quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.7% → 85.0%

Time: 12.7s

Alternatives: 7

Speedup: 11.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.6 \cdot 10^{-38}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.6e-38)
   (* -0.5 (/ c b_2))
   (if (<= b_2 6.5e+62)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.6e-38) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6.5e+62) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.6d-38)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 6.5d+62) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.6e-38) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6.5e+62) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.6e-38:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 6.5e+62:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.6e-38)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 6.5e+62)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.6e-38)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 6.5e+62)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.6e-38], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6.5e+62], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.6000000000000001e-38

   1. Initial program 18.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 88.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -3.6000000000000001e-38 < b_2 < 6.5000000000000003e62

   1. Initial program 81.7%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 6.5000000000000003e62 < b_2

   1. Initial program 62.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 96.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ 96.1%

         \[\leadsto \color{blue}{\frac{-2 \cdot b\_2}{a}} \]

      2. *-commutative 96.1%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   5. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{b\_2 \cdot -2}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.6 \cdot 10^{-38}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.6 \cdot 10^{-43}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.6e-43)
   (* -0.5 (/ c b_2))
   (if (<= b_2 3.7e-81)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.6e-43) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3.7e-81) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-6.6d-43)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 3.7d-81) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.6e-43) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3.7e-81) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.6e-43:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 3.7e-81:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.6e-43)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 3.7e-81)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.6e-43)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 3.7e-81)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.6e-43], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.7e-81], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.60000000000000031e-43

   1. Initial program 18.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 88.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -6.60000000000000031e-43 < b_2 < 3.69999999999999986e-81

   1. Initial program 75.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0 70.8%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* 70.8%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. neg-mul-1 70.8%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

   5. Simplified 70.8%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]

   if 3.69999999999999986e-81 < b_2

   1. Initial program 73.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 83.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.6 \cdot 10^{-43}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-309) (* -0.5 (/ c b_2)) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-309) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-309)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-309) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-309:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-309)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-309)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-309], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999988e-309

   1. Initial program 35.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 67.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -1.9999999999999988e-309 < b_2

   1. Initial program 73.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 67.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ 67.9%

         \[\leadsto \color{blue}{\frac{-2 \cdot b\_2}{a}} \]

      2. *-commutative 67.9%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   5. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{b\_2 \cdot -2}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 67.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-309) (* -0.5 (/ c b_2)) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-309) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-309)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-309) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-309:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-309)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-309)
		tmp = -0.5 * (c / b_2);
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-309], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999988e-309

   1. Initial program 35.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 67.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -1.9999999999999988e-309 < b_2

   1. Initial program 73.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-sub 73.8%

         \[\leadsto \color{blue}{\frac{-b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

      2. neg-mul-1 73.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. associate-/l* 73.8%

         \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      4. add-sqr-sqrt 73.7%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      5. sqrt-prod 73.6%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2}}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      6. sqr-neg 73.6%

         \[\leadsto -1 \cdot \frac{\sqrt{\color{blue}{\left(-b\_2\right) \cdot \left(-b\_2\right)}}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      7. sqrt-unprod 0.0%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      8. add-sqr-sqrt 25.6%

         \[\leadsto -1 \cdot \frac{\color{blue}{-b\_2}}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      9. fmm-def 25.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{-b\_2}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      11. sqrt-unprod 73.6%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      12. sqr-neg 73.6%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      13. sqrt-prod 73.7%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

      14. add-sqr-sqrt 73.8%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{b\_2}}{a}, -\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\right) \]

   4. Applied egg-rr 68.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{b\_2}{a}, -\frac{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}{a}\right)} \]

   5. Taylor expanded in b_2 around inf 67.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   6. Step-by-step derivation

      1. metadata-eval 67.9%

         \[\leadsto \color{blue}{\left(-2\right)} \cdot \frac{b\_2}{a} \]

      2. distribute-lft-neg-in 67.9%

         \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      3. associate-*r/ 67.9%

         \[\leadsto -\color{blue}{\frac{2 \cdot b\_2}{a}} \]

      4. *-commutative 67.9%

         \[\leadsto -\frac{\color{blue}{b\_2 \cdot 2}}{a} \]

      5. associate-*r/ 67.7%

         \[\leadsto -\color{blue}{b\_2 \cdot \frac{2}{a}} \]

      6. distribute-rgt-neg-in 67.7%

         \[\leadsto \color{blue}{b\_2 \cdot \left(-\frac{2}{a}\right)} \]

      7. distribute-neg-frac 67.7%

         \[\leadsto b\_2 \cdot \color{blue}{\frac{-2}{a}} \]

      8. metadata-eval 67.7%

         \[\leadsto b\_2 \cdot \frac{\color{blue}{-2}}{a} \]

   7. Simplified 67.7%

      \[\leadsto \color{blue}{b\_2 \cdot \frac{-2}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 35.5% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -0.5 (/ c b_2)))

C

double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

Python

def code(a, b_2, c):
	return -0.5 * (c / b_2)

Julia

function code(a, b_2, c)
	return Float64(-0.5 * Float64(c / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -0.5 * (c / b_2);
end

