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

Average Accuracy: 61.2% → 99.4%

Time: 6.0s

Precision: binary64

Cost: 836

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-287}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(z \cdot \frac{z}{y}\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-287)
   (* x (- (* 0.5 (* z (/ z y))) y))
   (* x (+ y (/ (* z -0.5) (/ 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 <= -5e-287) {
		tmp = x * ((0.5 * (z * (z / y))) - y);
	} else {
		tmp = x * (y + ((z * -0.5) / (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 <= (-5d-287)) then
        tmp = x * ((0.5d0 * (z * (z / y))) - y)
    else
        tmp = x * (y + ((z * (-0.5d0)) / (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 <= -5e-287) {
		tmp = x * ((0.5 * (z * (z / y))) - y);
	} else {
		tmp = x * (y + ((z * -0.5) / (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 <= -5e-287:
		tmp = x * ((0.5 * (z * (z / y))) - y)
	else:
		tmp = x * (y + ((z * -0.5) / (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 <= -5e-287)
		tmp = Float64(x * Float64(Float64(0.5 * Float64(z * Float64(z / y))) - y));
	else
		tmp = Float64(x * Float64(y + Float64(Float64(z * -0.5) / 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 <= -5e-287)
		tmp = x * ((0.5 * (z * (z / y))) - y);
	else
		tmp = x * (y + ((z * -0.5) / (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, -5e-287], N[(x * N[(N[(0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + N[(N[(z * -0.5), $MachinePrecision] / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	61.2%
Target	99.1%
Herbie	99.4%

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

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000025e-287

   1. Initial program 61.1%

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

   2. Taylor expanded in y around -inf 95.6%

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

   3. Simplified 95.6%

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

      [Start] 95.6	\[ x \cdot \left(0.5 \cdot \frac{{z}^{2}}{y} + -1 \cdot y\right) \]
      mul-1-neg [=>] 95.6	\[ x \cdot \left(0.5 \cdot \frac{{z}^{2}}{y} + \color{blue}{\left(-y\right)}\right) \]
      unsub-neg [=>] 95.6	\[ x \cdot \color{blue}{\left(0.5 \cdot \frac{{z}^{2}}{y} - y\right)} \]
      associate-*r/ [=>] 95.6	\[ x \cdot \left(\color{blue}{\frac{0.5 \cdot {z}^{2}}{y}} - y\right) \]
      unpow2 [=>] 95.6	\[ x \cdot \left(\frac{0.5 \cdot \color{blue}{\left(z \cdot z\right)}}{y} - y\right) \]

   4. Taylor expanded in z around 0 95.6%

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

   5. Simplified 99.5%

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

      [Start] 95.6	\[ x \cdot \left(0.5 \cdot \frac{{z}^{2}}{y} - y\right) \]
      unpow2 [=>] 95.6	\[ x \cdot \left(0.5 \cdot \frac{\color{blue}{z \cdot z}}{y} - y\right) \]
      associate-*r/ [=>] 95.6	\[ x \cdot \left(\color{blue}{\frac{0.5 \cdot \left(z \cdot z\right)}{y}} - y\right) \]
      associate-*r* [=>] 95.6	\[ x \cdot \left(\frac{\color{blue}{\left(0.5 \cdot z\right) \cdot z}}{y} - y\right) \]
      associate-*r/ [<=] 99.5	\[ x \cdot \left(\color{blue}{\left(0.5 \cdot z\right) \cdot \frac{z}{y}} - y\right) \]
      associate-*l* [=>] 99.5	\[ x \cdot \left(\color{blue}{0.5 \cdot \left(z \cdot \frac{z}{y}\right)} - y\right) \]

   if -5.00000000000000025e-287 < y

   1. Initial program 61.2%

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

   2. Taylor expanded in y around inf 95.1%

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

   3. Simplified 95.1%

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

      [Start] 95.1	\[ x \cdot \left(y + -0.5 \cdot \frac{{z}^{2}}{y}\right) \]
      unpow2 [=>] 95.1	\[ x \cdot \left(y + -0.5 \cdot \frac{\color{blue}{z \cdot z}}{y}\right) \]

   4. Applied egg-rr 99.2%

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

      [Start] 95.1	\[ x \cdot \left(y + -0.5 \cdot \frac{z \cdot z}{y}\right) \]
      *-commutative [=>] 95.1	\[ x \cdot \left(y + \color{blue}{\frac{z \cdot z}{y} \cdot -0.5}\right) \]
      associate-/l* [=>] 99.2	\[ x \cdot \left(y + \color{blue}{\frac{z}{\frac{y}{z}}} \cdot -0.5\right) \]
      associate-*l/ [=>] 99.2	\[ x \cdot \left(y + \color{blue}{\frac{z \cdot -0.5}{\frac{y}{z}}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-287}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(z \cdot \frac{z}{y}\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-287}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

Alternative 2
Accuracy	98.8%
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-287}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Accuracy	51.8%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023131 
(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)))))