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

Average Accuracy: 61.0% → 99.4%

Time: 8.8s

Precision: binary64

Cost: 836

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-290}:\\ \;\;\;\;\left(0.5 \cdot \frac{z}{\frac{y}{z}} - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + z \cdot \left(z \cdot \frac{-0.5}{y}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-290)
   (* (- (* 0.5 (/ z (/ y z))) y) x)
   (* x (+ y (* z (* z (/ -0.5 y)))))))

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 <= -4e-290) {
		tmp = ((0.5 * (z / (y / z))) - y) * x;
	} else {
		tmp = x * (y + (z * (z * (-0.5 / y))));
	}
	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 <= (-4d-290)) then
        tmp = ((0.5d0 * (z / (y / z))) - y) * x
    else
        tmp = x * (y + (z * (z * ((-0.5d0) / y))))
    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 <= -4e-290) {
		tmp = ((0.5 * (z / (y / z))) - y) * x;
	} else {
		tmp = x * (y + (z * (z * (-0.5 / y))));
	}
	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 <= -4e-290:
		tmp = ((0.5 * (z / (y / z))) - y) * x
	else:
		tmp = x * (y + (z * (z * (-0.5 / y))))
	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 <= -4e-290)
		tmp = Float64(Float64(Float64(0.5 * Float64(z / Float64(y / z))) - y) * x);
	else
		tmp = Float64(x * Float64(y + Float64(z * Float64(z * Float64(-0.5 / y)))));
	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 <= -4e-290)
		tmp = ((0.5 * (z / (y / z))) - y) * x;
	else
		tmp = x * (y + (z * (z * (-0.5 / y))));
	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, -4e-290], N[(N[(N[(0.5 * N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(y + N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	61.0%
Target	99.0%
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 < -4.0000000000000003e-290

   1. Initial program 62.4%

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

   2. Taylor expanded in y around -inf 94.8%

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

   3. Simplified 99.5%

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

      [Start] 94.8	\[ x \cdot \left(0.5 \cdot \frac{{z}^{2}}{y} + -1 \cdot y\right) \]
      fma-def [=>] 94.8	\[ x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{{z}^{2}}{y}, -1 \cdot y\right)} \]
      unpow2 [=>] 94.8	\[ x \cdot \mathsf{fma}\left(0.5, \frac{\color{blue}{z \cdot z}}{y}, -1 \cdot y\right) \]
      associate-/l* [=>] 99.5	\[ x \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{z}{\frac{y}{z}}}, -1 \cdot y\right) \]
      mul-1-neg [=>] 99.5	\[ x \cdot \mathsf{fma}\left(0.5, \frac{z}{\frac{y}{z}}, \color{blue}{-y}\right) \]

   4. Taylor expanded in x around 0 94.8%

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

   5. Simplified 99.5%

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

      [Start] 94.8	\[ \left(0.5 \cdot \frac{{z}^{2}}{y} - y\right) \cdot x \]
      unpow2 [=>] 94.8	\[ \left(0.5 \cdot \frac{\color{blue}{z \cdot z}}{y} - y\right) \cdot x \]
      associate-/l* [=>] 99.5	\[ \left(0.5 \cdot \color{blue}{\frac{z}{\frac{y}{z}}} - y\right) \cdot x \]

   if -4.0000000000000003e-290 < y

   1. Initial program 59.7%

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

   2. Taylor expanded in y around inf 94.9%

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

   3. Simplified 99.4%

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

      [Start] 94.9	\[ x \cdot \left(y + -0.5 \cdot \frac{{z}^{2}}{y}\right) \]
      +-commutative [=>] 94.9	\[ x \cdot \color{blue}{\left(-0.5 \cdot \frac{{z}^{2}}{y} + y\right)} \]
      fma-def [=>] 94.9	\[ x \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{z}^{2}}{y}, y\right)} \]
      unpow2 [=>] 94.9	\[ x \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{z \cdot z}}{y}, y\right) \]
      associate-/l* [=>] 99.4	\[ x \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{z}{\frac{y}{z}}}, y\right) \]

   4. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ x \cdot \mathsf{fma}\left(-0.5, \frac{z}{\frac{y}{z}}, y\right) \]
      fma-udef [=>] 99.4	\[ x \cdot \color{blue}{\left(-0.5 \cdot \frac{z}{\frac{y}{z}} + y\right)} \]
      distribute-lft-in [=>] 99.4	\[ \color{blue}{x \cdot \left(-0.5 \cdot \frac{z}{\frac{y}{z}}\right) + x \cdot y} \]
      +-commutative [=>] 99.4	\[ \color{blue}{x \cdot y + x \cdot \left(-0.5 \cdot \frac{z}{\frac{y}{z}}\right)} \]
      *-commutative [<=] 99.4	\[ \color{blue}{y \cdot x} + x \cdot \left(-0.5 \cdot \frac{z}{\frac{y}{z}}\right) \]
      *-commutative [=>] 99.4	\[ y \cdot x + x \cdot \color{blue}{\left(\frac{z}{\frac{y}{z}} \cdot -0.5\right)} \]
      associate-/r/ [=>] 99.4	\[ y \cdot x + x \cdot \left(\color{blue}{\left(\frac{z}{y} \cdot z\right)} \cdot -0.5\right) \]
      *-commutative [=>] 99.4	\[ y \cdot x + x \cdot \left(\color{blue}{\left(z \cdot \frac{z}{y}\right)} \cdot -0.5\right) \]

   5. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ y \cdot x + x \cdot \left(\left(z \cdot \frac{z}{y}\right) \cdot -0.5\right) \]
      *-commutative [=>] 99.4	\[ \color{blue}{x \cdot y} + x \cdot \left(\left(z \cdot \frac{z}{y}\right) \cdot -0.5\right) \]
      distribute-lft-out [=>] 99.4	\[ \color{blue}{x \cdot \left(y + \left(z \cdot \frac{z}{y}\right) \cdot -0.5\right)} \]
      associate-*l* [=>] 99.4	\[ x \cdot \left(y + \color{blue}{z \cdot \left(\frac{z}{y} \cdot -0.5\right)}\right) \]
      div-inv [=>] 99.4	\[ x \cdot \left(y + z \cdot \left(\color{blue}{\left(z \cdot \frac{1}{y}\right)} \cdot -0.5\right)\right) \]
      associate-*l* [=>] 99.4	\[ x \cdot \left(y + z \cdot \color{blue}{\left(z \cdot \left(\frac{1}{y} \cdot -0.5\right)\right)}\right) \]
      associate-*l/ [=>] 99.4	\[ x \cdot \left(y + z \cdot \left(z \cdot \color{blue}{\frac{1 \cdot -0.5}{y}}\right)\right) \]
      metadata-eval [=>] 99.4	\[ x \cdot \left(y + z \cdot \left(z \cdot \frac{\color{blue}{-0.5}}{y}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-290}:\\ \;\;\;\;\left(0.5 \cdot \frac{z}{\frac{y}{z}} - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + z \cdot \left(z \cdot \frac{-0.5}{y}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-297}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + z \cdot \left(z \cdot \frac{-0.5}{y}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	98.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-297}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Accuracy	53.3%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023138 
(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)))))