quadm (p42, negative)

Percentage Accurate: 50.5% → 85.2%

Time: 10.4s

Alternatives: 9

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.25 \cdot 10^{-83}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left(0 - b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.25e-83)
   (- 0.0 (/ c b))
   (if (<= b 4.2e+70)
     (/ (- (- 0.0 b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.25e-83) {
		tmp = 0.0 - (c / b);
	} else if (b <= 4.2e+70) {
		tmp = ((0.0 - b) - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.25d-83)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 4.2d+70) then
        tmp = ((0.0d0 - b) - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.25e-83) {
		tmp = 0.0 - (c / b);
	} else if (b <= 4.2e+70) {
		tmp = ((0.0 - b) - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.25e-83:
		tmp = 0.0 - (c / b)
	elif b <= 4.2e+70:
		tmp = ((0.0 - b) - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.25e-83)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 4.2e+70)
		tmp = Float64(Float64(Float64(0.0 - b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.25e-83)
		tmp = 0.0 - (c / b);
	elseif (b <= 4.2e+70)
		tmp = ((0.0 - b) - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.25e-83], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.2e+70], N[(N[(N[(0.0 - b), $MachinePrecision] - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.25e-83

   1. Initial program 17.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 91.0

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 91.0%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -3.25e-83 < b < 4.20000000000000015e70

   1. Initial program 78.4%

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

   if 4.20000000000000015e70 < b

   1. Initial program 58.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 97.3

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

   5. Simplified 97.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.25 \cdot 10^{-83}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left(0 - b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-85}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\left(0 - b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.2e-85)
   (- 0.0 (/ c b))
   (if (<= b 9.5e-132)
     (/ (- (- 0.0 b) (sqrt (* c (* a -4.0)))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.2e-85) {
		tmp = 0.0 - (c / b);
	} else if (b <= 9.5e-132) {
		tmp = ((0.0 - b) - sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9.2d-85)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 9.5d-132) then
        tmp = ((0.0d0 - b) - sqrt((c * (a * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.2e-85) {
		tmp = 0.0 - (c / b);
	} else if (b <= 9.5e-132) {
		tmp = ((0.0 - b) - Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.2e-85:
		tmp = 0.0 - (c / b)
	elif b <= 9.5e-132:
		tmp = ((0.0 - b) - math.sqrt((c * (a * -4.0)))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.2e-85)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 9.5e-132)
		tmp = Float64(Float64(Float64(0.0 - b) - sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.2e-85)
		tmp = 0.0 - (c / b);
	elseif (b <= 9.5e-132)
		tmp = ((0.0 - b) - sqrt((c * (a * -4.0)))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.2e-85], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.5e-132], N[(N[(N[(0.0 - b), $MachinePrecision] - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.2000000000000001e-85

   1. Initial program 17.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 91.0

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 91.0%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -9.2000000000000001e-85 < b < 9.49999999999999987e-132

   1. Initial program 69.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 68.2

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

   5. Simplified 68.2%

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

   if 9.49999999999999987e-132 < b

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 87.3

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

   5. Simplified 87.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-85}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\left(0 - b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{-82}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.22e-82)
   (- 0.0 (/ c b))
   (if (<= b 8.2e-134)
     (/ (- b (sqrt (* c (* a -4.0)))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.22e-82) {
		tmp = 0.0 - (c / b);
	} else if (b <= 8.2e-134) {
		tmp = (b - sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= (-1.22d-82)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 8.2d-134) then
        tmp = (b - sqrt((c * (a * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.22e-82) {
		tmp = 0.0 - (c / b);
	} else if (b <= 8.2e-134) {
		tmp = (b - Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.22e-82:
		tmp = 0.0 - (c / b)
	elif b <= 8.2e-134:
		tmp = (b - math.sqrt((c * (a * -4.0)))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.22e-82)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 8.2e-134)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.22e-82)
		tmp = 0.0 - (c / b);
	elseif (b <= 8.2e-134)
		tmp = (b - sqrt((c * (a * -4.0)))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.22e-82], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.2e-134], N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.22000000000000001e-82

   1. Initial program 17.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 91.0

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 91.0%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -1.22000000000000001e-82 < b < 8.2000000000000004e-134

   1. Initial program 69.7%

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

   4. Applied egg-rr 66.6%

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

   5. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 66.6

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

   7. Simplified 66.6%

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-/r* N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 66.7

