Quadratic roots, full range

Percentage Accurate: 52.1% → 85.9%

Time: 15.5s

Alternatives: 10

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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, 1.0× 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: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-92}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e-61)
   (- (/ c b) (/ b a))
   (if (<= b 1.7e-92)
     (* (- b (sqrt (* c (* a -4.0)))) (/ -0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-61) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.7e-92) {
		tmp = (b - sqrt((c * (a * -4.0)))) * (-0.5 / 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 <= (-7.6d-61)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.7d-92) then
        tmp = (b - sqrt((c * (a * (-4.0d0))))) * ((-0.5d0) / 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 <= -7.6e-61) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.7e-92) {
		tmp = (b - Math.sqrt((c * (a * -4.0)))) * (-0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.6e-61:
		tmp = (c / b) - (b / a)
	elif b <= 1.7e-92:
		tmp = (b - math.sqrt((c * (a * -4.0)))) * (-0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e-61)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.7e-92)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.6e-61)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.7e-92)
		tmp = (b - sqrt((c * (a * -4.0)))) * (-0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e-61], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7e-92], N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.59999999999999961e-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 -7.59999999999999961e-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. *-commutative 75.2%

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

      2. sqrt-prod 40.2%

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

   9. Applied egg-rr 40.2%

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

       1. frac-2neg 40.2%

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

       2. div-inv 40.2%

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

       3. distribute-neg-in 40.2%

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

       4. add-sqr-sqrt 23.5%

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

       5. sqrt-unprod 40.2%

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

       6. sqr-neg 40.2%

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

       7. sqrt-unprod 16.9%

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

       8. add-sqr-sqrt 39.4%

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

       9. sub-neg 39.4%

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

       10. add-sqr-sqrt 22.5%

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

       11. sqrt-unprod 39.5%

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

       12. sqr-neg 39.5%

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

       13. sqrt-unprod 16.7%

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

       14. add-sqr-sqrt 40.2%

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

       15. *-commutative 40.2%

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

       16. sqrt-prod 75.2%

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

       17. distribute-rgt-neg-in 75.2%

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

       18. metadata-eval 75.2%

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

   11. Applied egg-rr 75.2%

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

       1. associate-*r* 75.2%

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

       2. *-commutative 75.2%

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

       3. rem-square-sqrt 0.0%

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

       4. unpow2 0.0%

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

       5. associate-*l* 0.0%

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

       6. unpow2 0.0%

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

       7. rem-square-sqrt 75.2%

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

       8. *-commutative 75.2%

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

       9. associate-/r* 75.2%

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

       10. metadata-eval 75.2%

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

   13. Simplified 75.2%

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

   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 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-92}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
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

      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. +-commutative 75.2%

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

      2. unsub-neg 75.2%

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

   9. Applied egg-rr 75.2%

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

       1. *-commutative 75.2%

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

       2. rem-square-sqrt 0.0%

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

       3. unpow2 0.0%

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

       4. associate-*l* 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 75.2%

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

   11. Simplified 75.2%

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

   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, 12.8× 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, 19.1× 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

      1. *-commutative 10.4%

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

   3. Simplified 10.4%

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

   6. Applied egg-rr 27.6%

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

      1. unpow-1 27.6%

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

   8. Simplified 27.6%

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

   9. Taylor expanded in b around inf 0.0%

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

       1. associate-*r/ 0.0%

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

       2. *-commutative 0.0%

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

       3. times-frac 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 91.7%

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

       6. metadata-eval 91.7%

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

   11. Simplified 91.7%

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

       1. expm1-log1p-u 88.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{-1 \cdot \frac{b}{c}}\right)\right)} \]

       2. expm1-udef 40.8%

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

       3. add-sqr-sqrt 17.3%

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

       4. sqrt-unprod 40.9%

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

       5. mul-1-neg 40.9%

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

       6. mul-1-neg 40.9%

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

       7. sqr-neg 40.9%

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

       8. sqrt-unprod 23.6%

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

       9. add-sqr-sqrt 35.7%

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

       10. clear-num 35.7%

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

   13. Applied egg-rr 35.7%

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

       1. expm1-def 34.8%

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

       2. expm1-log1p 34.9%

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

   15. Simplified 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, 19.1× 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

