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

Average Error: 10.3 → 0.3

Time: 7.4s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-38} \lor \neg \left(z \leq 5.6 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2e-38) (not (<= z 5.6e+54)))
   (/ x (/ z (+ (- y z) 1.0)))
   (/ (fma x (- y z) x) 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 <= -2e-38) || !(z <= 5.6e+54)) {
		tmp = x / (z / ((y - z) + 1.0));
	} else {
		tmp = fma(x, (y - z), x) / 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 <= -2e-38) || !(z <= 5.6e+54))
		tmp = Float64(x / Float64(z / Float64(Float64(y - z) + 1.0)));
	else
		tmp = Float64(fma(x, Float64(y - z), x) / 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[Or[LessEqual[z, -2e-38], N[Not[LessEqual[z, 5.6e+54]], $MachinePrecision]], N[(x / N[(z / N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y - z), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	10.3
Target	0.5
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if z < -1.9999999999999999e-38 or 5.6000000000000003e54 < z

   1. Initial program 16.9

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

   2. Simplified 0.1

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

      [Start] 16.9	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

   if -1.9999999999999999e-38 < z < 5.6000000000000003e54

   1. Initial program 0.6

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

   2. Simplified 0.6

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

      [Start] 0.6	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      distribute-lft-in [=>] 0.6	\[ \frac{\color{blue}{x \cdot \left(y - z\right) + x \cdot 1}}{z} \]
      *-rgt-identity [=>] 0.6	\[ \frac{x \cdot \left(y - z\right) + \color{blue}{x}}{z} \]
      fma-def [=>] 0.6	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y - z, x\right)}}{z} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-38} \lor \neg \left(z \leq 5.6 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-38} \lor \neg \left(z \leq 1.4 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	841

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -2 \cdot 10^{-39} \lor \neg \left(z \leq 10^{+55}\right):\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]

Alternative 3
Error	12.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+154} \lor \neg \left(y \leq 5.8 \cdot 10^{+98}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 4
Error	12.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5
Error	19.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -250000000000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6
Error	32.9
Cost	128

\[-x \]

Alternative 7
Error	62.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023073 
(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))