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

Average Error: 12.9 → 2.0

Time: 6.3s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-20}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-200}:\\ \;\;\;\;x - \frac{1}{y} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-20)
   (- x (/ x (/ y z)))
   (if (<= y 7.2e-200) (- x (* (/ 1.0 y) (* x z))) (/ x (/ y (- y z))))))

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 (y <= -5e-20) {
		tmp = x - (x / (y / z));
	} else if (y <= 7.2e-200) {
		tmp = x - ((1.0 / y) * (x * z));
	} else {
		tmp = x / (y / (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 * (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 (y <= (-5d-20)) then
        tmp = x - (x / (y / z))
    else if (y <= 7.2d-200) then
        tmp = x - ((1.0d0 / y) * (x * z))
    else
        tmp = x / (y / (y - z))
    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 (y <= -5e-20) {
		tmp = x - (x / (y / z));
	} else if (y <= 7.2e-200) {
		tmp = x - ((1.0 / y) * (x * z));
	} else {
		tmp = x / (y / (y - z));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= -5e-20:
		tmp = x - (x / (y / z))
	elif y <= 7.2e-200:
		tmp = x - ((1.0 / y) * (x * z))
	else:
		tmp = x / (y / (y - z))
	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 (y <= -5e-20)
		tmp = Float64(x - Float64(x / Float64(y / z)));
	elseif (y <= 7.2e-200)
		tmp = Float64(x - Float64(Float64(1.0 / y) * Float64(x * z)));
	else
		tmp = Float64(x / Float64(y / Float64(y - z)));
	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 (y <= -5e-20)
		tmp = x - (x / (y / z));
	elseif (y <= 7.2e-200)
		tmp = x - ((1.0 / y) * (x * z));
	else
		tmp = x / (y / (y - z));
	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[y, -5e-20], N[(x - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e-200], N[(x - N[(N[(1.0 / y), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	12.9
Target	3.4
Herbie	2.0

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

Derivation

1. Split input into 3 regimes

2. if y < -4.9999999999999999e-20

   1. Initial program 16.5

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

   2. Simplified 0.1

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

      [Start] 16.5	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-*r/ [<=] 0.1	\[ \color{blue}{x \cdot \frac{y - z}{y}} \]
      div-sub [=>] 0.1	\[ x \cdot \color{blue}{\left(\frac{y}{y} - \frac{z}{y}\right)} \]
      distribute-rgt-out-- [<=] 0.1	\[ \color{blue}{\frac{y}{y} \cdot x - \frac{z}{y} \cdot x} \]
      *-inverses [=>] 0.1	\[ \color{blue}{1} \cdot x - \frac{z}{y} \cdot x \]
      *-lft-identity [=>] 0.1	\[ \color{blue}{x} - \frac{z}{y} \cdot x \]
      associate-*l/ [=>] 5.4	\[ x - \color{blue}{\frac{z \cdot x}{y}} \]
      *-commutative [<=] 5.4	\[ x - \frac{\color{blue}{x \cdot z}}{y} \]
      associate-/l* [=>] 0.1	\[ x - \color{blue}{\frac{x}{\frac{y}{z}}} \]

   if -4.9999999999999999e-20 < y < 7.2000000000000003e-200

   1. Initial program 8.3

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

   2. Simplified 8.8

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

      [Start] 8.3	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-*r/ [<=] 9.5	\[ \color{blue}{x \cdot \frac{y - z}{y}} \]
      div-sub [=>] 9.5	\[ x \cdot \color{blue}{\left(\frac{y}{y} - \frac{z}{y}\right)} \]
      distribute-rgt-out-- [<=] 9.5	\[ \color{blue}{\frac{y}{y} \cdot x - \frac{z}{y} \cdot x} \]
      *-inverses [=>] 9.5	\[ \color{blue}{1} \cdot x - \frac{z}{y} \cdot x \]
      *-lft-identity [=>] 9.5	\[ \color{blue}{x} - \frac{z}{y} \cdot x \]
      associate-*l/ [=>] 5.1	\[ x - \color{blue}{\frac{z \cdot x}{y}} \]
      *-commutative [<=] 5.1	\[ x - \frac{\color{blue}{x \cdot z}}{y} \]
      associate-/l* [=>] 8.8	\[ x - \color{blue}{\frac{x}{\frac{y}{z}}} \]

   3. Applied egg-rr 5.2

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

   if 7.2000000000000003e-200 < y

   1. Initial program 13.0

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

   2. Simplified 1.5

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

      [Start] 13.0	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-/l* [=>] 1.5	\[ \color{blue}{\frac{x}{\frac{y}{y - z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-20}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-200}:\\ \;\;\;\;x - \frac{1}{y} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.4
Cost	1480

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x\\ \mathbf{elif}\;t_0 \leq -500000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 2
Error	20.1
Cost	1177

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+45}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -950:\\ \;\;\;\;\frac{y \cdot x}{y}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-166} \lor \neg \left(y \leq -4.5 \cdot 10^{-213}\right) \land y \leq 7.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	20.3
Cost	913

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+45}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-168} \lor \neg \left(y \leq -3.2 \cdot 10^{-214}\right) \land y \leq 5.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	3.0
Cost	448

\[x - \frac{x}{\frac{y}{z}} \]

Alternative 5
Error	25.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023187 
(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))