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

Average Error: 10.3 → 0.1

Time: 6.6s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ t_1 := \frac{x}{\frac{z}{t_0}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-32}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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)) (t_1 (/ x (/ z t_0))))
   (if (<= z -1e-40) t_1 (if (<= z 1e-32) (* t_0 (/ x z)) t_1))))

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 t_1 = x / (z / t_0);
	double tmp;
	if (z <= -1e-40) {
		tmp = t_1;
	} else if (z <= 1e-32) {
		tmp = t_0 * (x / z);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (y - z) + 1.0d0
    t_1 = x / (z / t_0)
    if (z <= (-1d-40)) then
        tmp = t_1
    else if (z <= 1d-32) then
        tmp = t_0 * (x / z)
    else
        tmp = t_1
    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 t_1 = x / (z / t_0);
	double tmp;
	if (z <= -1e-40) {
		tmp = t_1;
	} else if (z <= 1e-32) {
		tmp = t_0 * (x / z);
	} else {
		tmp = t_1;
	}
	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
	t_1 = x / (z / t_0)
	tmp = 0
	if z <= -1e-40:
		tmp = t_1
	elif z <= 1e-32:
		tmp = t_0 * (x / z)
	else:
		tmp = t_1
	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)
	t_1 = Float64(x / Float64(z / t_0))
	tmp = 0.0
	if (z <= -1e-40)
		tmp = t_1;
	elseif (z <= 1e-32)
		tmp = Float64(t_0 * Float64(x / z));
	else
		tmp = t_1;
	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;
	t_1 = x / (z / t_0);
	tmp = 0.0;
	if (z <= -1e-40)
		tmp = t_1;
	elseif (z <= 1e-32)
		tmp = t_0 * (x / z);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e-40], t$95$1, If[LessEqual[z, 1e-32], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	10.3
Target	0.5
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999993e-41 or 1.00000000000000006e-32 < z

   1. Initial program 15.3

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

   2. Applied egg-rr 13.4

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

   3. Applied egg-rr 0.1

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

   if -9.9999999999999993e-41 < z < 1.00000000000000006e-32

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;z \leq 10^{-32}:\\ \;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	840

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-32}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	20.2
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -23.46080384702203:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.6294013420020339 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3
Error	20.2
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -23.46080384702203:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.6294013420020339 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Error	4.1
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}} - x\\ \mathbf{if}\;y \leq -3845.333807120885:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.798750198545802 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.9
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}} - x\\ \mathbf{if}\;z \leq -23.46080384702203:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.051299704929457:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	12.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+48}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 7
Error	19.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -23.46080384702203:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 0.051299704929457:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 8
Error	33.2
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022300 
(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))