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

Average Accuracy: 84.1% → 98.3%

Time: 9.8s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \mathbf{elif}\;z \leq 1650000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.5e+34)
   (- (* y (/ x z)) x)
   (if (<= z 1650000.0) (/ (fma x (- y z) x) z) (- (/ x (/ z y)) 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 <= -4.5e+34) {
		tmp = (y * (x / z)) - x;
	} else if (z <= 1650000.0) {
		tmp = fma(x, (y - z), x) / z;
	} else {
		tmp = (x / (z / y)) - 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 <= -4.5e+34)
		tmp = Float64(Float64(y * Float64(x / z)) - x);
	elseif (z <= 1650000.0)
		tmp = Float64(fma(x, Float64(y - z), x) / z);
	else
		tmp = Float64(Float64(x / Float64(z / y)) - 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[LessEqual[z, -4.5e+34], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[z, 1650000.0], N[(N[(x * N[(y - z), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]]

Target

Original	84.1%
Target	99.2%
Herbie	98.3%

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

Derivation

1. Split input into 3 regimes

2. if z < -4.5e34

   1. Initial program 72.1%

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

   2. Simplified 90.5%

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

      [Start] 72.1	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-*r/ [<=] 99.9	\[ \color{blue}{x \cdot \frac{\left(y - z\right) + 1}{z}} \]
      +-commutative [=>] 99.9	\[ x \cdot \frac{\color{blue}{1 + \left(y - z\right)}}{z} \]
      associate-+r- [=>] 99.9	\[ x \cdot \frac{\color{blue}{\left(1 + y\right) - z}}{z} \]
      div-sub [=>] 99.9	\[ x \cdot \color{blue}{\left(\frac{1 + y}{z} - \frac{z}{z}\right)} \]
      *-inverses [=>] 99.9	\[ x \cdot \left(\frac{1 + y}{z} - \color{blue}{1}\right) \]
      distribute-rgt-out-- [<=] 99.9	\[ \color{blue}{\frac{1 + y}{z} \cdot x - 1 \cdot x} \]
      *-lft-identity [=>] 99.9	\[ \frac{1 + y}{z} \cdot x - \color{blue}{x} \]
      *-commutative [=>] 99.9	\[ \color{blue}{x \cdot \frac{1 + y}{z}} - x \]
      associate-*r/ [=>] 90.5	\[ \color{blue}{\frac{x \cdot \left(1 + y\right)}{z}} - x \]
      *-commutative [=>] 90.5	\[ \frac{\color{blue}{\left(1 + y\right) \cdot x}}{z} - x \]
      +-commutative [=>] 90.5	\[ \frac{\color{blue}{\left(y + 1\right)} \cdot x}{z} - x \]
      distribute-lft1-in [<=] 90.5	\[ \frac{\color{blue}{y \cdot x + x}}{z} - x \]
      *-commutative [=>] 90.5	\[ \frac{\color{blue}{x \cdot y} + x}{z} - x \]
      fma-def [=>] 90.5	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, x\right)}}{z} - x \]

   3. Taylor expanded in y around inf 90.5%

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

   4. Simplified 95.1%

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

      [Start] 90.5	\[ \frac{y \cdot x}{z} - x \]
      *-lft-identity [<=] 90.5	\[ \frac{\color{blue}{1 \cdot \left(y \cdot x\right)}}{z} - x \]
      associate-*l/ [<=] 90.5	\[ \color{blue}{\frac{1}{z} \cdot \left(y \cdot x\right)} - x \]
      *-commutative [=>] 90.5	\[ \frac{1}{z} \cdot \color{blue}{\left(x \cdot y\right)} - x \]
      associate-*r* [=>] 95.1	\[ \color{blue}{\left(\frac{1}{z} \cdot x\right) \cdot y} - x \]
      *-commutative [=>] 95.1	\[ \color{blue}{y \cdot \left(\frac{1}{z} \cdot x\right)} - x \]
      associate-*l/ [=>] 95.1	\[ y \cdot \color{blue}{\frac{1 \cdot x}{z}} - x \]
      metadata-eval [<=] 95.1	\[ y \cdot \frac{\color{blue}{\left(--1\right)} \cdot x}{z} - x \]
      distribute-lft-neg-in [<=] 95.1	\[ y \cdot \frac{\color{blue}{--1 \cdot x}}{z} - x \]
      mul-1-neg [=>] 95.1	\[ y \cdot \frac{-\color{blue}{\left(-x\right)}}{z} - x \]
      remove-double-neg [=>] 95.1	\[ y \cdot \frac{\color{blue}{x}}{z} - x \]

