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

Average Accuracy: 80.0% → 99.3%

Time: 6.8s

Precision: binary64

Cost: 8332

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x - x \cdot \frac{z}{y}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{-z}{y}, x\right)\\ \mathbf{elif}\;t_0 \leq 9.08 \cdot 10^{+271}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (- y z)) y)))
   (if (<= t_0 (- INFINITY))
     (- x (* x (/ z y)))
     (if (<= t_0 -1e-78)
       t_0
       (if (<= t_0 2e+34)
         (fma x (/ (- z) y) x)
         (if (<= t_0 9.08e+271) 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 * (y - z)) / y;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x - (x * (z / y));
	} else if (t_0 <= -1e-78) {
		tmp = t_0;
	} else if (t_0 <= 2e+34) {
		tmp = fma(x, (-z / y), x);
	} else if (t_0 <= 9.08e+271) {
		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(Float64(x * Float64(y - z)) / y)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x - Float64(x * Float64(z / y)));
	elseif (t_0 <= -1e-78)
		tmp = t_0;
	elseif (t_0 <= 2e+34)
		tmp = fma(x, Float64(Float64(-z) / y), x);
	elseif (t_0 <= 9.08e+271)
		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[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x - N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e-78], t$95$0, If[LessEqual[t$95$0, 2e+34], N[(x * N[((-z) / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 9.08e+271], t$95$0, N[(x - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	80.0%
Target	94.7%
Herbie	99.3%

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 0.0%

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

   2. Simplified 99.9%

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

      [Start] 0.0	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-*r/ [<=] 99.9	\[ \color{blue}{x \cdot \frac{y - z}{y}} \]
      div-sub [=>] 99.9	\[ x \cdot \color{blue}{\left(\frac{y}{y} - \frac{z}{y}\right)} \]
      distribute-rgt-out-- [<=] 99.9	\[ \color{blue}{\frac{y}{y} \cdot x - \frac{z}{y} \cdot x} \]
      *-inverses [=>] 99.9	\[ \color{blue}{1} \cdot x - \frac{z}{y} \cdot x \]
      *-lft-identity [=>] 99.9	\[ \color{blue}{x} - \frac{z}{y} \cdot x \]
      associate-*l/ [=>] 66.2	\[ x - \color{blue}{\frac{z \cdot x}{y}} \]
      *-commutative [<=] 66.2	\[ x - \frac{\color{blue}{x \cdot z}}{y} \]
      associate-/l* [=>] 99.9	\[ x - \color{blue}{\frac{x}{\frac{y}{z}}} \]

   3. Applied egg-rr 99.9%

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

      [Start] 99.9	\[ x - \frac{x}{\frac{y}{z}} \]
      associate-/l* [<=] 66.2	\[ x - \color{blue}{\frac{x \cdot z}{y}} \]
      *-commutative [=>] 66.2	\[ x - \frac{\color{blue}{z \cdot x}}{y} \]
      associate-*l/ [<=] 99.9	\[ x - \color{blue}{\frac{z}{y} \cdot x} \]

   if -inf.0 < (/.f64 (*.f64 x (-.f64 y z)) y) < -9.99999999999999999e-79 or 1.99999999999999989e34 < (/.f64 (*.f64 x (-.f64 y z)) y) < 9.0800000000000003e271

   1. Initial program 99.5%

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

   if -9.99999999999999999e-79 < (/.f64 (*.f64 x (-.f64 y z)) y) < 1.99999999999999989e34

   1. Initial program 88.6%

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

   2. Simplified 99.8%

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

      [Start] 88.6	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      *-commutative [=>] 88.6	\[ \frac{\color{blue}{\left(y - z\right) \cdot x}}{y} \]
      associate-*l/ [<=] 99.8	\[ \color{blue}{\frac{y - z}{y} \cdot x} \]
      div-sub [=>] 99.8	\[ \color{blue}{\left(\frac{y}{y} - \frac{z}{y}\right)} \cdot x \]
      sub-neg [=>] 99.8	\[ \color{blue}{\left(\frac{y}{y} + \left(-\frac{z}{y}\right)\right)} \cdot x \]
      +-commutative [=>] 99.8	\[ \color{blue}{\left(\left(-\frac{z}{y}\right) + \frac{y}{y}\right)} \cdot x \]
      *-inverses [=>] 99.8	\[ \left(\left(-\frac{z}{y}\right) + \color{blue}{1}\right) \cdot x \]
      distribute-lft1-in [<=] 99.8	\[ \color{blue}{\left(-\frac{z}{y}\right) \cdot x + x} \]
      *-commutative [=>] 99.8	\[ \color{blue}{x \cdot \left(-\frac{z}{y}\right)} + x \]
      fma-def [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(x, -\frac{z}{y}, x\right)} \]
      distribute-neg-frac [=>] 99.8	\[ \mathsf{fma}\left(x, \color{blue}{\frac{-z}{y}}, x\right) \]

   if 9.0800000000000003e271 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 20.7%

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

   2. Simplified 96.1%

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

      [Start] 20.7	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-*r/ [<=] 96.0	\[ \color{blue}{x \cdot \frac{y - z}{y}} \]
      div-sub [=>] 96.0	\[ x \cdot \color{blue}{\left(\frac{y}{y} - \frac{z}{y}\right)} \]
      distribute-rgt-out-- [<=] 96.0	\[ \color{blue}{\frac{y}{y} \cdot x - \frac{z}{y} \cdot x} \]
      *-inverses [=>] 96.0	\[ \color{blue}{1} \cdot x - \frac{z}{y} \cdot x \]
      *-lft-identity [=>] 96.0	\[ \color{blue}{x} - \frac{z}{y} \cdot x \]
      associate-*l/ [=>] 76.2	\[ x - \color{blue}{\frac{z \cdot x}{y}} \]
      *-commutative [<=] 76.2	\[ x - \frac{\color{blue}{x \cdot z}}{y} \]
      associate-/l* [=>] 96.1	\[ x - \color{blue}{\frac{x}{\frac{y}{z}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;x - x \cdot \frac{z}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -1 \cdot 10^{-78}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{-z}{y}, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 9.08 \cdot 10^{+271}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	2512

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ t_1 := x - x \cdot \frac{z}{y}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 9.08 \cdot 10^{+271}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 2
Accuracy	68.2%
Cost	1444

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+80} \lor \neg \left(z \leq -0.1\right) \land \left(z \leq -2.6 \cdot 10^{-30} \lor \neg \left(z \leq 1.02 \cdot 10^{-117}\right) \land \left(z \leq 2 \cdot 10^{-94} \lor \neg \left(z \leq 1.05 \cdot 10^{+77}\right) \land \left(z \leq 1.5 \cdot 10^{+96} \lor \neg \left(z \leq 1.75 \cdot 10^{+167}\right)\right)\right)\right):\\ \;\;\;\;z \cdot \left(-\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	68.1%
Cost	1440

\[\begin{array}{l} t_0 := z \cdot \left(-\frac{x}{y}\right)\\ t_1 := x \cdot \frac{-z}{y}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -46:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+168}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	67.9%
Cost	1440

\[\begin{array}{l} t_0 := \frac{x \cdot \left(-z\right)}{y}\\ t_1 := x \cdot \frac{-z}{y}\\ \mathbf{if}\;z \leq -9 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -0.15:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-30}:\\ \;\;\;\;z \cdot \left(-\frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+75}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+167}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	87.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+203}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+124}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	94.8%
Cost	448

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

Alternative 7
Accuracy	95.2%
Cost	448

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

Alternative 8
Accuracy	59.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(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))