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

   9. Applied egg-rr 66.7%

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

   if 8.2000000000000004e-134 < b

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 87.3

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

   5. Simplified 87.3%

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

Alternative 4: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-82}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.55e-82)
   (- 0.0 (/ c b))
   (if (<= b 9.5e-132)
     (* (/ 0.5 a) (- b (sqrt (* (* c a) -4.0))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-82) {
		tmp = 0.0 - (c / b);
	} else if (b <= 9.5e-132) {
		tmp = (0.5 / a) * (b - sqrt(((c * a) * -4.0)));
	} else {
		tmp = (c / b) - (b / a);
	}
	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-82)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 9.5d-132) then
        tmp = (0.5d0 / a) * (b - sqrt(((c * a) * (-4.0d0))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-82) {
		tmp = 0.0 - (c / b);
	} else if (b <= 9.5e-132) {
		tmp = (0.5 / a) * (b - Math.sqrt(((c * a) * -4.0)));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.55e-82:
		tmp = 0.0 - (c / b)
	elif b <= 9.5e-132:
		tmp = (0.5 / a) * (b - math.sqrt(((c * a) * -4.0)))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.55e-82)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 9.5e-132)
		tmp = Float64(Float64(0.5 / a) * Float64(b - sqrt(Float64(Float64(c * a) * -4.0))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.55e-82)
		tmp = 0.0 - (c / b);
	elseif (b <= 9.5e-132)
		tmp = (0.5 / a) * (b - sqrt(((c * a) * -4.0)));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.55e-82], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.5e-132], N[(N[(0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.55e-82

   1. Initial program 17.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 91.0

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 91.0%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -1.55e-82 < b < 9.49999999999999987e-132

   1. Initial program 69.7%

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

   4. Applied egg-rr 66.6%

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

   5. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 66.6

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

   7. Simplified 66.6%

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

   if 9.49999999999999987e-132 < b

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 87.3

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

   5. Simplified 87.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-82}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - \sqrt{\left(c \cdot a\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (- 0.0 (/ c b)) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = (c / b) - (b / a);
	}
	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 = 0.0d0 - (c / b)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = 0.0 - (c / b)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(0.0 - Float64(c / b));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 36.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 66.1

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 66.1%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 69.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 73.5

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

   5. Simplified 73.5%

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

Alternative 6: 67.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (- 0.0 (/ c b)) (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = 0.0 - (b / a);
	}
	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 = 0.0d0 - (c / b)
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = 0.0 - (c / b)
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(0.0 - Float64(c / b));
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 36.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

      4. /-lowering-/.f64 66.1

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

   5. Simplified 66.1%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 69.6%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

      4. /-lowering-/.f64 73.1

         \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

   5. Simplified 73.1%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 73.1

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

   7. Applied egg-rr 73.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -23000000000000:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -23000000000000.0) (/ c b) (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -23000000000000.0) {
		tmp = c / b;
	} else {
		tmp = 0.0 - (b / a);
	}
	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 <= (-23000000000000.0d0)) then
        tmp = c / b
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -23000000000000.0) {
		tmp = c / b;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -23000000000000.0:
		tmp = c / b
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -23000000000000.0)
		tmp = Float64(c / b);
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -23000000000000.0)
		tmp = c / b;
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -23000000000000.0], N[(c / b), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.3e13

   1. Initial program 17.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 2.5

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

   5. Simplified 2.5%

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

   6. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 31.1

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

   8. Simplified 31.1%

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

   if -2.3e13 < b

   1. Initial program 65.5%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

      4. /-lowering-/.f64 53.4

         \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 53.4

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

   7. Applied egg-rr 53.4%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -23000000000000:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.0% accurate, 4.2× 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 54.8%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. /-lowering-/.f64 N/A

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

   5. --lowering--.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. /-lowering-/.f64 41.9

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

5. Simplified 41.9%

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

6. Taylor expanded in c around inf

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

   1. /-lowering-/.f64 9.2

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

8. Simplified 9.2%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
9. Add Preprocessing

Alternative 9: 2.5% accurate, 4.2× 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 54.8%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

   4. /-lowering-/.f64 42.1

      \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

5. Simplified 42.1%

   \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
6. Step-by-step derivation

   1. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {\left(\frac{b}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - {\left(\frac{b}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   3. sub0-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left({\left(\frac{b}{a}\right)}^{3}\right)}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   4. cube-neg N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(\frac{b}{a}\right)\right)}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   5. distribute-neg-frac N/A

      \[\leadsto \frac{{\color{blue}{\left(\frac{\mathsf{neg}\left(b\right)}{a}\right)}}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   6. cube-div N/A

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{neg}\left(b\right)\right)}^{3}}{{a}^{3}}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{{\left(\mathsf{neg}\left(b\right)\right)}^{\color{blue}{\left(2 \cdot \frac{3}{2}\right)}}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{{\left(\mathsf{neg}\left(b\right)\right)}^{\left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot 3\right)}\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   9. pow-pow N/A

      \[\leadsto \frac{\frac{\color{blue}{{\left({\left(\mathsf{neg}\left(b\right)\right)}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   10. pow2 N/A

       \[\leadsto \frac{\frac{{\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   11. sqr-neg N/A

       \[\leadsto \frac{\frac{{\color{blue}{\left(b \cdot b\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{{\color{blue}{\left({b}^{2}\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   13. pow-pow N/A

       \[\leadsto \frac{\frac{\color{blue}{{b}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\frac{{b}^{\left(2 \cdot \color{blue}{\frac{3}{2}}\right)}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{{b}^{\color{blue}{3}}}{{a}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   16. cube-div N/A

       \[\leadsto \frac{\color{blue}{{\left(\frac{b}{a}\right)}^{3}}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

   17. metadata-eval N/A

       \[\leadsto \frac{{\left(\frac{b}{a}\right)}^{3}}{\color{blue}{0} + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

7. Applied egg-rr 2.3%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024195 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

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