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

Average Error: 10.4 → 0.6

Time: 7.7s

Precision: binary64

Cost: 841

Math

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

↓

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-170} \lor \neg \left(x \leq 5 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]

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 (or (<= x -9.5e-170) (not (<= x 5e-61)))
     (/ x (/ z t_0))
     (/ (* x t_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 t_0 = (y - z) + 1.0;
	double tmp;
	if ((x <= -9.5e-170) || !(x <= 5e-61)) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x * t_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) :: t_0
    real(8) :: tmp
    t_0 = (y - z) + 1.0d0
    if ((x <= (-9.5d-170)) .or. (.not. (x <= 5d-61))) then
        tmp = x / (z / t_0)
    else
        tmp = (x * t_0) / 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 ((x <= -9.5e-170) || !(x <= 5e-61)) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x * t_0) / 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 (x <= -9.5e-170) or not (x <= 5e-61):
		tmp = x / (z / t_0)
	else:
		tmp = (x * t_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)
	t_0 = Float64(Float64(y - z) + 1.0)
	tmp = 0.0
	if ((x <= -9.5e-170) || !(x <= 5e-61))
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(Float64(x * t_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)
	t_0 = (y - z) + 1.0;
	tmp = 0.0;
	if ((x <= -9.5e-170) || ~((x <= 5e-61)))
		tmp = x / (z / t_0);
	else
		tmp = (x * t_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_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -9.5e-170], N[Not[LessEqual[x, 5e-61]], $MachinePrecision]], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	10.4
Target	0.4
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000001e-170 or 4.9999999999999999e-61 < x

   1. Initial program 17.1

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

   2. Simplified 1.0

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

      [Start] 17.1	\[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
      associate-/l* [=>] 1.0	\[ \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

   if -9.5000000000000001e-170 < x < 4.9999999999999999e-61

   1. Initial program 0.1

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-170} \lor \neg \left(x \leq 5 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.1
Cost	7492

\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;\frac{x \cdot t_0}{z} \leq 10^{+297}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-12} \lor \neg \left(z \leq 7 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]

Alternative 3
Error	20.3
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0076:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Error	4.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2800000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]

Alternative 5
Error	1.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0076 \lor \neg \left(z \leq 820000000000\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]

Alternative 6
Error	3.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2800000000:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + -1\right)\\ \end{array} \]

Alternative 7
Error	3.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2800000000:\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} - x\\ \end{array} \]

Alternative 8
Error	12.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+95}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 9
Error	12.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+95}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 10
Error	12.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+27}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 11
Error	11.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+94}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternative 12
Error	19.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0076:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 0.086:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 13
Error	32.9
Cost	128

\[-x \]

Alternative 14
Error	62.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023034 
(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))