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

Average Accuracy: 87.3% → 99.8%

Time: 8.5s

Precision: binary64

Cost: 7112

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;x \leq 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.35e-14)
   (/ x (/ z (+ (- y z) 1.0)))
   (if (<= x 1e-149) (- (/ (fma x y x) z) x) (- (* (/ x z) (+ y 1.0)) 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 (x <= -5.35e-14) {
		tmp = x / (z / ((y - z) + 1.0));
	} else if (x <= 1e-149) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = ((x / z) * (y + 1.0)) - 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 (x <= -5.35e-14)
		tmp = Float64(x / Float64(z / Float64(Float64(y - z) + 1.0)));
	elseif (x <= 1e-149)
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	else
		tmp = Float64(Float64(Float64(x / z) * Float64(y + 1.0)) - 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[x, -5.35e-14], N[(x / N[(z / N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-149], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]]

Target

Original	87.3%
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 x < -5.3499999999999999e-14

   1. Initial program 80.9%

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

   2. Simplified 100.0%

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

      [Start] 80.9	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 100.0	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

   if -5.3499999999999999e-14 < x < 9.99999999999999979e-150

   1. Initial program 99.8%

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

   2. Simplified 99.9%

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

      [Start] 99.8	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      +-commutative [=>] 99.8	\[ \frac{x \cdot \color{blue}{\left(1 + \left(y - z\right)\right)}}{z} \]
      sub-neg [=>] 99.8	\[ \frac{x \cdot \left(1 + \color{blue}{\left(y + \left(-z\right)\right)}\right)}{z} \]
      +-commutative [=>] 99.8	\[ \frac{x \cdot \left(1 + \color{blue}{\left(\left(-z\right) + y\right)}\right)}{z} \]
      associate-+r+ [=>] 99.8	\[ \frac{x \cdot \color{blue}{\left(\left(1 + \left(-z\right)\right) + y\right)}}{z} \]
      unsub-neg [=>] 99.8	\[ \frac{x \cdot \left(\color{blue}{\left(1 - z\right)} + y\right)}{z} \]
      associate-+l- [=>] 99.8	\[ \frac{x \cdot \color{blue}{\left(1 - \left(z - y\right)\right)}}{z} \]
      distribute-lft-out-- [<=] 99.8	\[ \frac{\color{blue}{x \cdot 1 - x \cdot \left(z - y\right)}}{z} \]
      *-rgt-identity [=>] 99.8	\[ \frac{\color{blue}{x} - x \cdot \left(z - y\right)}{z} \]
      distribute-rgt-out-- [<=] 99.8	\[ \frac{x - \color{blue}{\left(z \cdot x - y \cdot x\right)}}{z} \]
      sub-neg [=>] 99.8	\[ \frac{x - \color{blue}{\left(z \cdot x + \left(-y \cdot x\right)\right)}}{z} \]
      +-commutative [=>] 99.8	\[ \frac{x - \color{blue}{\left(\left(-y \cdot x\right) + z \cdot x\right)}}{z} \]
      associate--r+ [=>] 99.8	\[ \frac{\color{blue}{\left(x - \left(-y \cdot x\right)\right) - z \cdot x}}{z} \]
      div-sub [=>] 99.8	\[ \color{blue}{\frac{x - \left(-y \cdot x\right)}{z} - \frac{z \cdot x}{z}} \]

   if 9.99999999999999979e-150 < x

   1. Initial program 87.2%

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

   2. Simplified 98.8%

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

      [Start] 87.2	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 98.8	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

   3. Applied egg-rr 99.8%

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

      [Start] 98.8	\[ \frac{x}{\frac{z}{\left(y - z\right) + 1}} \]
      associate-/r/ [=>] 99.8	\[ \color{blue}{\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)} \]

   4. Taylor expanded in y around 0 82.8%

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

   5. Simplified 99.9%

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

      [Start] 82.8	\[ \frac{y \cdot x}{z} + \frac{\left(1 - z\right) \cdot x}{z} \]
      associate-/l* [=>] 91.6	\[ \frac{y \cdot x}{z} + \color{blue}{\frac{1 - z}{\frac{z}{x}}} \]
      div-sub [=>] 87.5	\[ \frac{y \cdot x}{z} + \color{blue}{\left(\frac{1}{\frac{z}{x}} - \frac{z}{\frac{z}{x}}\right)} \]
      associate-/r/ [=>] 87.5	\[ \frac{y \cdot x}{z} + \left(\color{blue}{\frac{1}{z} \cdot x} - \frac{z}{\frac{z}{x}}\right) \]
      associate-*l/ [=>] 87.6	\[ \frac{y \cdot x}{z} + \left(\color{blue}{\frac{1 \cdot x}{z}} - \frac{z}{\frac{z}{x}}\right) \]
      *-lft-identity [=>] 87.6	\[ \frac{y \cdot x}{z} + \left(\frac{\color{blue}{x}}{z} - \frac{z}{\frac{z}{x}}\right) \]
      associate-+r- [=>] 87.5	\[ \color{blue}{\left(\frac{y \cdot x}{z} + \frac{x}{z}\right) - \frac{z}{\frac{z}{x}}} \]
      associate-*r/ [<=] 88.4	\[ \left(\color{blue}{y \cdot \frac{x}{z}} + \frac{x}{z}\right) - \frac{z}{\frac{z}{x}} \]
      *-lft-identity [<=] 88.4	\[ \left(y \cdot \frac{x}{z} + \color{blue}{1 \cdot \frac{x}{z}}\right) - \frac{z}{\frac{z}{x}} \]
      distribute-rgt-in [<=] 94.7	\[ \color{blue}{\frac{x}{z} \cdot \left(y + 1\right)} - \frac{z}{\frac{z}{x}} \]
      +-commutative [<=] 94.7	\[ \frac{x}{z} \cdot \color{blue}{\left(1 + y\right)} - \frac{z}{\frac{z}{x}} \]
      associate-/r/ [=>] 99.9	\[ \frac{x}{z} \cdot \left(1 + y\right) - \color{blue}{\frac{z}{z} \cdot x} \]
      *-inverses [=>] 99.9	\[ \frac{x}{z} \cdot \left(1 + y\right) - \color{blue}{1} \cdot x \]
      *-lft-identity [=>] 99.9	\[ \frac{x}{z} \cdot \left(1 + y\right) - \color{blue}{x} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;x \leq 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	1865

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+33} \lor \neg \left(t_1 \leq 2 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+62} \lor \neg \left(z \leq 9 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Accuracy	65.4%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3200000000:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Accuracy	95.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -63 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 5
Accuracy	95.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -63 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 6
Accuracy	98.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.92 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \]

Alternative 7
Accuracy	95.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -63:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \end{array} \]

Alternative 8
Accuracy	83.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -75000000000 \lor \neg \left(y \leq 2.35 \cdot 10^{+151}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 9
Accuracy	83.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -50000000000000:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 10
Accuracy	98.1%
Cost	576

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

Alternative 11
Accuracy	65.2%
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 12
Accuracy	39.4%
Cost	128

\[-x \]

Alternative 13
Accuracy	3.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023160 
(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))