Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A

Average Error: 28.4 → 6.9

Time: 18.5s

Precision: binary64

Cost: 15560

Math

\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

↓

\[\begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 (- INFINITY))
     (* 0.5 y)
     (if (<= t_0 5e+299)
       (* 0.5 (+ y (/ (- (pow x 2.0) (pow z 2.0)) y)))
       (* 0.5 y)))))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

↓

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.5 * y;
	} else if (t_0 <= 5e+299) {
		tmp = 0.5 * (y + ((pow(x, 2.0) - pow(z, 2.0)) / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

↓

public static double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.5 * y;
	} else if (t_0 <= 5e+299) {
		tmp = 0.5 * (y + ((Math.pow(x, 2.0) - Math.pow(z, 2.0)) / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

↓

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.5 * y
	elif t_0 <= 5e+299:
		tmp = 0.5 * (y + ((math.pow(x, 2.0) - math.pow(z, 2.0)) / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.5 * y);
	elseif (t_0 <= 5e+299)
		tmp = Float64(0.5 * Float64(y + Float64(Float64((x ^ 2.0) - (z ^ 2.0)) / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 0.5 * y;
	elseif (t_0 <= 5e+299)
		tmp = 0.5 * (y + (((x ^ 2.0) - (z ^ 2.0)) / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$0, 5e+299], N[(0.5 * N[(y + N[(N[(N[Power[x, 2.0], $MachinePrecision] - N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]

Target

Original	28.4
Target	0.2
Herbie	6.9

\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y 2)) < -inf.0 or 5.0000000000000003e299 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y 2))

   1. Initial program 63.3

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf 13.8

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y 2)) < 5.0000000000000003e299

   1. Initial program 2.7

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around 0 1.9

      \[\leadsto \color{blue}{0.5 \cdot y + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]

   3. Simplified 1.9

      \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
      Proof

      [Start] 1.9	\[ 0.5 \cdot y + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} \]
      rational.json-simplify-1 [=>] 1.9	\[ \color{blue}{0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} + 0.5 \cdot y} \]
      rational.json-simplify-2 [=>] 1.9	\[ 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y} + \color{blue}{y \cdot 0.5} \]
      rational.json-simplify-51 [=>] 1.9	\[ \color{blue}{0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -\infty:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 5 \cdot 10^{+299}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternatives

Alternative 1
Error	7.4
Cost	3016

\[\begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 2
Error	7.3
Cost	1224

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+148}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x + \left(y \cdot y - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - y \cdot -0.5\right)\\ \end{array} \]

Alternative 3
Error	7.3
Cost	1224

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{0.5}{\frac{y}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - y \cdot -0.5\right)\\ \end{array} \]

Alternative 4
Error	26.7
Cost	192

\[0.5 \cdot y \]

Reproduce

herbie shell --seed 2023068 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))