Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Average Error: 10.3 → 0.1

Time: 6.9s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -0.11 \lor \neg \left(z \leq 20500000\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.11) (not (<= z 20500000.0)))
   (/ x (/ z (+ (- y z) 1.0)))
   (- (/ (fma x y x) z) x)))

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

↓

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.11) || !(z <= 20500000.0)) {
		tmp = x / (z / ((y - z) + 1.0));
	} else {
		tmp = (fma(x, y, x) / z) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z)
end

↓

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.11) || !(z <= 20500000.0))
		tmp = Float64(x / Float64(z / Float64(Float64(y - z) + 1.0)));
	else
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := If[Or[LessEqual[z, -0.11], N[Not[LessEqual[z, 20500000.0]], $MachinePrecision]], N[(x / N[(z / N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision]]

Target

Original	10.3
Target	0.4
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if z < -0.110000000000000001 or 2.05e7 < z

   1. Initial program 17.0

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

   2. Simplified 0.1

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

      [Start] 17.0	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

   if -0.110000000000000001 < z < 2.05e7

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-*r/ [<=] 8.7	\[ \color{blue}{x \cdot \frac{\left(y - z\right) + 1}{z}} \]
      +-commutative [=>] 8.7	\[ x \cdot \frac{\color{blue}{1 + \left(y - z\right)}}{z} \]
      associate-+r- [=>] 8.7	\[ x \cdot \frac{\color{blue}{\left(1 + y\right) - z}}{z} \]
      div-sub [=>] 8.7	\[ x \cdot \color{blue}{\left(\frac{1 + y}{z} - \frac{z}{z}\right)} \]
      *-inverses [=>] 8.7	\[ x \cdot \left(\frac{1 + y}{z} - \color{blue}{1}\right) \]
      distribute-rgt-out-- [<=] 8.7	\[ \color{blue}{\frac{1 + y}{z} \cdot x - 1 \cdot x} \]
      *-lft-identity [=>] 8.7	\[ \frac{1 + y}{z} \cdot x - \color{blue}{x} \]
      *-commutative [=>] 8.7	\[ \color{blue}{x \cdot \frac{1 + y}{z}} - x \]
      associate-*r/ [=>] 0.1	\[ \color{blue}{\frac{x \cdot \left(1 + y\right)}{z}} - x \]
      *-commutative [=>] 0.1	\[ \frac{\color{blue}{\left(1 + y\right) \cdot x}}{z} - x \]
      +-commutative [=>] 0.1	\[ \frac{\color{blue}{\left(y + 1\right)} \cdot x}{z} - x \]
      distribute-lft1-in [<=] 0.1	\[ \frac{\color{blue}{y \cdot x + x}}{z} - x \]
      *-commutative [=>] 0.1	\[ \frac{\color{blue}{x \cdot y} + x}{z} - x \]
      fma-def [=>] 0.1	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, x\right)}}{z} - x \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.11 \lor \neg \left(z \leq 20500000\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \end{array} \]

Alternatives

Alternative 1
Error	12.3
Cost	849

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{+75}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+19} \lor \neg \left(y \leq 1.55 \cdot 10^{+100}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 2
Error	12.3
Cost	848

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.75 \cdot 10^{+72}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-15} \lor \neg \left(z \leq 5 \cdot 10^{-56}\right):\\ \;\;\;\;x \cdot \frac{y + \left(1 - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 4
Error	0.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-17} \lor \neg \left(z \leq 1.3 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 5
Error	0.1
Cost	841

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -20 \lor \neg \left(z \leq 100000\right):\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]

Alternative 6
Error	2.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 7
Error	9.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+32}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 0.0155:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 8
Error	19.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -600000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 9
Error	32.9
Cost	128

\[-x \]

Alternative 10
Error	62.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))