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

Average Error: 10.2 → 0.1

Time: 3.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ t_1 := \left(\frac{x}{z} + y \cdot \frac{x}{z}\right) - x\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+236}:\\ \;\;\;\;\left(\frac{x}{z} + \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (- y z) 1.0)) z))
        (t_1 (- (+ (/ x z) (* y (/ x z))) x)))
   (if (<= t_0 -5e+217)
     t_1
     (if (<= t_0 5e+236) (- (+ (/ x z) (/ (* x y) z)) x) 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 = (x * ((y - z) + 1.0)) / z;
	double t_1 = ((x / z) + (y * (x / z))) - x;
	double tmp;
	if (t_0 <= -5e+217) {
		tmp = t_1;
	} else if (t_0 <= 5e+236) {
		tmp = ((x / z) + ((x * y) / z)) - x;
	} 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 = (x * ((y - z) + 1.0d0)) / z
    t_1 = ((x / z) + (y * (x / z))) - x
    if (t_0 <= (-5d+217)) then
        tmp = t_1
    else if (t_0 <= 5d+236) then
        tmp = ((x / z) + ((x * y) / z)) - x
    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 = (x * ((y - z) + 1.0)) / z;
	double t_1 = ((x / z) + (y * (x / z))) - x;
	double tmp;
	if (t_0 <= -5e+217) {
		tmp = t_1;
	} else if (t_0 <= 5e+236) {
		tmp = ((x / z) + ((x * y) / z)) - x;
	} 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 = (x * ((y - z) + 1.0)) / z
	t_1 = ((x / z) + (y * (x / z))) - x
	tmp = 0
	if t_0 <= -5e+217:
		tmp = t_1
	elif t_0 <= 5e+236:
		tmp = ((x / z) + ((x * y) / z)) - x
	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(x * Float64(Float64(y - z) + 1.0)) / z)
	t_1 = Float64(Float64(Float64(x / z) + Float64(y * Float64(x / z))) - x)
	tmp = 0.0
	if (t_0 <= -5e+217)
		tmp = t_1;
	elseif (t_0 <= 5e+236)
		tmp = Float64(Float64(Float64(x / z) + Float64(Float64(x * y) / z)) - x);
	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 = (x * ((y - z) + 1.0)) / z;
	t_1 = ((x / z) + (y * (x / z))) - x;
	tmp = 0.0;
	if (t_0 <= -5e+217)
		tmp = t_1;
	elseif (t_0 <= 5e+236)
		tmp = ((x / z) + ((x * y) / z)) - x;
	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[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x / z), $MachinePrecision] + N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+217], t$95$1, If[LessEqual[t$95$0, 5e+236], N[(N[(N[(x / z), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], t$95$1]]]]

Target

Original	10.2
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 (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -5.00000000000000041e217 or 4.9999999999999997e236 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

   1. Initial program 40.2

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

   2. Simplified 13.5

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]

   3. Taylor expanded in y around 0 13.5

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

   4. Applied egg-rr 0.1

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

   if -5.00000000000000041e217 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 4.9999999999999997e236

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x} \]

   3. Taylor expanded in y around 0 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -5 \cdot 10^{+217}:\\ \;\;\;\;\left(\frac{x}{z} + y \cdot \frac{x}{z}\right) - x\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 5 \cdot 10^{+236}:\\ \;\;\;\;\left(\frac{x}{z} + \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} + y \cdot \frac{x}{z}\right) - x\\ \end{array} \]

Reproduce

herbie shell --seed 2022207 
(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))