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

Average Error: 9.9 → 2.0

Time: 5.9s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}} - x\\ \mathbf{if}\;z \leq -1.3712974317740827 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 39557020297.2278:\\ \;\;\;\;\frac{x + x \cdot \left(y - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ y (/ z x)) x)))
   (if (<= z -1.3712974317740827e+50)
     t_0
     (if (<= z 39557020297.2278) (/ (+ x (* x (- y z))) z) t_0))))

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 t_0 = (y / (z / x)) - x;
	double tmp;
	if (z <= -1.3712974317740827e+50) {
		tmp = t_0;
	} else if (z <= 39557020297.2278) {
		tmp = (x + (x * (y - z))) / z;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (y / (z / x)) - x
    if (z <= (-1.3712974317740827d+50)) then
        tmp = t_0
    else if (z <= 39557020297.2278d0) then
        tmp = (x + (x * (y - z))) / z
    else
        tmp = t_0
    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 t_0 = (y / (z / x)) - x;
	double tmp;
	if (z <= -1.3712974317740827e+50) {
		tmp = t_0;
	} else if (z <= 39557020297.2278) {
		tmp = (x + (x * (y - z))) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (y / (z / x)) - x
	tmp = 0
	if z <= -1.3712974317740827e+50:
		tmp = t_0
	elif z <= 39557020297.2278:
		tmp = (x + (x * (y - z))) / z
	else:
		tmp = t_0
	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)
	t_0 = Float64(Float64(y / Float64(z / x)) - x)
	tmp = 0.0
	if (z <= -1.3712974317740827e+50)
		tmp = t_0;
	elseif (z <= 39557020297.2278)
		tmp = Float64(Float64(x + Float64(x * Float64(y - z))) / z);
	else
		tmp = t_0;
	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)
	t_0 = (y / (z / x)) - x;
	tmp = 0.0;
	if (z <= -1.3712974317740827e+50)
		tmp = t_0;
	elseif (z <= 39557020297.2278)
		tmp = (x + (x * (y - z))) / z;
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[z, -1.3712974317740827e+50], t$95$0, If[LessEqual[z, 39557020297.2278], N[(N[(x + N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Target

Original	9.9
Target	0.4
Herbie	2.0

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

Derivation

1. Split input into 2 regimes

2. if z < -1.3712974317740827e50 or 39557020297.227798 < z

   1. Initial program 17.8

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

   2. Simplified 5.4

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

   3. Taylor expanded in y around inf 5.5

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

   4. Simplified 3.0

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

   5. Applied egg-rr 3.3

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

   if -1.3712974317740827e50 < z < 39557020297.227798

   1. Initial program 0.5

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

   2. Applied egg-rr 0.5

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3712974317740827 \cdot 10^{+50}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{elif}\;z \leq 39557020297.2278:\\ \;\;\;\;\frac{x + x \cdot \left(y - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \end{array} \]

Alternatives

Alternative 1
Error	21.9
Cost	984

\[\begin{array}{l} t_0 := \frac{y \cdot x}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -1.56 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-260}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.00031:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 2
Error	2.5
Cost	712

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}} - x\\ \mathbf{if}\;y \leq -27.340922656872998:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0008738801419590957:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.8
Cost	712

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}} - x\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x + y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	2.8
Cost	712

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

Alternative 5
Error	11.3
Cost	584

\[\begin{array}{l} t_0 := \frac{y \cdot x}{z}\\ \mathbf{if}\;y \leq -3.374842221325893 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	11.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -3.374842221325893 \cdot 10^{+58}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternative 7
Error	11.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -3.374842221325893 \cdot 10^{+58}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternative 8
Error	18.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 0.00031:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 9
Error	33.3
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022325 
(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))