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

Average Error: 19.84% → 0.81%

Time: 6.7s

Precision: binary64

Cost: 8332

Math

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

↓

\[\begin{array}{l} t_0 := x - \frac{x \cdot z}{y}\\ t_1 := \frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y - z}{\frac{y}{x}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{-z}{y}, x\right)\\ \mathbf{elif}\;t_1 \leq 10^{+258}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (/ (* x z) y))) (t_1 (/ (* x (- y z)) y)))
   (if (<= t_1 (- INFINITY))
     (/ (- y z) (/ y x))
     (if (<= t_1 -2e+18)
       t_0
       (if (<= t_1 2e-121)
         (fma x (/ (- z) y) x)
         (if (<= t_1 1e+258) t_0 (- x (/ x (/ 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 t_0 = x - ((x * z) / y);
	double t_1 = (x * (y - z)) / y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y - z) / (y / x);
	} else if (t_1 <= -2e+18) {
		tmp = t_0;
	} else if (t_1 <= 2e-121) {
		tmp = fma(x, (-z / y), x);
	} else if (t_1 <= 1e+258) {
		tmp = t_0;
	} else {
		tmp = x - (x / (y / z));
	}
	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(x - Float64(Float64(x * z) / y))
	t_1 = Float64(Float64(x * Float64(y - z)) / y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y - z) / Float64(y / x));
	elseif (t_1 <= -2e+18)
		tmp = t_0;
	elseif (t_1 <= 2e-121)
		tmp = fma(x, Float64(Float64(-z) / y), x);
	elseif (t_1 <= 1e+258)
		tmp = t_0;
	else
		tmp = Float64(x - Float64(x / Float64(y / z)));
	end
	return 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[(x - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y - z), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+18], t$95$0, If[LessEqual[t$95$1, 2e-121], N[(x * N[((-z) / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+258], t$95$0, N[(x - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	19.84%
Target	4.91%
Herbie	0.81%

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -inf.0

   1. Initial program 100

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

   2. Simplified 0.21

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

      [Start] 100	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      *-commutative [=>] 100	\[ \frac{\color{blue}{\left(y - z\right) \cdot x}}{y} \]
      associate-/l* [=>] 0.21	\[ \color{blue}{\frac{y - z}{\frac{y}{x}}} \]

   if -inf.0 < (/.f64 (*.f64 x (-.f64 y z)) y) < -2e18 or 2e-121 < (/.f64 (*.f64 x (-.f64 y z)) y) < 1.00000000000000006e258

   1. Initial program 0.49

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

   2. Simplified 8.36

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

      [Start] 0.49	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-*r/ [<=] 8.96	\[ \color{blue}{x \cdot \frac{y - z}{y}} \]
      div-sub [=>] 8.95	\[ x \cdot \color{blue}{\left(\frac{y}{y} - \frac{z}{y}\right)} \]
      distribute-rgt-out-- [<=] 8.94	\[ \color{blue}{\frac{y}{y} \cdot x - \frac{z}{y} \cdot x} \]
      *-inverses [=>] 8.94	\[ \color{blue}{1} \cdot x - \frac{z}{y} \cdot x \]
      *-lft-identity [=>] 8.94	\[ \color{blue}{x} - \frac{z}{y} \cdot x \]
      associate-*l/ [=>] 0.36	\[ x - \color{blue}{\frac{z \cdot x}{y}} \]
      *-commutative [<=] 0.36	\[ x - \frac{\color{blue}{x \cdot z}}{y} \]
      associate-/l* [=>] 8.36	\[ x - \color{blue}{\frac{x}{\frac{y}{z}}} \]

   3. Taylor expanded in x around 0 0.36

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

   if -2e18 < (/.f64 (*.f64 x (-.f64 y z)) y) < 2e-121

   1. Initial program 12.21

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

   2. Simplified 0.15

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

      [Start] 12.21	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      *-commutative [=>] 12.21	\[ \frac{\color{blue}{\left(y - z\right) \cdot x}}{y} \]
      associate-*l/ [<=] 0.17	\[ \color{blue}{\frac{y - z}{y} \cdot x} \]
      div-sub [=>] 0.17	\[ \color{blue}{\left(\frac{y}{y} - \frac{z}{y}\right)} \cdot x \]
      sub-neg [=>] 0.17	\[ \color{blue}{\left(\frac{y}{y} + \left(-\frac{z}{y}\right)\right)} \cdot x \]
      +-commutative [=>] 0.17	\[ \color{blue}{\left(\left(-\frac{z}{y}\right) + \frac{y}{y}\right)} \cdot x \]
      *-inverses [=>] 0.17	\[ \left(\left(-\frac{z}{y}\right) + \color{blue}{1}\right) \cdot x \]
      distribute-lft1-in [<=] 0.15	\[ \color{blue}{\left(-\frac{z}{y}\right) \cdot x + x} \]
      *-commutative [=>] 0.15	\[ \color{blue}{x \cdot \left(-\frac{z}{y}\right)} + x \]
      fma-def [=>] 0.15	\[ \color{blue}{\mathsf{fma}\left(x, -\frac{z}{y}, x\right)} \]
      distribute-neg-frac [=>] 0.15	\[ \mathsf{fma}\left(x, \color{blue}{\frac{-z}{y}}, x\right) \]

   if 1.00000000000000006e258 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 76.22

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

   2. Simplified 5.83

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

      [Start] 76.22	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-*r/ [<=] 5.95	\[ \color{blue}{x \cdot \frac{y - z}{y}} \]
      div-sub [=>] 5.95	\[ x \cdot \color{blue}{\left(\frac{y}{y} - \frac{z}{y}\right)} \]
      distribute-rgt-out-- [<=] 5.94	\[ \color{blue}{\frac{y}{y} \cdot x - \frac{z}{y} \cdot x} \]
      *-inverses [=>] 5.94	\[ \color{blue}{1} \cdot x - \frac{z}{y} \cdot x \]
      *-lft-identity [=>] 5.94	\[ \color{blue}{x} - \frac{z}{y} \cdot x \]
      associate-*l/ [=>] 26.21	\[ x - \color{blue}{\frac{z \cdot x}{y}} \]
      *-commutative [<=] 26.21	\[ x - \frac{\color{blue}{x \cdot z}}{y} \]
      associate-/l* [=>] 5.83	\[ x - \color{blue}{\frac{x}{\frac{y}{z}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 0.81

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;\frac{y - z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -2 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{-z}{y}, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 10^{+258}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.87%
Cost	2513

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{+18} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 10^{+258}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 2
Error	0.86%
Cost	2513

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{y - z}{\frac{y}{x}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{+18} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 10^{+258}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 3
Error	5.05%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-15} \lor \neg \left(y \leq 4 \cdot 10^{-198}\right):\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4
Error	12.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+171}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+192}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	29.81%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -3950000 \lor \neg \left(z \leq 2.36 \cdot 10^{+58}\right):\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	29.84%
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -1050000:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \end{array} \]

Alternative 7
Error	39.63%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023115 
(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))