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

Average Error: 10.3 → 0.1

Time: 11.2s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_0 := 1 + \left(y - z\right)\\ t_1 := \frac{x}{\frac{z}{t_0}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-46}:\\ \;\;\;\;t_0 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (- y z))) (t_1 (/ x (/ z t_0))))
   (if (<= z -1e-15) t_1 (if (<= z 1e-46) (* 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 = 1.0 + (y - z);
	double t_1 = x / (z / t_0);
	double tmp;
	if (z <= -1e-15) {
		tmp = t_1;
	} else if (z <= 1e-46) {
		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 = 1.0d0 + (y - z)
    t_1 = x / (z / t_0)
    if (z <= (-1d-15)) then
        tmp = t_1
    else if (z <= 1d-46) 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 = 1.0 + (y - z);
	double t_1 = x / (z / t_0);
	double tmp;
	if (z <= -1e-15) {
		tmp = t_1;
	} else if (z <= 1e-46) {
		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 = 1.0 + (y - z)
	t_1 = x / (z / t_0)
	tmp = 0
	if z <= -1e-15:
		tmp = t_1
	elif z <= 1e-46:
		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(1.0 + Float64(y - z))
	t_1 = Float64(x / Float64(z / t_0))
	tmp = 0.0
	if (z <= -1e-15)
		tmp = t_1;
	elseif (z <= 1e-46)
		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 = 1.0 + (y - z);
	t_1 = x / (z / t_0);
	tmp = 0.0;
	if (z <= -1e-15)
		tmp = t_1;
	elseif (z <= 1e-46)
		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[(1.0 + N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e-15], t$95$1, If[LessEqual[z, 1e-46], N[(t$95$0 * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	10.3
Target	0.4
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 < -1.0000000000000001e-15 or 1.00000000000000002e-46 < z

   1. Initial program 15.7

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

   2. Applied egg-rr 0.2

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

   3. Applied egg-rr 0.1

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

   if -1.0000000000000001e-15 < z < 1.00000000000000002e-46

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \mathbf{elif}\;z \leq 10^{-46}:\\ \;\;\;\;\left(1 + \left(y - z\right)\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	7112

\[\begin{array}{l} t_0 := \frac{z}{1 + \left(y - z\right)}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+178}:\\ \;\;\;\;x \cdot \frac{1}{t_0}\\ \mathbf{elif}\;x \leq 1.09185764049456 \cdot 10^{-145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_0}\\ \end{array} \]

Alternative 2
Error	21.4
Cost	1112

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;z \leq -6.0011780056835 \cdot 10^{+69}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{-152}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 5.280853505289251 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3
Error	21.3
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -6.0011780056835 \cdot 10^{+69}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{-31}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{-152}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 5.280853505289251 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Error	21.7
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.8785255008240393 \cdot 10^{+85}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{-152}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 5.280853505289251 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 5
Error	21.7
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.8785255008240393 \cdot 10^{+85}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{-152}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 5.280853505289251 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 6
Error	0.1
Cost	840

\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-46}:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	2.4
Cost	712

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}} - x\\ \mathbf{if}\;y \leq -149.5986447389087:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.777014375517274 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	3.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -58212442611471720:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{elif}\;z \leq 0.000565460407112389:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} - x\\ \end{array} \]

Alternative 9
Error	3.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -58212442611471720:\\ \;\;\;\;y \cdot \frac{x}{z} - x\\ \mathbf{elif}\;z \leq 0.000565460407112389:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} - x\\ \end{array} \]

Alternative 10
Error	12.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -5.004291982121553 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 11
Error	19.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -58212442611471720:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 0.000565460407112389:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 12
Error	33.2
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022316 
(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))