The quadratic formula (r1)

Percentage Accurate: 63.6% → 87.7%

Time: 19.7s

Alternatives: 8

Speedup: 9.7×

• could not determine a ground truth

  1. a = -3.3887297905792814e-221
  2. b = -9.757583861191998e-223
  3. c = 1.6794026275760462e-183

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: 63.6% 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: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+69}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e+69)
   (- (/ c b) (/ b a))
   (if (<= b 5e+130)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (* (/ 0.5 a) (- b b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e+69) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5e+130) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (0.5 / a) * (b - 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 <= (-7d+69)) then
        tmp = (c / b) - (b / a)
    else if (b <= 5d+130) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (0.5d0 / a) * (b - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e+69) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5e+130) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (0.5 / a) * (b - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7e+69:
		tmp = (c / b) - (b / a)
	elif b <= 5e+130:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (0.5 / a) * (b - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e+69)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 5e+130)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(0.5 / a) * Float64(b - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7e+69)
		tmp = (c / b) - (b / a);
	elseif (b <= 5e+130)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (0.5 / a) * (b - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e+69], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+130], 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[(N[(0.5 / a), $MachinePrecision] * N[(b - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.99999999999999974e69

   1. Initial program 58.2%

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

      1. *-commutative 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in b around -inf 94.2%

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

      1. +-commutative 94.2%

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

      2. mul-1-neg 94.2%

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

      3. unsub-neg 94.2%

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

   7. Simplified 94.2%

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

   if -6.99999999999999974e69 < b < 4.9999999999999996e130

   1. Initial program 88.2%

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

   if 4.9999999999999996e130 < b

   1. Initial program 3.9%

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

      1. *-commutative 3.9%

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

   3. Simplified 3.9%

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

   6. Applied egg-rr 3.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a} + \left(-b \cdot \frac{0.5}{a}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 3.2%

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

      2. distribute-rgt-out-- 3.9%

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

   8. Simplified 3.9%

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

   9. Taylor expanded in a around 0 95.6%

      \[\leadsto \frac{0.5}{a} \cdot \left(\color{blue}{b} - b\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+69}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))))
   (if (<= b -1.1e-19)
     (- (/ c b) (/ b a))
     (if (<= b -1.12e-38)
       t_0
       (if (<= b -8.2e-102)
         (/ b (- a))
         (if (<= b 8.5e-88) t_0 (* (/ 0.5 a) (- b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.1e-19) {
		tmp = (c / b) - (b / a);
	} else if (b <= -1.12e-38) {
		tmp = t_0;
	} else if (b <= -8.2e-102) {
		tmp = b / -a;
	} else if (b <= 8.5e-88) {
		tmp = t_0;
	} else {
		tmp = (0.5 / a) * (b - 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    if (b <= (-1.1d-19)) then
        tmp = (c / b) - (b / a)
    else if (b <= (-1.12d-38)) then
        tmp = t_0
    else if (b <= (-8.2d-102)) then
        tmp = b / -a
    else if (b <= 8.5d-88) then
        tmp = t_0
    else
        tmp = (0.5d0 / a) * (b - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.1e-19) {
		tmp = (c / b) - (b / a);
	} else if (b <= -1.12e-38) {
		tmp = t_0;
	} else if (b <= -8.2e-102) {
		tmp = b / -a;
	} else if (b <= 8.5e-88) {
		tmp = t_0;
	} else {
		tmp = (0.5 / a) * (b - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	tmp = 0
	if b <= -1.1e-19:
		tmp = (c / b) - (b / a)
	elif b <= -1.12e-38:
		tmp = t_0
	elif b <= -8.2e-102:
		tmp = b / -a
	elif b <= 8.5e-88:
		tmp = t_0
	else:
		tmp = (0.5 / a) * (b - b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -1.1e-19)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -1.12e-38)
		tmp = t_0;
	elseif (b <= -8.2e-102)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 8.5e-88)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 / a) * Float64(b - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	tmp = 0.0;
	if (b <= -1.1e-19)
		tmp = (c / b) - (b / a);
	elseif (b <= -1.12e-38)
		tmp = t_0;
	elseif (b <= -8.2e-102)
		tmp = b / -a;
	elseif (b <= 8.5e-88)
		tmp = t_0;
	else
		tmp = (0.5 / a) * (b - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.1e-19], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.12e-38], t$95$0, If[LessEqual[b, -8.2e-102], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 8.5e-88], t$95$0, N[(N[(0.5 / a), $MachinePrecision] * N[(b - b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.0999999999999999e-19

