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

Average Error: 25.2 → 0.5

Time: 4.1s

Precision: binary64

Cost: 964

Math

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

↓

\[\begin{array}{l} t_0 := \frac{z}{\frac{y}{z}}\\ \mathbf{if}\;y \leq -8.235311207396689 \cdot 10^{-250}:\\ \;\;\;\;0.5 \cdot \left(t_0 \cdot x\right) - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + t_0 \cdot -0.5\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ z (/ y z))))
   (if (<= y -8.235311207396689e-250)
     (- (* 0.5 (* t_0 x)) (* y x))
     (* x (+ y (* t_0 -0.5))))))

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 t_0 = z / (y / z);
	double tmp;
	if (y <= -8.235311207396689e-250) {
		tmp = (0.5 * (t_0 * x)) - (y * x);
	} else {
		tmp = x * (y + (t_0 * -0.5));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = z / (y / z)
    if (y <= (-8.235311207396689d-250)) then
        tmp = (0.5d0 * (t_0 * x)) - (y * x)
    else
        tmp = x * (y + (t_0 * (-0.5d0)))
    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 t_0 = z / (y / z);
	double tmp;
	if (y <= -8.235311207396689e-250) {
		tmp = (0.5 * (t_0 * x)) - (y * x);
	} else {
		tmp = x * (y + (t_0 * -0.5));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = z / (y / z)
	tmp = 0
	if y <= -8.235311207396689e-250:
		tmp = (0.5 * (t_0 * x)) - (y * x)
	else:
		tmp = x * (y + (t_0 * -0.5))
	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)
	t_0 = Float64(z / Float64(y / z))
	tmp = 0.0
	if (y <= -8.235311207396689e-250)
		tmp = Float64(Float64(0.5 * Float64(t_0 * x)) - Float64(y * x));
	else
		tmp = Float64(x * Float64(y + Float64(t_0 * -0.5)));
	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)
	t_0 = z / (y / z);
	tmp = 0.0;
	if (y <= -8.235311207396689e-250)
		tmp = (0.5 * (t_0 * x)) - (y * x);
	else
		tmp = x * (y + (t_0 * -0.5));
	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_] := Block[{t$95$0 = N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.235311207396689e-250], N[(N[(0.5 * N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	25.2
Target	0.6
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if y < -8.23531120739668927e-250

   1. Initial program 24.7

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

   2. Taylor expanded in y around -inf 3.9

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

   3. Simplified 0.3

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{z}{\frac{y}{z}} \cdot x\right) - y \cdot x} \]
      Proof
      (-.f64 (*.f64 1/2 (*.f64 (/.f64 z (/.f64 y z)) x)) (*.f64 y x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 1/2 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z z) y)) x)) (*.f64 y x)): 16 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 1/2 (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 z z) (/.f64 y x)))) (*.f64 y x)): 12 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 z z) x) y))) (*.f64 y x)): 0 points increase in error, 12 points decrease in error
      (-.f64 (*.f64 1/2 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) x) y)) (*.f64 y x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 1/2 (/.f64 (*.f64 (pow.f64 z 2) x) y)) (neg.f64 (*.f64 y x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (/.f64 (*.f64 (pow.f64 z 2) x) y)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 y x)))): 0 points increase in error, 0 points decrease in error

   if -8.23531120739668927e-250 < y

   1. Initial program 25.6

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

   2. Taylor expanded in y around inf 3.6

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

   3. Simplified 3.6

      \[\leadsto x \cdot \color{blue}{\left(y + -0.5 \cdot \frac{z \cdot z}{y}\right)} \]
      Proof
      (+.f64 y (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (*.f64 -1/2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) y))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in z around 0 3.6

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

   5. Simplified 0.6

      \[\leadsto x \cdot \left(y + -0.5 \cdot \color{blue}{\frac{z}{\frac{y}{z}}}\right) \]
      Proof
      (/.f64 z (/.f64 y z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z z) y)): 33 points increase in error, 29 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) y): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.235311207396689 \cdot 10^{-250}:\\ \;\;\;\;0.5 \cdot \left(\frac{z}{\frac{y}{z}} \cdot x\right) - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z}{\frac{y}{z}} \cdot -0.5\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -8.235311207396689 \cdot 10^{-250}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z}{\frac{y}{z}} \cdot -0.5\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	836

\[\begin{array}{l} t_0 := \frac{z}{\frac{y}{z}}\\ \mathbf{if}\;y \leq 1.436388288155802 \cdot 10^{-292}:\\ \;\;\;\;x \cdot \left(0.5 \cdot t_0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + t_0 \cdot -0.5\right)\\ \end{array} \]

Alternative 3
Error	0.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq 1.436388288155802 \cdot 10^{-292}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Error	30.2
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2022329 
(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)))))