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

Average Accuracy: 83.9% → 99.8%

Time: 7.7s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\left(\frac{y}{z} + -1\right) \cdot x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e+28)
   (* (+ (/ y z) -1.0) x)
   (if (<= z 5e+54) (- (/ (fma x y x) z) x) (/ x (/ z (+ 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 <= -1e+28) {
		tmp = ((y / z) + -1.0) * x;
	} else if (z <= 5e+54) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = x / (z / (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 <= -1e+28)
		tmp = Float64(Float64(Float64(y / z) + -1.0) * x);
	elseif (z <= 5e+54)
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	else
		tmp = Float64(x / Float64(z / 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, -1e+28], N[(N[(N[(y / z), $MachinePrecision] + -1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 5e+54], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(x / N[(z / N[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	83.9%
Target	99.4%
Herbie	99.8%

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

Derivation

1. Split input into 3 regimes

2. if z < -9.99999999999999958e27

   1. Initial program 71.6%

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

   2. Simplified 90.4%

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

      [Start] 71.6	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      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/ [=>] 90.4	\[ \color{blue}{\frac{x \cdot \left(1 + y\right)}{z}} - x \]
      *-commutative [=>] 90.4	\[ \frac{\color{blue}{\left(1 + y\right) \cdot x}}{z} - x \]
      +-commutative [=>] 90.4	\[ \frac{\color{blue}{\left(y + 1\right)} \cdot x}{z} - x \]
      distribute-lft1-in [<=] 90.4	\[ \frac{\color{blue}{y \cdot x + x}}{z} - x \]
      *-commutative [=>] 90.4	\[ \frac{\color{blue}{x \cdot y} + x}{z} - x \]
      fma-def [=>] 90.4	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, x\right)}}{z} - x \]

   3. Taylor expanded in y around inf 90.4%

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

   4. Simplified 96.2%

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

      [Start] 90.4	\[ \frac{y \cdot x}{z} - x \]
      *-lft-identity [<=] 90.4	\[ \frac{\color{blue}{1 \cdot \left(y \cdot x\right)}}{z} - x \]
      associate-*l/ [<=] 90.3	\[ \color{blue}{\frac{1}{z} \cdot \left(y \cdot x\right)} - x \]
      *-commutative [=>] 90.3	\[ \frac{1}{z} \cdot \color{blue}{\left(x \cdot y\right)} - x \]
      associate-*r* [=>] 96.2	\[ \color{blue}{\left(\frac{1}{z} \cdot x\right) \cdot y} - x \]
      *-commutative [=>] 96.2	\[ \color{blue}{y \cdot \left(\frac{1}{z} \cdot x\right)} - x \]
      associate-*l/ [=>] 96.2	\[ y \cdot \color{blue}{\frac{1 \cdot x}{z}} - x \]
      metadata-eval [<=] 96.2	\[ y \cdot \frac{\color{blue}{\left(--1\right)} \cdot x}{z} - x \]
      distribute-lft-neg-in [<=] 96.2	\[ y \cdot \frac{\color{blue}{--1 \cdot x}}{z} - x \]
      mul-1-neg [=>] 96.2	\[ y \cdot \frac{-\color{blue}{\left(-x\right)}}{z} - x \]
      remove-double-neg [=>] 96.2	\[ y \cdot \frac{\color{blue}{x}}{z} - x \]

   5. Taylor expanded in x around 0 99.9%

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

   if -9.99999999999999958e27 < z < 5.00000000000000005e54

   1. Initial program 99.2%

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

   2. Simplified 99.7%

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

      [Start] 99.2	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-*r/ [<=] 88.2	\[ \color{blue}{x \cdot \frac{\left(y - z\right) + 1}{z}} \]
      +-commutative [=>] 88.2	\[ x \cdot \frac{\color{blue}{1 + \left(y - z\right)}}{z} \]
      associate-+r- [=>] 88.2	\[ x \cdot \frac{\color{blue}{\left(1 + y\right) - z}}{z} \]
      div-sub [=>] 88.2	\[ x \cdot \color{blue}{\left(\frac{1 + y}{z} - \frac{z}{z}\right)} \]
      *-inverses [=>] 88.2	\[ x \cdot \left(\frac{1 + y}{z} - \color{blue}{1}\right) \]
      distribute-rgt-out-- [<=] 88.2	\[ \color{blue}{\frac{1 + y}{z} \cdot x - 1 \cdot x} \]
      *-lft-identity [=>] 88.2	\[ \frac{1 + y}{z} \cdot x - \color{blue}{x} \]
      *-commutative [=>] 88.2	\[ \color{blue}{x \cdot \frac{1 + y}{z}} - x \]
      associate-*r/ [=>] 99.6	\[ \color{blue}{\frac{x \cdot \left(1 + y\right)}{z}} - x \]
      *-commutative [=>] 99.6	\[ \frac{\color{blue}{\left(1 + y\right) \cdot x}}{z} - x \]
      +-commutative [=>] 99.6	\[ \frac{\color{blue}{\left(y + 1\right)} \cdot x}{z} - x \]
      distribute-lft1-in [<=] 99.6	\[ \frac{\color{blue}{y \cdot x + x}}{z} - x \]
      *-commutative [=>] 99.6	\[ \frac{\color{blue}{x \cdot y} + x}{z} - x \]
      fma-def [=>] 99.7	\[ \frac{\color{blue}{\mathsf{fma}\left(x, y, x\right)}}{z} - x \]

   if 5.00000000000000005e54 < z

   1. Initial program 68.5%

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

   2. Simplified 99.9%

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

      [Start] 68.5	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 99.9	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\left(\frac{y}{z} + -1\right) \cdot x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	68.0%
Cost	1245

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-237}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-227}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+24} \lor \neg \left(z \leq 2.1 \cdot 10^{+125}\right) \land z \leq 6.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 2
Accuracy	69.5%
Cost	980

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-236}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-15} \lor \neg \left(z \leq 1.22 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot \frac{y + \left(1 - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 4
Accuracy	99.8%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{y + \left(1 - z\right)}{z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array} \]

Alternative 5
Accuracy	99.8%
Cost	840

\[\begin{array}{l} t_0 := 1 + \left(y - z\right)\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+16}:\\ \;\;\;\;\left(\frac{y}{z} + -1\right) \cdot x\\ \mathbf{elif}\;z \leq 800000000:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 6
Accuracy	92.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -21000000000 \lor \neg \left(y \leq 2.45 \cdot 10^{-32}\right):\\ \;\;\;\;\left(\frac{y}{z} + -1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 7
Accuracy	98.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.112\right):\\ \;\;\;\;\left(\frac{y}{z} + -1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 8
Accuracy	81.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+129} \lor \neg \left(y \leq 2.3 \cdot 10^{+17}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 9
Accuracy	81.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+129} \lor \neg \left(y \leq 2.3 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 10
Accuracy	81.1%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+129} \lor \neg \left(y \leq 6 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 11
Accuracy	69.6%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 0.039:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 12
Accuracy	47.5%
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023122 
(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))