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

Average Error: 9.4 → 0.3

Time: 7.3s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;z \leq 0.00076:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e-73)
   (/ x (/ z (+ (- y z) 1.0)))
   (if (<= z 0.00076) (* (/ x z) (+ y 1.0)) (* x (/ (- y (+ z -1.0)) 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 <= -1e-73) {
		tmp = x / (z / ((y - z) + 1.0));
	} else if (z <= 0.00076) {
		tmp = (x / z) * (y + 1.0);
	} else {
		tmp = x * ((y - (z + -1.0)) / 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 <= (-1d-73)) then
        tmp = x / (z / ((y - z) + 1.0d0))
    else if (z <= 0.00076d0) then
        tmp = (x / z) * (y + 1.0d0)
    else
        tmp = x * ((y - (z + (-1.0d0))) / 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 <= -1e-73) {
		tmp = x / (z / ((y - z) + 1.0));
	} else if (z <= 0.00076) {
		tmp = (x / z) * (y + 1.0);
	} else {
		tmp = x * ((y - (z + -1.0)) / 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 <= -1e-73:
		tmp = x / (z / ((y - z) + 1.0))
	elif z <= 0.00076:
		tmp = (x / z) * (y + 1.0)
	else:
		tmp = x * ((y - (z + -1.0)) / 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 <= -1e-73)
		tmp = Float64(x / Float64(z / Float64(Float64(y - z) + 1.0)));
	elseif (z <= 0.00076)
		tmp = Float64(Float64(x / z) * Float64(y + 1.0));
	else
		tmp = Float64(x * Float64(Float64(y - Float64(z + -1.0)) / 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 <= -1e-73)
		tmp = x / (z / ((y - z) + 1.0));
	elseif (z <= 0.00076)
		tmp = (x / z) * (y + 1.0);
	else
		tmp = x * ((y - (z + -1.0)) / 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[LessEqual[z, -1e-73], N[(x / N[(z / N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.00076], N[(N[(x / z), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	9.4
Target	0.4
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 3 regimes

2. if z < -9.99999999999999997e-74

   1. Initial program 12.9

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

   2. Simplified 0.3

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

      [Start] 12.9	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 0.3	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

   if -9.99999999999999997e-74 < z < 7.6000000000000004e-4

   1. Initial program 0.1

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

   2. Taylor expanded in z around 0 0.5

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

   3. Simplified 0.5

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

      [Start] 0.5	\[ \frac{\left(1 + y\right) \cdot x}{z} \]
      +-commutative [<=] 0.5	\[ \frac{\color{blue}{\left(y + 1\right)} \cdot x}{z} \]
      *-commutative [=>] 0.5	\[ \frac{\color{blue}{x \cdot \left(y + 1\right)}}{z} \]
      associate-*l/ [<=] 0.5	\[ \color{blue}{\frac{x}{z} \cdot \left(y + 1\right)} \]
      +-commutative [=>] 0.5	\[ \frac{x}{z} \cdot \color{blue}{\left(1 + y\right)} \]

   if 7.6000000000000004e-4 < z

   1. Initial program 15.4

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

   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 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;z \leq 0.00076:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-73} \lor \neg \left(z \leq 0.00076\right):\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 2
Error	9.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-9} \lor \neg \left(z \leq 0.38\right):\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 3
Error	0.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.98 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right)\\ \end{array} \]

Alternative 4
Error	11.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+14} \lor \neg \left(y \leq 3.6 \cdot 10^{+91}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 5
Error	19.5
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 6
Error	33.6
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023046 
(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))