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

Average Error: 10.5 → 0.3

Time: 7.4s

Precision: binary64

Cost: 969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-86} \lor \neg \left(z \leq 9 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{1}{y + \left(1 - z\right)}}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.8e-86) (not (<= z 9e+15)))
   (/ x (/ z (+ (- y z) 1.0)))
   (/ (/ x (/ 1.0 (+ y (- 1.0 z)))) 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 <= -1.8e-86) || !(z <= 9e+15)) {
		tmp = x / (z / ((y - z) + 1.0));
	} else {
		tmp = (x / (1.0 / (y + (1.0 - z)))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * ((y - z) + 1.0d0)) / z
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.8d-86)) .or. (.not. (z <= 9d+15))) then
        tmp = x / (z / ((y - z) + 1.0d0))
    else
        tmp = (x / (1.0d0 / (y + (1.0d0 - z)))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.8e-86) || !(z <= 9e+15)) {
		tmp = x / (z / ((y - z) + 1.0));
	} else {
		tmp = (x / (1.0 / (y + (1.0 - z)))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * ((y - z) + 1.0)) / z

↓

def code(x, y, z):
	tmp = 0
	if (z <= -1.8e-86) or not (z <= 9e+15):
		tmp = x / (z / ((y - z) + 1.0))
	else:
		tmp = (x / (1.0 / (y + (1.0 - z)))) / 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 <= -1.8e-86) || !(z <= 9e+15))
		tmp = Float64(x / Float64(z / Float64(Float64(y - z) + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(1.0 / Float64(y + Float64(1.0 - z)))) / z);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * ((y - z) + 1.0)) / z;
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.8e-86) || ~((z <= 9e+15)))
		tmp = x / (z / ((y - z) + 1.0));
	else
		tmp = (x / (1.0 / (y + (1.0 - z)))) / z;
	end
	tmp_2 = 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, -1.8e-86], N[Not[LessEqual[z, 9e+15]], $MachinePrecision]], N[(x / N[(z / N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 / N[(y + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	10.5
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 < -1.79999999999999983e-86 or 9e15 < z

   1. Initial program 15.8

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

   2. Simplified 0.3

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

      [Start] 15.8	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 0.3	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

   if -1.79999999999999983e-86 < z < 9e15

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{1}{y - \left(z + -1\right)}}}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-86} \lor \neg \left(z \leq 9 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{1}{y + \left(1 - z\right)}}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.5
Cost	977

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+102}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+79} \lor \neg \left(z \leq -9500000\right) \land z \leq 3.1 \cdot 10^{-13}:\\ \;\;\;\;\frac{y + 1}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 2
Error	9.4
Cost	977

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+102}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{y + 1}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -14800000 \lor \neg \left(z \leq 3.1 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-86} \lor \neg \left(z \leq 9 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	841

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{-86} \lor \neg \left(z \leq 5 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]

Alternative 5
Error	19.7
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+102}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+79}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6
Error	12.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{\frac{z}{x}}\\ \end{array} \]

Alternative 7
Error	12.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+120} \lor \neg \left(y \leq 2.05 \cdot 10^{+67}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 8
Error	19.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 9
Error	33.4
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023125 
(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))