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

Percentage Accurate: 67.9% → 98.2%

Time: 12.7s

Precision: binary64

Cost: 964

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-259}:\\ \;\;\;\;0.5 \cdot \left(\frac{z}{\frac{y}{z}} \cdot x\right) - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-259) (- (* 0.5 (* (/ z (/ y z)) x)) (* y x)) (* y x)))

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 <= -5e-259) {
		tmp = (0.5 * ((z / (y / z)) * x)) - (y * x);
	} else {
		tmp = y * x;
	}
	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 <= (-5d-259)) then
        tmp = (0.5d0 * ((z / (y / z)) * x)) - (y * x)
    else
        tmp = y * x
    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 <= -5e-259) {
		tmp = (0.5 * ((z / (y / z)) * x)) - (y * x);
	} else {
		tmp = y * x;
	}
	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 <= -5e-259:
		tmp = (0.5 * ((z / (y / z)) * x)) - (y * x)
	else:
		tmp = y * x
	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 <= -5e-259)
		tmp = Float64(Float64(0.5 * Float64(Float64(z / Float64(y / z)) * x)) - Float64(y * x));
	else
		tmp = Float64(y * x);
	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 <= -5e-259)
		tmp = (0.5 * ((z / (y / z)) * x)) - (y * x);
	else
		tmp = y * x;
	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, -5e-259], N[(N[(0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]

Target

Original	67.9%
Target	99.2%
Herbie	98.2%

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

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999977e-259

   1. Initial program 72.6%

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

   2. Taylor expanded in y around -inf 90.5%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{z}{\frac{y}{z}} \cdot x\right) - y \cdot x} \]
      Step-by-step derivation

      [Start] 90.5	\[ 0.5 \cdot \frac{{z}^{2} \cdot x}{y} + -1 \cdot \left(y \cdot x\right) \]
      mul-1-neg [=>] 90.5	\[ 0.5 \cdot \frac{{z}^{2} \cdot x}{y} + \color{blue}{\left(-y \cdot x\right)} \]
      unsub-neg [=>] 90.5	\[ \color{blue}{0.5 \cdot \frac{{z}^{2} \cdot x}{y} - y \cdot x} \]
      unpow2 [=>] 90.5	\[ 0.5 \cdot \frac{\color{blue}{\left(z \cdot z\right)} \cdot x}{y} - y \cdot x \]
      associate-/l* [=>] 90.5	\[ 0.5 \cdot \color{blue}{\frac{z \cdot z}{\frac{y}{x}}} - y \cdot x \]
      associate-/r/ [=>] 92.9	\[ 0.5 \cdot \color{blue}{\left(\frac{z \cdot z}{y} \cdot x\right)} - y \cdot x \]
      associate-/l* [=>] 99.2	\[ 0.5 \cdot \left(\color{blue}{\frac{z}{\frac{y}{z}}} \cdot x\right) - y \cdot x \]

   if -4.99999999999999977e-259 < y

   1. Initial program 64.2%

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

   2. Taylor expanded in y around inf 99.5%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-259}:\\ \;\;\;\;0.5 \cdot \left(\frac{z}{\frac{y}{z}} \cdot x\right) - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-295}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-295}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Accuracy	53.1%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023160 
(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)))))