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

Average Error: 15.4% → 0.99%

Time: 6.5s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{elif}\;z \leq 100000000000:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + \left(1 - z\right)}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- y z) 1.0)))
   (if (<= z -7.5e+102)
     (/ x (/ z t_0))
     (if (<= z 100000000000.0) (/ (* x t_0) z) (* x (/ (+ 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 t_0 = (y - z) + 1.0;
	double tmp;
	if (z <= -7.5e+102) {
		tmp = x / (z / t_0);
	} else if (z <= 100000000000.0) {
		tmp = (x * t_0) / z;
	} else {
		tmp = x * ((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) :: t_0
    real(8) :: tmp
    t_0 = (y - z) + 1.0d0
    if (z <= (-7.5d+102)) then
        tmp = x / (z / t_0)
    else if (z <= 100000000000.0d0) then
        tmp = (x * t_0) / z
    else
        tmp = x * ((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 t_0 = (y - z) + 1.0;
	double tmp;
	if (z <= -7.5e+102) {
		tmp = x / (z / t_0);
	} else if (z <= 100000000000.0) {
		tmp = (x * t_0) / z;
	} else {
		tmp = x * ((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):
	t_0 = (y - z) + 1.0
	tmp = 0
	if z <= -7.5e+102:
		tmp = x / (z / t_0)
	elif z <= 100000000000.0:
		tmp = (x * t_0) / z
	else:
		tmp = x * ((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)
	t_0 = Float64(Float64(y - z) + 1.0)
	tmp = 0.0
	if (z <= -7.5e+102)
		tmp = Float64(x / Float64(z / t_0));
	elseif (z <= 100000000000.0)
		tmp = Float64(Float64(x * t_0) / z);
	else
		tmp = Float64(x * Float64(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)
	t_0 = (y - z) + 1.0;
	tmp = 0.0;
	if (z <= -7.5e+102)
		tmp = x / (z / t_0);
	elseif (z <= 100000000000.0)
		tmp = (x * t_0) / z;
	else
		tmp = x * ((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_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[z, -7.5e+102], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 100000000000.0], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(N[(y + N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	15.4%
Target	0.79%
Herbie	0.99%

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

Derivation

1. Split input into 3 regimes

2. if z < -7.5e102

   1. Initial program 33.04

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

   2. Simplified 0.07

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

      [Start] 33.04	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 0.07	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

   if -7.5e102 < z < 1e11

   1. Initial program 1.87

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

   if 1e11 < z

   1. Initial program 26.37

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

   2. Applied egg-rr 0.1

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

4. Final simplification 0.99

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;z \leq 100000000000:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + \left(1 - z\right)}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	34.88%
Cost	852

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+86}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 2
Error	18.76%
Cost	849

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+200} \lor \neg \left(y \leq 4.5 \cdot 10^{+214}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3
Error	18.83%
Cost	848

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+200}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+214}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	18.81%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+20}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+200}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+214}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5
Error	16.78%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+85}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 3.85:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6
Error	0.35%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-6} \lor \neg \left(z \leq 8.2 \cdot 10^{-35}\right):\\ \;\;\;\;x \cdot \frac{y + \left(1 - z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \]

Alternative 7
Error	0.35%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + \left(1 - z\right)}{z}\\ \end{array} \]

Alternative 8
Error	31.22%
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 9
Error	52.35%
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023089 
(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))