Wolfram

code[a_, b$95$2_, c_] := N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.9%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in b_2 around -inf 30.7%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
4. Add Preprocessing

Alternative 6: 15.3% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ c (- b_2)))

C

double code(double a, double b_2, double c) {
	return c / -b_2;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = c / -b_2
end function

Java

public static double code(double a, double b_2, double c) {
	return c / -b_2;
}

Python

def code(a, b_2, c):
	return c / -b_2

Julia

function code(a, b_2, c)
	return Float64(c / Float64(-b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = c / -b_2;
end

Wolfram

code[a_, b$95$2_, c_] := N[(c / (-b$95$2)), $MachinePrecision]

Derivation

1. Initial program 56.9%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-2neg 56.9%

      \[\leadsto \color{blue}{\frac{-\left(\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)}{-a}} \]

   2. div-inv 56.8%

      \[\leadsto \color{blue}{\left(-\left(\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a}} \]

4. Applied egg-rr 56.3%

   \[\leadsto \color{blue}{\left(b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)\right) \cdot \frac{1}{-a}} \]
5. Step-by-step derivation

   1. distribute-rgt-neg-out 56.3%

      \[\leadsto \left(b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{-a \cdot c}}\right)\right) \cdot \frac{1}{-a} \]

   2. distribute-lft-neg-in 56.3%

      \[\leadsto \left(b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{\left(-a\right) \cdot c}}\right)\right) \cdot \frac{1}{-a} \]

6. Simplified 56.3%

   \[\leadsto \color{blue}{\left(b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{\left(-a\right) \cdot c}\right)\right) \cdot \frac{1}{-a}} \]

7. Taylor expanded in a around 0 56.3%

   \[\leadsto \left(b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{\left(-a\right) \cdot c}\right)\right) \cdot \color{blue}{\frac{-1}{a}} \]

8. Taylor expanded in b_2 around -inf 0.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b\_2}} \]

10. Simplified 13.4%

    \[\leadsto \color{blue}{\frac{-c}{b\_2}} \]

11. Final simplification 13.4%

    \[\leadsto \frac{c}{-b\_2} \]
12. Add Preprocessing

Alternative 7: 3.6% accurate, 112.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 -2.0)

C

double code(double a, double b_2, double c) {
	return -2.0;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = -2.0d0
end function

Java

public static double code(double a, double b_2, double c) {
	return -2.0;
}

Python

def code(a, b_2, c):
	return -2.0

Julia

function code(a, b_2, c)
	return -2.0
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -2.0;
end

Wolfram

code[a_, b$95$2_, c_] := -2.0

Derivation

1. Initial program 56.9%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-2neg 56.9%

      \[\leadsto \color{blue}{\frac{-\left(\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)}{-a}} \]

   2. div-inv 56.8%

      \[\leadsto \color{blue}{\left(-\left(\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a}} \]

4. Applied egg-rr 56.3%

   \[\leadsto \color{blue}{\left(b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)\right) \cdot \frac{1}{-a}} \]
5. Step-by-step derivation

   1. distribute-rgt-neg-out 56.3%

      \[\leadsto \left(b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{-a \cdot c}}\right)\right) \cdot \frac{1}{-a} \]

   2. distribute-lft-neg-in 56.3%

      \[\leadsto \left(b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{\left(-a\right) \cdot c}}\right)\right) \cdot \frac{1}{-a} \]

6. Simplified 56.3%

   \[\leadsto \color{blue}{\left(b\_2 + \mathsf{hypot}\left(b\_2, \sqrt{\left(-a\right) \cdot c}\right)\right) \cdot \frac{1}{-a}} \]

7. Taylor expanded in b_2 around inf 39.3%

   \[\leadsto \left(b\_2 + \color{blue}{b\_2}\right) \cdot \frac{1}{-a} \]
8. Step-by-step derivation

   1. expm1-log1p-u 19.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(b\_2 + b\_2\right) \cdot \frac{1}{-a}\right)\right)} \]

   2. expm1-undefine 17.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(b\_2 + b\_2\right) \cdot \frac{1}{-a}\right)} - 1} \]

   3. un-div-inv 17.9%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{b\_2 + b\_2}{-a}}\right)} - 1 \]

   4. flip-+ 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{b\_2 - b\_2}}}{-a}\right)} - 1 \]

   5. +-inverses 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0}}{b\_2 - b\_2}}{-a}\right)} - 1 \]

   6. +-inverses 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{0}{\color{blue}{0}}}{-a}\right)} - 1 \]

   7. add-sqr-sqrt 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}\right)} - 1 \]

   8. sqrt-unprod 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{\color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}}\right)} - 1 \]

   9. sqr-neg 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{\sqrt{\color{blue}{a \cdot a}}}\right)} - 1 \]

   10. sqrt-unprod 0.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}\right)} - 1 \]

   11. add-sqr-sqrt 0.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{\color{blue}{a}}\right)} - 1 \]

9. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0}{0}}{a}\right)} - 1} \]

11. Simplified 3.8%

    \[\leadsto \color{blue}{-2} \]
12. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

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

Reproduce

herbie shell --seed 2024157 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, 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_2 0) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))