   1. *-commutative 51.3%

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

3. Simplified 51.3%

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

   1. +-commutative 51.3%

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

   2. add-sqr-sqrt 49.7%

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

   3. fma-def 49.4%

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

6. Applied egg-rr 53.7%

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

   1. div-sub 49.3%

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

   2. sub-neg 49.3%

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

   3. div-inv 48.0%

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

   4. *-commutative 48.0%

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

   5. associate-/r* 48.0%

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

   6. metadata-eval 48.0%

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

   7. div-inv 49.2%

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

   8. *-commutative 49.2%

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

   9. associate-/r* 49.2%

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

   10. metadata-eval 49.2%

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

8. Applied egg-rr 49.2%

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

   1. sub-neg 49.2%

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

   2. distribute-rgt-out-- 53.6%

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

10. Simplified 53.6%

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

11. Taylor expanded in b around -inf 35.1%

    \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot b\right)} \]
12. Step-by-step derivation

    1. *-commutative 35.1%

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

13. Simplified 35.1%

    \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(b \cdot -2\right)} \]
14. Step-by-step derivation

    1. expm1-log1p-u 20.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{a} \cdot \left(b \cdot -2\right)\right)\right)} \]

    2. expm1-udef 15.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5}{a} \cdot \left(b \cdot -2\right)\right)} - 1} \]

    3. associate-*l/ 15.8%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.5 \cdot \left(b \cdot -2\right)}{a}}\right)} - 1 \]

    4. *-commutative 15.8%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5 \cdot \color{blue}{\left(-2 \cdot b\right)}}{a}\right)} - 1 \]

    5. associate-*r* 15.8%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(0.5 \cdot -2\right) \cdot b}}{a}\right)} - 1 \]

    6. metadata-eval 15.8%

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

    7. neg-mul-1 15.8%

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

    8. add-sqr-sqrt 14.2%

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

    9. sqrt-unprod 11.7%

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

    10. sqr-neg 11.7%

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

    11. sqrt-unprod 1.5%

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

    12. add-sqr-sqrt 2.2%

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

15. Applied egg-rr 2.2%

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

    1. expm1-def 1.9%

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

    2. expm1-log1p 2.4%

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

17. Simplified 2.4%

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

18. Final simplification 2.4%

    \[\leadsto \frac{b}{a} \]
19. 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

   1. *-commutative 51.3%

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

3. Simplified 51.3%

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

6. Applied egg-rr 53.5%

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

   1. unpow-1 53.5%

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

8. Simplified 53.5%

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

9. Taylor expanded in b around inf 0.0%

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

    1. associate-*r/ 0.0%

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

    2. *-commutative 0.0%

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

    3. times-frac 0.0%

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

    4. unpow2 0.0%

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

    5. rem-square-sqrt 34.4%

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

    6. metadata-eval 34.4%

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

11. Simplified 34.4%

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

    1. expm1-log1p-u 32.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{-1 \cdot \frac{b}{c}}\right)\right)} \]

    2. expm1-udef 14.0%

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

    3. add-sqr-sqrt 6.9%

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

    4. sqrt-unprod 14.6%

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

    5. mul-1-neg 14.6%

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

    6. mul-1-neg 14.6%

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

    7. sqr-neg 14.6%

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

    8. sqrt-unprod 7.9%

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

    9. add-sqr-sqrt 11.8%

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

    10. clear-num 11.8%

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

13. Applied egg-rr 11.8%

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

    1. expm1-def 11.5%

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

    2. expm1-log1p 12.1%

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

15. Simplified 12.1%

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

16. Final simplification 12.1%

    \[\leadsto \frac{c}{b} \]
17. 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)))