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

Average Error: 9.8 → 0.1

Time: 18.3s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{elif}\;z \leq 0.02:\\ \;\;\;\;\left(\frac{y \cdot x}{z} + \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \left(z + -1\right)}{z} \cdot x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -4e+17)
   (- (* (/ y z) x) x)
   (if (<= z 0.02)
     (- (+ (/ (* y x) z) (/ x z)) x)
     (* (/ (- y (+ z -1.0)) z) x))))

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 <= -4e+17) {
		tmp = ((y / z) * x) - x;
	} else if (z <= 0.02) {
		tmp = (((y * x) / z) + (x / z)) - x;
	} else {
		tmp = ((y - (z + -1.0)) / z) * x;
	}
	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 <= (-4d+17)) then
        tmp = ((y / z) * x) - x
    else if (z <= 0.02d0) then
        tmp = (((y * x) / z) + (x / z)) - x
    else
        tmp = ((y - (z + (-1.0d0))) / z) * x
    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 <= -4e+17) {
		tmp = ((y / z) * x) - x;
	} else if (z <= 0.02) {
		tmp = (((y * x) / z) + (x / z)) - x;
	} else {
		tmp = ((y - (z + -1.0)) / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * ((y - z) + 1.0)) / z

↓

def code(x, y, z):
	tmp = 0
	if z <= -4e+17:
		tmp = ((y / z) * x) - x
	elif z <= 0.02:
		tmp = (((y * x) / z) + (x / z)) - x
	else:
		tmp = ((y - (z + -1.0)) / z) * x
	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 <= -4e+17)
		tmp = Float64(Float64(Float64(y / z) * x) - x);
	elseif (z <= 0.02)
		tmp = Float64(Float64(Float64(Float64(y * x) / z) + Float64(x / z)) - x);
	else
		tmp = Float64(Float64(Float64(y - Float64(z + -1.0)) / z) * x);
	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 <= -4e+17)
		tmp = ((y / z) * x) - x;
	elseif (z <= 0.02)
		tmp = (((y * x) / z) + (x / z)) - x;
	else
		tmp = ((y - (z + -1.0)) / z) * x;
	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, -4e+17], N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[z, 0.02], N[(N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] + N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(y - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]]]

Target

Original	9.8
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 3 regimes

2. if z < -4e17

   1. Initial program 17.5

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

   2. Taylor expanded in z around 0 5.4

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

   3. Simplified 5.4

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

   4. Taylor expanded in y around 0 5.4

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

   5. Taylor expanded in y around inf 5.4

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}} - x \]

   6. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x} - x \]

   if -4e17 < z < 0.0200000000000000004

   1. Initial program 0.2

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

   2. Taylor expanded in z around 0 0.1

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

   3. Simplified 0.1

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

   4. Taylor expanded in y around 0 0.1

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

   if 0.0200000000000000004 < z

   1. Initial program 15.3

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

   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.

Alternatives

Alternative 1
Error	0.1
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\left(-x\right) + \frac{\left(1 + y\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \left(z + -1\right)}{z} \cdot x\\ \end{array} \]

Alternative 2
Error	20.7
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{z} \cdot y\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{+55}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.06 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-151}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 230000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 3
Error	20.7
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+57}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 420000:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Error	0.1
Cost	840

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot x - x\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - \left(z - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - \left(z - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \left(z + -1\right)}{z} \cdot x\\ \end{array} \]

Alternative 6
Error	0.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \left(z + -1\right)}{z} \cdot x\\ \end{array} \]

Alternative 7
Error	4.3
Cost	712

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot x - x\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	0.9
Cost	712

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

Alternative 9
Error	12.3
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{z} \cdot y\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	12.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+16}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Alternative 11
Error	19.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00185:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1450:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 12
Error	33.0
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023010 
(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))