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.7d-92) then
        tmp = (0.5d0 / a) * (b + sqrt((a * (c * (-4.0d0)))))
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-61) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.7e-92) {
		tmp = (0.5 / a) * (b + Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e-61:
		tmp = (c / b) - (b / a)
	elif b <= 1.7e-92:
		tmp = (0.5 / a) * (b + math.sqrt((a * (c * -4.0))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-61)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.7e-92)
		tmp = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e-61)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.7e-92)
		tmp = (0.5 / a) * (b + sqrt((a * (c * -4.0))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-61], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7e-92], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5e-61

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in b around -inf 88.3%

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

      1. +-commutative 88.3%

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

      2. mul-1-neg 88.3%

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

      3. unsub-neg 88.3%

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

   7. Simplified 88.3%

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

   if -4.5e-61 < b < 1.7000000000000001e-92

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in b around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. associate-*r* 75.2%

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

   7. Simplified 75.2%

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

      1. clear-num 75.1%

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

      2. inv-pow 75.1%

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

      3. +-commutative 75.1%

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

      4. sqrt-prod 40.2%

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

      5. fma-def 40.2%

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

      6. add-sqr-sqrt 23.4%

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

      7. sqrt-unprod 40.2%

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

      8. sqr-neg 40.2%

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

      9. sqrt-prod 17.0%

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

      10. add-sqr-sqrt 39.4%

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

   9. Applied egg-rr 39.4%

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

       1. unpow-1 39.4%

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

   11. Simplified 39.4%

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

       1. associate-/r/ 39.4%

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

       2. fma-udef 39.4%

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

       3. sqrt-prod 73.7%

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

       4. distribute-lft-in 73.7%

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

       5. *-commutative 73.7%

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

       6. associate-/r* 73.7%

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

       7. metadata-eval 73.7%

          \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \sqrt{a \cdot \left(c \cdot -4\right)} + \frac{1}{a \cdot 2} \cdot b \]

       8. *-commutative 73.7%

          \[\leadsto \frac{0.5}{a} \cdot \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)}} + \frac{1}{a \cdot 2} \cdot b \]

       9. *-commutative 73.7%

          \[\leadsto \frac{0.5}{a} \cdot \sqrt{a \cdot \left(-4 \cdot c\right)} + \frac{1}{\color{blue}{2 \cdot a}} \cdot b \]

       10. associate-/r* 73.7%

           \[\leadsto \frac{0.5}{a} \cdot \sqrt{a \cdot \left(-4 \cdot c\right)} + \color{blue}{\frac{\frac{1}{2}}{a}} \cdot b \]

       11. metadata-eval 73.7%

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

   13. Applied egg-rr 73.7%

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

       1. distribute-lft-out 73.7%

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

       2. +-commutative 73.7%

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

       3. *-commutative 73.7%

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

   15. Simplified 73.7%

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

   if 1.7000000000000001e-92 < b

   1. Initial program 13.8%

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

      1. *-commutative 13.8%

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

   3. Simplified 13.8%

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

   5. Taylor expanded in b around inf 85.4%

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

      1. mul-1-neg 85.4%

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

      2. distribute-neg-frac 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-63}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.36 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.55e-63)
   (- (/ c b) (/ b a))
   (if (<= b 1.36e-93)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-63) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.36e-93) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.55d-63)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.36d-93) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-63) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.36e-93) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.55e-63:
		tmp = (c / b) - (b / a)
	elif b <= 1.36e-93:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.55e-63)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.36e-93)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.55e-63)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.36e-93)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.55e-63], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.36e-93], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.54999999999999992e-63

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in b around -inf 88.3%

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

      1. +-commutative 88.3%

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

      2. mul-1-neg 88.3%

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

      3. unsub-neg 88.3%

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

   7. Simplified 88.3%

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

   if -1.54999999999999992e-63 < b < 1.3599999999999999e-93

   1. Initial program 80.1%

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

   3. Simplified 80.1%

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

      1. add-sqr-sqrt 79.6%

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

      2. pow2 79.6%

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

      3. pow1/2 79.6%

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

      4. sqrt-pow1 79.7%

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

      5. pow2 79.7%

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

      6. metadata-eval 79.7%

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

   6. Applied egg-rr 79.7%

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

   7. Taylor expanded in a around inf 37.3%

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

   9. Simplified 75.2%

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

   if 1.3599999999999999e-93 < b

   1. Initial program 13.8%

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

      1. *-commutative 13.8%

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

   3. Simplified 13.8%

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

   5. Taylor expanded in b around inf 85.4%

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

      1. mul-1-neg 85.4%

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

      2. distribute-neg-frac 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-63}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.36 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-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 <= (-4d-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 <= -4e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-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 <= -4e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-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 < -3.999999999999988e-310

