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

Average Error: 10.4 → 0.1

Time: 7.0s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -14000000:\\ \;\;\;\;\frac{y + \left(1 - z\right)}{z} \cdot x\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -14000000.0)
   (* (/ (+ y (- 1.0 z)) z) x)
   (if (<= z 6.7e+15) (/ (fma x (- y z) x) z) (* x (+ -1.0 (/ y z))))))

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 <= -14000000.0) {
		tmp = ((y + (1.0 - z)) / z) * x;
	} else if (z <= 6.7e+15) {
		tmp = fma(x, (y - z), x) / z;
	} else {
		tmp = x * (-1.0 + (y / z));
	}
	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 <= -14000000.0)
		tmp = Float64(Float64(Float64(y + Float64(1.0 - z)) / z) * x);
	elseif (z <= 6.7e+15)
		tmp = Float64(fma(x, Float64(y - z), x) / z);
	else
		tmp = Float64(x * Float64(-1.0 + Float64(y / z)));
	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[LessEqual[z, -14000000.0], N[(N[(N[(y + N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 6.7e+15], N[(N[(x * N[(y - z), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	10.4
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 3 regimes

2. if z < -1.4e7

   1. Initial program 17.4

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

   2. Applied egg-rr 0.1

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

   if -1.4e7 < z < 6.7e15

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      distribute-lft-in [=>] 0.2	\[ \frac{\color{blue}{x \cdot \left(y - z\right) + x \cdot 1}}{z} \]
      *-rgt-identity [=>] 0.2	\[ \frac{x \cdot \left(y - z\right) + \color{blue}{x}}{z} \]
      fma-def [=>] 0.2	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y - z, x\right)}}{z} \]

   if 6.7e15 < z

   1. Initial program 17.7

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

   2. Simplified 5.7

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

      [Start] 17.7	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-*r/ [<=] 0.1	\[ \color{blue}{x \cdot \frac{\left(y - z\right) + 1}{z}} \]
      +-commutative [=>] 0.1	\[ x \cdot \frac{\color{blue}{1 + \left(y - z\right)}}{z} \]
      associate-+r- [=>] 0.1	\[ x \cdot \frac{\color{blue}{\left(1 + y\right) - z}}{z} \]
      div-sub [=>] 0.1	\[ x \cdot \color{blue}{\left(\frac{1 + y}{z} - \frac{z}{z}\right)} \]
      *-inverses [=>] 0.1	\[ x \cdot \left(\frac{1 + y}{z} - \color{blue}{1}\right) \]
      distribute-rgt-out-- [<=] 0.1	\[ \color{blue}{\frac{1 + y}{z} \cdot x - 1 \cdot x} \]
      *-lft-identity [=>] 0.1	\[ \frac{1 + y}{z} \cdot x - \color{blue}{x} \]
      *-commutative [=>] 0.1	\[ \color{blue}{x \cdot \frac{1 + y}{z}} - x \]
      associate-*r/ [=>] 5.7	\[ \color{blue}{\frac{x \cdot \left(1 + y\right)}{z}} - x \]
      *-commutative [=>] 5.7	\[ \frac{\color{blue}{\left(1 + y\right) \cdot x}}{z} - x \]
      +-commutative [=>] 5.7	\[ \frac{\color{blue}{\left(y + 1\right)} \cdot x}{z} - x \]
      distribute-lft1-in [<=] 5.7	\[ \frac{\color{blue}{y \cdot x + x}}{z} - x \]
      *-commutative [=>] 5.7	\[ \frac{\color{blue}{x \cdot y} + x}{z} - x \]
      fma-def [=>] 5.7	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, x\right)}}{z} - x \]

   3. Taylor expanded in y around inf 5.7

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

   4. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{\left(\frac{y}{z} - 1\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -14000000:\\ \;\;\;\;\frac{y + \left(1 - z\right)}{z} \cdot x\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{-15} \lor \neg \left(z \leq 1.05 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{y + \left(1 - z\right)}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -20000000:\\ \;\;\;\;\frac{y + \left(1 - z\right)}{z} \cdot x\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \end{array} \]

Alternative 3
Error	4.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -400000 \lor \neg \left(y \leq 0.0285\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 4
Error	1.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 5
Error	20.0
Cost	588

\[\begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+25}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6
Error	11.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+107} \lor \neg \left(y \leq 9 \cdot 10^{+47}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 7
Error	19.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 8
Error	33.0
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023187 
(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))