   if -4.5e34 < z < 1.65e6

   1. Initial program 99.5%

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

   2. Simplified 99.5%

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

      [Start] 99.5	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      distribute-lft-in [=>] 99.5	\[ \frac{\color{blue}{x \cdot \left(y - z\right) + x \cdot 1}}{z} \]
      *-rgt-identity [=>] 99.5	\[ \frac{x \cdot \left(y - z\right) + \color{blue}{x}}{z} \]
      fma-def [=>] 99.5	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y - z, x\right)}}{z} \]

   if 1.65e6 < z

   1. Initial program 72.6%

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

   2. Simplified 90.6%

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

      [Start] 72.6	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-*r/ [<=] 99.9	\[ \color{blue}{x \cdot \frac{\left(y - z\right) + 1}{z}} \]
      +-commutative [=>] 99.9	\[ x \cdot \frac{\color{blue}{1 + \left(y - z\right)}}{z} \]
      associate-+r- [=>] 99.9	\[ x \cdot \frac{\color{blue}{\left(1 + y\right) - z}}{z} \]
      div-sub [=>] 99.9	\[ x \cdot \color{blue}{\left(\frac{1 + y}{z} - \frac{z}{z}\right)} \]
      *-inverses [=>] 99.9	\[ x \cdot \left(\frac{1 + y}{z} - \color{blue}{1}\right) \]
      distribute-rgt-out-- [<=] 99.9	\[ \color{blue}{\frac{1 + y}{z} \cdot x - 1 \cdot x} \]
      *-lft-identity [=>] 99.9	\[ \frac{1 + y}{z} \cdot x - \color{blue}{x} \]
      *-commutative [=>] 99.9	\[ \color{blue}{x \cdot \frac{1 + y}{z}} - x \]
      associate-*r/ [=>] 90.6	\[ \color{blue}{\frac{x \cdot \left(1 + y\right)}{z}} - x \]
      *-commutative [=>] 90.6	\[ \frac{\color{blue}{\left(1 + y\right) \cdot x}}{z} - x \]
      +-commutative [=>] 90.6	\[ \frac{\color{blue}{\left(y + 1\right)} \cdot x}{z} - x \]
      distribute-lft1-in [<=] 90.6	\[ \frac{\color{blue}{y \cdot x + x}}{z} - x \]
      *-commutative [=>] 90.6	\[ \frac{\color{blue}{x \cdot y} + x}{z} - x \]
      fma-def [=>] 90.6	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, x\right)}}{z} - x \]

   3. Taylor expanded in y around inf 90.3%

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

   4. Simplified 95.7%

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

      [Start] 90.3	\[ \frac{y \cdot x}{z} - x \]
      *-lft-identity [<=] 90.3	\[ \frac{\color{blue}{1 \cdot \left(y \cdot x\right)}}{z} - x \]
      associate-*l/ [<=] 90.2	\[ \color{blue}{\frac{1}{z} \cdot \left(y \cdot x\right)} - x \]
      *-commutative [=>] 90.2	\[ \frac{1}{z} \cdot \color{blue}{\left(x \cdot y\right)} - x \]
      associate-*r* [=>] 95.6	\[ \color{blue}{\left(\frac{1}{z} \cdot x\right) \cdot y} - x \]
      *-commutative [=>] 95.6	\[ \color{blue}{y \cdot \left(\frac{1}{z} \cdot x\right)} - x \]
      associate-*l/ [=>] 95.7	\[ y \cdot \color{blue}{\frac{1 \cdot x}{z}} - x \]
      metadata-eval [<=] 95.7	\[ y \cdot \frac{\color{blue}{\left(--1\right)} \cdot x}{z} - x \]
      distribute-lft-neg-in [<=] 95.7	\[ y \cdot \frac{\color{blue}{--1 \cdot x}}{z} - x \]
      mul-1-neg [=>] 95.7	\[ y \cdot \frac{-\color{blue}{\left(-x\right)}}{z} - x \]
      remove-double-neg [=>] 95.7	\[ y \cdot \frac{\color{blue}{x}}{z} - x \]

   5. Taylor expanded in y around 0 90.3%

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

   6. Simplified 99.6%

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

      [Start] 90.3	\[ \frac{y \cdot x}{z} - x \]
      *-commutative [=>] 90.3	\[ \frac{\color{blue}{x \cdot y}}{z} - x \]
      associate-/l* [=>] 99.6	\[ \color{blue}{\frac{x}{\frac{z}{y}}} - x \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \mathbf{elif}\;z \leq 1650000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	68.3%
Cost	980

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-204}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3800000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 2
Accuracy	98.3%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + \left(1 - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

Alternative 3
Accuracy	96.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -27000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 4
Accuracy	97.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.97:\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x + y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \end{array} \]

Alternative 5
Accuracy	82.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -10800000 \lor \neg \left(y \leq 8.4 \cdot 10^{+55}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 6
Accuracy	69.8%
Cost	456

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

Alternative 7
Accuracy	48.2%
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023140 
(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))