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in b around -inf 70.1%

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

      1. +-commutative 70.1%

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

      2. mul-1-neg 70.1%

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

      3. unsub-neg 70.1%

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

   7. Simplified 70.1%

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

   if -3.999999999999988e-310 < b

   1. Initial program 30.8%

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

      1. *-commutative 30.8%

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

   3. Simplified 30.8%

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

   5. Taylor expanded in b around inf 65.6%

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

      1. mul-1-neg 65.6%

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

      2. distribute-neg-frac 65.6%

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

   7. Simplified 65.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 6.5e+62) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.5e+62) {
		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 <= 6.5d+62) 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 <= 6.5e+62) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.5e+62:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.5e+62)
		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 <= 6.5e+62)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.5e+62], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5000000000000003e62

   1. Initial program 67.3%

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

      1. *-commutative 67.3%

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

   3. Simplified 67.3%

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

   5. Taylor expanded in b around -inf 47.9%

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

      1. associate-*r/ 47.9%

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

      2. mul-1-neg 47.9%

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

   7. Simplified 47.9%

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

   if 6.5000000000000003e62 < b

   1. Initial program 10.4%

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

   3. Simplified 10.5%

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

      1. clear-num 10.5%

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

      2. inv-pow 10.5%

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

      3. *-commutative 10.5%

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

      4. *-un-lft-identity 10.5%

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

      5. times-frac 10.5%

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

      6. metadata-eval 10.5%

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

      7. sub-neg 10.5%

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

      8. pow2 10.5%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 4.4%

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

      11. sqr-neg 4.4%

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

      12. sqrt-prod 4.4%

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

      13. add-sqr-sqrt 4.4%

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

   6. Applied egg-rr 4.4%

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

      1. unpow-1 4.4%

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

      2. associate-*r/ 4.4%

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

      3. *-commutative 4.4%

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

      4. +-commutative 4.4%

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

   8. Simplified 4.4%

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

   9. Taylor expanded in b around -inf 34.9%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{-306}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.8e-306) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.8e-306) {
		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 <= 6.8d-306) 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 <= 6.8e-306) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.8e-306:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.8e-306)
		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 <= 6.8e-306)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.8e-306], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.7999999999999996e-306

   1. Initial program 72.9%

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

      1. *-commutative 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in b around -inf 68.5%

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

      1. associate-*r/ 68.5%

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

      2. mul-1-neg 68.5%

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

   7. Simplified 68.5%

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

   if 6.7999999999999996e-306 < b

   1. Initial program 30.3%

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

      1. *-commutative 30.3%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in b around inf 66.1%

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

      1. mul-1-neg 66.1%

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

      2. distribute-neg-frac 66.1%

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

   7. Simplified 66.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{-306}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 51.3%

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

3. Simplified 51.3%

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

   1. clear-num 51.2%

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

   2. inv-pow 51.2%

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

   3. *-commutative 51.2%

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

   4. *-un-lft-identity 51.2%

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

   5. times-frac 51.2%

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

   6. metadata-eval 51.2%

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

   7. sub-neg 51.2%

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

   8. pow2 51.2%

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

   9. add-sqr-sqrt 35.3%

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

   10. sqrt-unprod 49.1%

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

   11. sqr-neg 49.1%

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

   12. sqrt-prod 14.0%

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

   13. add-sqr-sqrt 32.2%

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

6. Applied egg-rr 32.2%

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

   1. unpow-1 32.2%

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

   2. associate-*r/ 32.2%

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

   3. *-commutative 32.2%

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

   4. +-commutative 32.2%

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

8. Simplified 32.2%

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

9. Taylor expanded in a around 0 2.4%

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

10. Final simplification 2.4%

    \[\leadsto \frac{b}{a} \]
11. Add Preprocessing

Alternative 10: 10.7% 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 51.3%

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

3. Simplified 51.3%

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

   1. clear-num 51.2%

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

   2. inv-pow 51.2%

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

   3. *-commutative 51.2%

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

   4. *-un-lft-identity 51.2%

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

   5. times-frac 51.2%

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

   6. metadata-eval 51.2%

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

   7. sub-neg 51.2%

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

   8. pow2 51.2%

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

   9. add-sqr-sqrt 35.3%

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

   10. sqrt-unprod 49.1%

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

   11. sqr-neg 49.1%

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

   12. sqrt-prod 14.0%

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

   13. add-sqr-sqrt 32.2%

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

6. Applied egg-rr 32.2%

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

   1. unpow-1 32.2%

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

   2. associate-*r/ 32.2%

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

   3. *-commutative 32.2%

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

   4. +-commutative 32.2%

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

8. Simplified 32.2%

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

9. Taylor expanded in b around -inf 12.1%

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

10. Final simplification 12.1%

    \[\leadsto \frac{c}{b} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))