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in b around -inf 89.9%

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

      1. +-commutative 89.9%

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

      2. mul-1-neg 89.9%

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

      3. unsub-neg 89.9%

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

   7. Simplified 89.9%

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

   if -1.0999999999999999e-19 < b < -1.1200000000000001e-38 or -8.2000000000000005e-102 < b < 8.4999999999999996e-88

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in b around 0 76.0%

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

      1. *-commutative 76.0%

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

      2. associate-*r* 76.0%

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

   7. Simplified 76.0%

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

   if -1.1200000000000001e-38 < b < -8.2000000000000005e-102

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in b around -inf 76.7%

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

      1. associate-*r/ 76.7%

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

      2. mul-1-neg 76.7%

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

   7. Simplified 76.7%

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

   if 8.4999999999999996e-88 < b

   1. Initial program 39.4%

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

      1. *-commutative 39.4%

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

   3. Simplified 39.4%

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

   6. Applied egg-rr 34.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a} + \left(-b \cdot \frac{0.5}{a}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 34.0%

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

      2. distribute-rgt-out-- 39.4%

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

   8. Simplified 39.4%

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

   9. Taylor expanded in a around 0 87.7%

      \[\leadsto \frac{0.5}{a} \cdot \left(\color{blue}{b} - b\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ b (- a)) (* (/ 0.5 a) (- b b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = (0.5 / a) * (b - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b / -a
    else
        tmp = (0.5d0 / a) * (b - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = (0.5 / a) * (b - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / -a
	else:
		tmp = (0.5 / a) * (b - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(Float64(0.5 / a) * Float64(b - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = b / -a;
	else
		tmp = (0.5 / a) * (b - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / (-a)), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(b - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in b around -inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. mul-1-neg 69.9%

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

   7. Simplified 69.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 53.8%

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

      1. *-commutative 53.8%

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

   3. Simplified 53.8%

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

   6. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a} + \left(-b \cdot \frac{0.5}{a}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 50.4%

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

      2. distribute-rgt-out-- 53.9%

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

   8. Simplified 53.9%

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

   9. Taylor expanded in a around 0 62.3%

      \[\leadsto \frac{0.5}{a} \cdot \left(\color{blue}{b} - b\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e-231) (- (/ c b) (/ b a)) (* (/ 0.5 a) (- b b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-231) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = (0.5 / a) * (b - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.2d-231)) then
        tmp = (c / b) - (b / a)
    else
        tmp = (0.5d0 / a) * (b - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-231) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = (0.5 / a) * (b - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e-231:
		tmp = (c / b) - (b / a)
	else:
		tmp = (0.5 / a) * (b - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e-231)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(0.5 / a) * Float64(b - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.2e-231)
		tmp = (c / b) - (b / a);
	else
		tmp = (0.5 / a) * (b - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e-231], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(b - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.20000000000000009e-231

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in b around -inf 74.8%

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

      1. +-commutative 74.8%

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

      2. mul-1-neg 74.8%

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

      3. unsub-neg 74.8%

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

   7. Simplified 74.8%

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

   if -2.20000000000000009e-231 < b

   1. Initial program 57.0%

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

      1. *-commutative 57.0%

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

   3. Simplified 57.0%

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

   6. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a} + \left(-b \cdot \frac{0.5}{a}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 53.7%

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

      2. distribute-rgt-out-- 57.0%

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

   8. Simplified 57.0%

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

   9. Taylor expanded in a around 0 58.3%

      \[\leadsto \frac{0.5}{a} \cdot \left(\color{blue}{b} - b\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ b (- a)) (/ -1.0 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = -1.0 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b / -a
    else
        tmp = (-1.0d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = -1.0 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / -a
	else:
		tmp = -1.0 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(-1.0 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = b / -a;
	else
		tmp = -1.0 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / (-a)), $MachinePrecision], N[(-1.0 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in b around -inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. mul-1-neg 69.9%

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

   7. Simplified 69.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 53.8%

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

      1. *-commutative 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in b around inf 35.2%

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

      1. add-cbrt-cube 49.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2} \cdot \frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2}\right) \cdot \frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2}}} \]

      2. pow1/3 46.6%

         \[\leadsto \color{blue}{{\left(\left(\frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2} \cdot \frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2}\right) \cdot \frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2}\right)}^{0.3333333333333333}} \]

