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

Average Error: 24.7 → 0.2

Time: 11.8s

Precision: binary64

Cost: 13764

Math

\[x \cdot \sqrt{y \cdot y - z \cdot z} \]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -3.027804635639142 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(\frac{z}{y} \cdot \left(z \cdot 0.5\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{\frac{\sqrt{y - z}}{\sqrt{y + z}}}\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (sqrt (- (* y y) (* z z)))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.027804635639142e-286)
   (* x (- (* (/ z y) (* z 0.5)) y))
   (* x (/ (- y z) (/ (sqrt (- y z)) (sqrt (+ y z)))))))

C

double code(double x, double y, double z) {
	return x * sqrt(((y * y) - (z * z)));
}

↓

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.027804635639142e-286) {
		tmp = x * (((z / y) * (z * 0.5)) - y);
	} else {
		tmp = x * ((y - z) / (sqrt((y - z)) / sqrt((y + z))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * sqrt(((y * y) - (z * z)))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.027804635639142d-286)) then
        tmp = x * (((z / y) * (z * 0.5d0)) - y)
    else
        tmp = x * ((y - z) / (sqrt((y - z)) / sqrt((y + z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return x * Math.sqrt(((y * y) - (z * z)));
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.027804635639142e-286) {
		tmp = x * (((z / y) * (z * 0.5)) - y);
	} else {
		tmp = x * ((y - z) / (Math.sqrt((y - z)) / Math.sqrt((y + z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	return x * math.sqrt(((y * y) - (z * z)))

↓

def code(x, y, z):
	tmp = 0
	if y <= -3.027804635639142e-286:
		tmp = x * (((z / y) * (z * 0.5)) - y)
	else:
		tmp = x * ((y - z) / (math.sqrt((y - z)) / math.sqrt((y + z))))
	return tmp

Julia

function code(x, y, z)
	return Float64(x * sqrt(Float64(Float64(y * y) - Float64(z * z))))
end

↓

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.027804635639142e-286)
		tmp = Float64(x * Float64(Float64(Float64(z / y) * Float64(z * 0.5)) - y));
	else
		tmp = Float64(x * Float64(Float64(y - z) / Float64(sqrt(Float64(y - z)) / sqrt(Float64(y + z)))));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * sqrt(((y * y) - (z * z)));
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.027804635639142e-286)
		tmp = x * (((z / y) * (z * 0.5)) - y);
	else
		tmp = x * ((y - z) / (sqrt((y - z)) / sqrt((y + z))));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := If[LessEqual[y, -3.027804635639142e-286], N[(x * N[(N[(N[(z / y), $MachinePrecision] * N[(z * 0.5), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y - z), $MachinePrecision] / N[(N[Sqrt[N[(y - z), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	24.7
Target	0.6
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y < 2.5816096488251695 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < -3.02780463563914185e-286

   1. Initial program 25.2

      \[x \cdot \sqrt{y \cdot y - z \cdot z} \]

   2. Taylor expanded in y around -inf 3.3

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \frac{{z}^{2}}{y} + -1 \cdot y\right)} \]

   3. Simplified 0.3

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{z}{\frac{y}{z}}, -y\right)} \]
      Proof

      [Start] 3.3	\[ x \cdot \left(0.5 \cdot \frac{{z}^{2}}{y} + -1 \cdot y\right) \]
      fma-def [=>] 3.3	\[ x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{z}^{2}}{y}, -1 \cdot y\right)} \]
      unpow2 [=>] 3.3	\[ x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{z \cdot z}}{y}, -1 \cdot y\right) \]
      associate-/l* [=>] 0.3	\[ x \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{z}{\frac{y}{z}}}, -1 \cdot y\right) \]
      mul-1-neg [=>] 0.3	\[ x \cdot \mathsf{fma}\left(0.5, \frac{z}{\frac{y}{z}}, \color{blue}{-y}\right) \]

   4. Taylor expanded in x around 0 3.3

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

   5. Simplified 0.3

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

      [Start] 3.3	\[ \left(0.5 \cdot \frac{{z}^{2}}{y} - y\right) \cdot x \]
      *-commutative [=>] 3.3	\[ \color{blue}{x \cdot \left(0.5 \cdot \frac{{z}^{2}}{y} - y\right)} \]
      *-commutative [=>] 3.3	\[ x \cdot \left(\color{blue}{\frac{{z}^{2}}{y} \cdot 0.5} - y\right) \]
      unpow2 [=>] 3.3	\[ x \cdot \left(\frac{\color{blue}{z \cdot z}}{y} \cdot 0.5 - y\right) \]
      associate-*l/ [<=] 0.3	\[ x \cdot \left(\color{blue}{\left(\frac{z}{y} \cdot z\right)} \cdot 0.5 - y\right) \]
      associate-*l* [=>] 0.3	\[ x \cdot \left(\color{blue}{\frac{z}{y} \cdot \left(z \cdot 0.5\right)} - y\right) \]

   if -3.02780463563914185e-286 < y

   1. Initial program 24.3

      \[x \cdot \sqrt{y \cdot y - z \cdot z} \]

   2. Applied egg-rr 0.5

      \[\leadsto x \cdot \color{blue}{\left(\sqrt{y - z} \cdot \sqrt{y + z}\right)} \]

   3. Applied egg-rr 15.6

      \[\leadsto x \cdot \color{blue}{\frac{\left(y - z\right) \cdot \sqrt{y + z}}{\sqrt{y - z}}} \]

   4. Simplified 0.1

      \[\leadsto x \cdot \color{blue}{\frac{y - z}{\frac{\sqrt{y - z}}{\sqrt{z + y}}}} \]
      Proof

      [Start] 15.6	\[ x \cdot \frac{\left(y - z\right) \cdot \sqrt{y + z}}{\sqrt{y - z}} \]
      associate-/l* [=>] 0.1	\[ x \cdot \color{blue}{\frac{y - z}{\frac{\sqrt{y - z}}{\sqrt{y + z}}}} \]
      +-commutative [=>] 0.1	\[ x \cdot \frac{y - z}{\frac{\sqrt{y - z}}{\sqrt{\color{blue}{z + y}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.027804635639142 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(\frac{z}{y} \cdot \left(z \cdot 0.5\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{\frac{\sqrt{y - z}}{\sqrt{y + z}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13508

\[\begin{array}{l} \mathbf{if}\;y \leq -3.027804635639142 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(\frac{z}{y} \cdot \left(z \cdot 0.5\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y - z} \cdot \sqrt{y + z}\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	7236

\[\begin{array}{l} \mathbf{if}\;y \leq -3.027804635639142 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(\frac{z}{y} \cdot \left(z \cdot 0.5\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.5 \cdot \left(x \cdot \frac{z}{\frac{y}{z}}\right)\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -3.027804635639142 \cdot 10^{-286}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -3.027804635639142 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(\frac{z}{y} \cdot \left(z \cdot 0.5\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\right)\\ \end{array} \]

Alternative 5
Error	0.7
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq -3.027804635639142 \cdot 10^{-286}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Error	30.3
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B"
  :precision binary64

  :herbie-target
  (if (< y 2.5816096488251695e-278) (- (* x y)) (* x (* (sqrt (+ y z)) (sqrt (- y z)))))

  (* x (sqrt (- (* y y) (* z z)))))