jeff quadratic root 1

Average Error: 20.0 → 6.5

Time: 5.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ t_2 := \frac{\left(-b\right) - b}{a \cdot 2}\\ t_3 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 10^{+293}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
   (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* c (* a -4.0)))))
        (t_1
         (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ (* c 2.0) (- t_0 b))))
        (t_2 (/ (- (- b) b) (* a 2.0)))
        (t_3 (if (>= b 0.0) t_2 (- (/ c b)))))
   (if (<= t_1 (- INFINITY))
     (if (>= b 0.0) t_2 (/ (* c 2.0) (- (sqrt (* (* a c) -4.0)) b)))
     (if (<= t_1 -5e-302)
       t_1
       (if (<= t_1 0.0) t_3 (if (<= t_1 1e+293) t_1 t_3))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + sqrt(((b * b) - ((4.0 * a) * c))));
	}
	return tmp;
}

↓

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (a * 2.0);
	} else {
		tmp = (c * 2.0) / (t_0 - b);
	}
	double t_1 = tmp;
	double t_2 = (-b - b) / (a * 2.0);
	double tmp_1;
	if (b >= 0.0) {
		tmp_1 = t_2;
	} else {
		tmp_1 = -(c / b);
	}
	double t_3 = tmp_1;
	double tmp_3;
	if (t_1 <= -((double) INFINITY)) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_2;
		} else {
			tmp_4 = (c * 2.0) / (sqrt(((a * c) * -4.0)) - b);
		}
		tmp_3 = tmp_4;
	} else if (t_1 <= -5e-302) {
		tmp_3 = t_1;
	} else if (t_1 <= 0.0) {
		tmp_3 = t_3;
	} else if (t_1 <= 1e+293) {
		tmp_3 = t_1;
	} else {
		tmp_3 = t_3;
	}
	return tmp_3;
}

Java

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

↓

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) + (c * (a * -4.0))));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (a * 2.0);
	} else {
		tmp = (c * 2.0) / (t_0 - b);
	}
	double t_1 = tmp;
	double t_2 = (-b - b) / (a * 2.0);
	double tmp_1;
	if (b >= 0.0) {
		tmp_1 = t_2;
	} else {
		tmp_1 = -(c / b);
	}
	double t_3 = tmp_1;
	double tmp_3;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_2;
		} else {
			tmp_4 = (c * 2.0) / (Math.sqrt(((a * c) * -4.0)) - b);
		}
		tmp_3 = tmp_4;
	} else if (t_1 <= -5e-302) {
		tmp_3 = t_1;
	} else if (t_1 <= 0.0) {
		tmp_3 = t_3;
	} else if (t_1 <= 1e+293) {
		tmp_3 = t_1;
	} else {
		tmp_3 = t_3;
	}
	return tmp_3;
}

Python

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

↓

def code(a, b, c):
	t_0 = math.sqrt(((b * b) + (c * (a * -4.0))))
	tmp = 0
	if b >= 0.0:
		tmp = (-b - t_0) / (a * 2.0)
	else:
		tmp = (c * 2.0) / (t_0 - b)
	t_1 = tmp
	t_2 = (-b - b) / (a * 2.0)
	tmp_1 = 0
	if b >= 0.0:
		tmp_1 = t_2
	else:
		tmp_1 = -(c / b)
	t_3 = tmp_1
	tmp_3 = 0
	if t_1 <= -math.inf:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = t_2
		else:
			tmp_4 = (c * 2.0) / (math.sqrt(((a * c) * -4.0)) - b)
		tmp_3 = tmp_4
	elif t_1 <= -5e-302:
		tmp_3 = t_1
	elif t_1 <= 0.0:
		tmp_3 = t_3
	elif t_1 <= 1e+293:
		tmp_3 = t_1
	else:
		tmp_3 = t_3
	return tmp_3

Julia

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

↓

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c * 2.0) / Float64(t_0 - b));
	end
	t_1 = tmp
	t_2 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0))
	tmp_1 = 0.0
	if (b >= 0.0)
		tmp_1 = t_2;
	else
		tmp_1 = Float64(-Float64(c / b));
	end
	t_3 = tmp_1
	tmp_3 = 0.0
	if (t_1 <= Float64(-Inf))
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = t_2;
		else
			tmp_4 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b));
		end
		tmp_3 = tmp_4;
	elseif (t_1 <= -5e-302)
		tmp_3 = t_1;
	elseif (t_1 <= 0.0)
		tmp_3 = t_3;
	elseif (t_1 <= 1e+293)
		tmp_3 = t_1;
	else
		tmp_3 = t_3;
	end
	return tmp_3
end

MATLAB

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

↓

function tmp_6 = code(a, b, c)
	t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - t_0) / (a * 2.0);
	else
		tmp = (c * 2.0) / (t_0 - b);
	end
	t_1 = tmp;
	t_2 = (-b - b) / (a * 2.0);
	tmp_2 = 0.0;
	if (b >= 0.0)
		tmp_2 = t_2;
	else
		tmp_2 = -(c / b);
	end
	t_3 = tmp_2;
	tmp_4 = 0.0;
	if (t_1 <= -Inf)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = t_2;
		else
			tmp_5 = (c * 2.0) / (sqrt(((a * c) * -4.0)) - b);
		end
		tmp_4 = tmp_5;
	elseif (t_1 <= -5e-302)
		tmp_4 = t_1;
	elseif (t_1 <= 0.0)
		tmp_4 = t_3;
	elseif (t_1 <= 1e+293)
		tmp_4 = t_1;
	else
		tmp_4 = t_3;
	end
	tmp_6 = tmp_4;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], 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], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

↓

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$2 = N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = If[GreaterEqual[b, 0.0], t$95$2, (-N[(c / b), $MachinePrecision])]}, If[LessEqual[t$95$1, (-Infinity)], If[GreaterEqual[b, 0.0], t$95$2, N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$1, -5e-302], t$95$1, If[LessEqual[t$95$1, 0.0], t$95$3, If[LessEqual[t$95$1, 1e+293], t$95$1, t$95$3]]]]]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded in b around inf 18.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Taylor expanded in b around 0 18.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}}\\ \end{array} \]

   if -inf.0 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < -5.00000000000000033e-302 or 0.0 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < 9.9999999999999992e292

   1. Initial program 2.8

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

   if -5.00000000000000033e-302 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < 0.0 or 9.9999999999999992e292 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))

   1. Initial program 43.3

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

   2. Taylor expanded in b around inf 31.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Taylor expanded in b around -inf 11.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq -5 \cdot 10^{-302}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq 10^{+293}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022159 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))