      3. pow3 46.6%

         \[\leadsto {\color{blue}{\left({\left(\frac{-2 \cdot \frac{a \cdot c}{b}}{a \cdot 2}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. *-commutative 46.6%

         \[\leadsto {\left({\left(\frac{-2 \cdot \frac{a \cdot c}{b}}{\color{blue}{2 \cdot a}}\right)}^{3}\right)}^{0.3333333333333333} \]

      5. times-frac 46.6%

         \[\leadsto {\left({\color{blue}{\left(\frac{-2}{2} \cdot \frac{\frac{a \cdot c}{b}}{a}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. metadata-eval 46.6%

         \[\leadsto {\left({\left(\color{blue}{-1} \cdot \frac{\frac{a \cdot c}{b}}{a}\right)}^{3}\right)}^{0.3333333333333333} \]

      7. associate-/l* 45.5%

         \[\leadsto {\left({\left(-1 \cdot \frac{\color{blue}{a \cdot \frac{c}{b}}}{a}\right)}^{3}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{{\left({\left(-1 \cdot \frac{a \cdot \frac{c}{b}}{a}\right)}^{3}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 48.1%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(-1 \cdot \frac{a \cdot \frac{c}{b}}{a}\right)}^{3}}} \]

      2. rem-cbrt-cube 34.2%

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

      3. clear-num 34.5%

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

      4. un-div-inv 34.5%

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

      5. *-un-lft-identity 34.5%

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

      6. *-commutative 34.5%

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

      7. times-frac 26.2%

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

      8. clear-num 26.2%

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

   9. Applied egg-rr 26.2%

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

       1. *-inverses 26.2%

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

       2. associate-/r* 26.2%

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

       3. /-rgt-identity 26.2%

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

   11. Simplified 26.2%

       \[\leadsto \color{blue}{\frac{-1}{\frac{b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.2% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in b around -inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. mul-1-neg 69.9%

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

   7. Simplified 69.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 53.8%

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

      1. *-commutative 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in b around inf 25.6%

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

      1. mul-1-neg 25.6%

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

      2. distribute-neg-frac2 25.6%

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

   7. Simplified 25.6%

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

4. Final simplification 48.6%

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

Alternative 7: 13.7% accurate, 29.0× 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 / Float64(-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 66.2%

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

   1. *-commutative 66.2%

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

3. Simplified 66.2%

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

5. Taylor expanded in b around inf 13.5%

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

   1. mul-1-neg 13.5%

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

   2. distribute-neg-frac2 13.5%

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

7. Simplified 13.5%

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

8. Final simplification 13.5%

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

Alternative 8: 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 66.2%

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

   1. *-commutative 66.2%

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

3. Simplified 66.2%

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

   1. *-un-lft-identity 66.2%

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

   2. *-un-lft-identity 66.2%

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

   3. prod-diff 66.2%

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

   4. *-commutative 66.2%

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

   5. *-un-lft-identity 66.2%

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

   6. fma-define 66.2%

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

   7. *-un-lft-identity 66.2%

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

   8. +-commutative 66.2%

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

   9. add-sqr-sqrt 40.3%

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

   10. sqrt-unprod 52.9%

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

   11. sqr-neg 52.9%

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

   12. sqrt-prod 12.6%

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

   13. add-sqr-sqrt 32.8%

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

   14. pow2 32.8%

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

   15. add-sqr-sqrt 20.9%

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

   16. sqrt-unprod 32.8%

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

   17. sqr-neg 32.8%

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

   18. sqrt-prod 12.6%

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

   19. add-sqr-sqrt 32.4%

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

   20. *-commutative 32.4%

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

   21. *-un-lft-identity 32.4%

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

6. Applied egg-rr 32.4%

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

   1. +-commutative 32.4%

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

   2. associate-+l+ 32.4%

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

   3. fma-undefine 32.4%

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

   4. *-rgt-identity 32.4%

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

8. Simplified 32.4%

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

9. Taylor expanded in b around -inf 2.4%

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

10. Final simplification 2.4%

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

Developer target: 70.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - t\_0}{2 \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* a (/ (- (- b) t_0) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b < 0.0d0) then
        tmp = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-b - t_0) / (2.0d0 * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b < 0.0:
		tmp = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-b - t_0) / (2.0 * a)))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b < 0.0)
		tmp = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024046 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

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