Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3

Average Accuracy: 81.0% → 96.1%

Time: 6.6s

Precision: binary64

Cost: 712

Math

\[\frac{x \cdot \left(y - z\right)}{y} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-274}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-62}:\\ \;\;\;\;x - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.45e-274)
   (/ x (/ y (- y z)))
   (if (<= x 2e-62) (- x (* z (/ x y))) (* x (- 1.0 (/ z y))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.45e-274) {
		tmp = x / (y / (y - z));
	} else if (x <= 2e-62) {
		tmp = x - (z * (x / y));
	} else {
		tmp = x * (1.0 - (z / 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 * (y - z)) / y
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 (x <= 1.45d-274) then
        tmp = x / (y / (y - z))
    else if (x <= 2d-62) then
        tmp = x - (z * (x / y))
    else
        tmp = x * (1.0d0 - (z / y))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.45e-274) {
		tmp = x / (y / (y - z));
	} else if (x <= 2e-62) {
		tmp = x - (z * (x / y));
	} else {
		tmp = x * (1.0 - (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * (y - z)) / y

↓

def code(x, y, z):
	tmp = 0
	if x <= 1.45e-274:
		tmp = x / (y / (y - z))
	elif x <= 2e-62:
		tmp = x - (z * (x / y))
	else:
		tmp = x * (1.0 - (z / y))
	return tmp

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.45e-274)
		tmp = Float64(x / Float64(y / Float64(y - z)));
	elseif (x <= 2e-62)
		tmp = Float64(x - Float64(z * Float64(x / y)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(z / y)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.45e-274)
		tmp = x / (y / (y - z));
	elseif (x <= 2e-62)
		tmp = x - (z * (x / y));
	else
		tmp = x * (1.0 - (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := If[LessEqual[x, 1.45e-274], N[(x / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-62], N[(x - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	81.0%
Target	95.2%
Herbie	96.1%

\[\begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if x < 1.44999999999999988e-274

   1. Initial program 80.6%

      \[\frac{x \cdot \left(y - z\right)}{y} \]

   2. Simplified 80.2%

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

      [Start] 80.6	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-*l/ [<=] 80.2	\[ \color{blue}{\frac{x}{y} \cdot \left(y - z\right)} \]

   3. Applied egg-rr 95.0%

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

      [Start] 80.2	\[ \frac{x}{y} \cdot \left(y - z\right) \]
      associate-*l/ [=>] 80.6	\[ \color{blue}{\frac{x \cdot \left(y - z\right)}{y}} \]
      associate-/l* [=>] 95.0	\[ \color{blue}{\frac{x}{\frac{y}{y - z}}} \]

   if 1.44999999999999988e-274 < x < 2.0000000000000001e-62

   1. Initial program 92.1%

      \[\frac{x \cdot \left(y - z\right)}{y} \]

   2. Simplified 94.5%

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

      [Start] 92.1	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-*l/ [<=] 74.9	\[ \color{blue}{\frac{x}{y} \cdot \left(y - z\right)} \]
      distribute-rgt-out-- [<=] 74.9	\[ \color{blue}{y \cdot \frac{x}{y} - z \cdot \frac{x}{y}} \]
      associate-*r/ [=>] 90.3	\[ \color{blue}{\frac{y \cdot x}{y}} - z \cdot \frac{x}{y} \]
      associate-*l/ [<=] 94.5	\[ \color{blue}{\frac{y}{y} \cdot x} - z \cdot \frac{x}{y} \]
      *-inverses [=>] 94.5	\[ \color{blue}{1} \cdot x - z \cdot \frac{x}{y} \]
      *-lft-identity [=>] 94.5	\[ \color{blue}{x} - z \cdot \frac{x}{y} \]

   if 2.0000000000000001e-62 < x

   1. Initial program 73.4%

      \[\frac{x \cdot \left(y - z\right)}{y} \]

   2. Simplified 90.2%

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

      [Start] 73.4	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-*l/ [<=] 89.1	\[ \color{blue}{\frac{x}{y} \cdot \left(y - z\right)} \]
      distribute-rgt-out-- [<=] 89.2	\[ \color{blue}{y \cdot \frac{x}{y} - z \cdot \frac{x}{y}} \]
      associate-*r/ [=>] 65.9	\[ \color{blue}{\frac{y \cdot x}{y}} - z \cdot \frac{x}{y} \]
      associate-*l/ [<=] 90.2	\[ \color{blue}{\frac{y}{y} \cdot x} - z \cdot \frac{x}{y} \]
      *-inverses [=>] 90.2	\[ \color{blue}{1} \cdot x - z \cdot \frac{x}{y} \]
      *-lft-identity [=>] 90.2	\[ \color{blue}{x} - z \cdot \frac{x}{y} \]

   3. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{\left(1 - \frac{z}{y}\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-274}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-62}:\\ \;\;\;\;x - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-273} \lor \neg \left(x \leq 2 \cdot 10^{-62}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{x}{y}\\ \end{array} \]

Alternative 2
Accuracy	67.9%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+83}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-108}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	65.8%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+83}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	66.2%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+83}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{-x}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	95.1%
Cost	448

\[x \cdot \left(1 - \frac{z}{y}\right) \]

Alternative 6
Accuracy	60.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023146 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))