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

Average Error: 19.96% → 2.53%

Time: 6.1s

Precision: binary64

Cost: 2512

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+289}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{elif}\;t_0 \leq -4 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+70}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (- y z)) y)))
   (if (<= t_0 -5e+289)
     (* x (- 1.0 (/ z y)))
     (if (<= t_0 -4e-81)
       t_0
       (if (<= t_0 4e+70) (/ x (/ y (- y z))) (if (<= t_0 5e+282) t_0 x))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (x * (y - z)) / y;
	double tmp;
	if (t_0 <= -5e+289) {
		tmp = x * (1.0 - (z / y));
	} else if (t_0 <= -4e-81) {
		tmp = t_0;
	} else if (t_0 <= 4e+70) {
		tmp = x / (y / (y - z));
	} else if (t_0 <= 5e+282) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (x * (y - z)) / y
    if (t_0 <= (-5d+289)) then
        tmp = x * (1.0d0 - (z / y))
    else if (t_0 <= (-4d-81)) then
        tmp = t_0
    else if (t_0 <= 4d+70) then
        tmp = x / (y / (y - z))
    else if (t_0 <= 5d+282) then
        tmp = t_0
    else
        tmp = x
    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 t_0 = (x * (y - z)) / y;
	double tmp;
	if (t_0 <= -5e+289) {
		tmp = x * (1.0 - (z / y));
	} else if (t_0 <= -4e-81) {
		tmp = t_0;
	} else if (t_0 <= 4e+70) {
		tmp = x / (y / (y - z));
	} else if (t_0 <= 5e+282) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x * (y - z)) / y
	tmp = 0
	if t_0 <= -5e+289:
		tmp = x * (1.0 - (z / y))
	elif t_0 <= -4e-81:
		tmp = t_0
	elif t_0 <= 4e+70:
		tmp = x / (y / (y - z))
	elif t_0 <= 5e+282:
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(x * Float64(y - z)) / y)
	tmp = 0.0
	if (t_0 <= -5e+289)
		tmp = Float64(x * Float64(1.0 - Float64(z / y)));
	elseif (t_0 <= -4e-81)
		tmp = t_0;
	elseif (t_0 <= 4e+70)
		tmp = Float64(x / Float64(y / Float64(y - z)));
	elseif (t_0 <= 5e+282)
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (x * (y - z)) / y;
	tmp = 0.0;
	if (t_0 <= -5e+289)
		tmp = x * (1.0 - (z / y));
	elseif (t_0 <= -4e-81)
		tmp = t_0;
	elseif (t_0 <= 4e+70)
		tmp = x / (y / (y - z));
	elseif (t_0 <= 5e+282)
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+289], N[(x * N[(1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -4e-81], t$95$0, If[LessEqual[t$95$0, 4e+70], N[(x / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+282], t$95$0, x]]]]]

Target

Original	19.96%
Target	5.08%
Herbie	2.53%

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -5.00000000000000031e289

   1. Initial program 89.52

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

   2. Simplified 5.49

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

      [Start] 89.52	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      *-commutative [=>] 89.52	\[ \frac{\color{blue}{\left(y - z\right) \cdot x}}{y} \]
      associate-*r/ [<=] 5.49	\[ \color{blue}{\left(y - z\right) \cdot \frac{x}{y}} \]

   3. Taylor expanded in x around 0 89.52

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

   4. Simplified 2.18

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

      [Start] 89.52	\[ \frac{\left(y - z\right) \cdot x}{y} \]
      associate-*l/ [<=] 2.18	\[ \color{blue}{\frac{y - z}{y} \cdot x} \]
      *-commutative [=>] 2.18	\[ \color{blue}{x \cdot \frac{y - z}{y}} \]
      div-sub [=>] 2.18	\[ x \cdot \color{blue}{\left(\frac{y}{y} - \frac{z}{y}\right)} \]
      *-inverses [=>] 2.18	\[ x \cdot \left(\color{blue}{1} - \frac{z}{y}\right) \]

   if -5.00000000000000031e289 < (/.f64 (*.f64 x (-.f64 y z)) y) < -3.9999999999999998e-81 or 4.00000000000000029e70 < (/.f64 (*.f64 x (-.f64 y z)) y) < 4.99999999999999978e282

   1. Initial program 0.37

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

   if -3.9999999999999998e-81 < (/.f64 (*.f64 x (-.f64 y z)) y) < 4.00000000000000029e70

   1. Initial program 10.47

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

   2. Simplified 0.4

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

      [Start] 10.47	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-/l* [=>] 0.4	\[ \color{blue}{\frac{x}{\frac{y}{y - z}}} \]

   if 4.99999999999999978e282 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 87.11

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

   2. Simplified 3.19

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

      [Start] 87.11	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-/l* [=>] 3.19	\[ \color{blue}{\frac{x}{\frac{y}{y - z}}} \]

   3. Taylor expanded in y around inf 24.04

      \[\leadsto \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 2.53

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -5 \cdot 10^{+289}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -4 \cdot 10^{-81}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 4 \cdot 10^{+70}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Error	5.55%
Cost	845

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+66} \lor \neg \left(z \leq 3.4 \cdot 10^{+193}\right) \land z \leq 7 \cdot 10^{+255}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]

Alternative 2
Error	5.27%
Cost	844

\[\begin{array}{l} t_0 := \left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+194}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+255}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]

Alternative 3
Error	29.68%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	29.41%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	5.42%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+211}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]

Alternative 6
Error	39.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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))