Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Average Error: 10.1 → 0.2

Time: 2.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{if}\;y \leq -5.192274197996505 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.8138946845704933 \cdot 10^{+56}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (/ x z)))))
   (if (<= y -5.192274197996505e+48)
     t_0
     (if (<= y 3.8138946845704933e+56) (- (+ y (/ x z)) (/ (* y x) z)) t_0))))

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

↓

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - (x / z));
	double tmp;
	if (y <= -5.192274197996505e+48) {
		tmp = t_0;
	} else if (y <= 3.8138946845704933e+56) {
		tmp = (y + (x / z)) - ((y * x) / z);
	} else {
		tmp = t_0;
	}
	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 - x))) / 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 * (1.0d0 - (x / z))
    if (y <= (-5.192274197996505d+48)) then
        tmp = t_0
    else if (y <= 3.8138946845704933d+56) then
        tmp = (y + (x / z)) - ((y * x) / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x + (y * (z - x))) / z;
}

↓

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - (x / z));
	double tmp;
	if (y <= -5.192274197996505e+48) {
		tmp = t_0;
	} else if (y <= 3.8138946845704933e+56) {
		tmp = (y + (x / z)) - ((y * x) / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = y * (1.0 - (x / z))
	tmp = 0
	if y <= -5.192274197996505e+48:
		tmp = t_0
	elif y <= 3.8138946845704933e+56:
		tmp = (y + (x / z)) - ((y * x) / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x + Float64(y * Float64(z - x))) / z)
end

↓

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - Float64(x / z)))
	tmp = 0.0
	if (y <= -5.192274197996505e+48)
		tmp = t_0;
	elseif (y <= 3.8138946845704933e+56)
		tmp = Float64(Float64(y + Float64(x / z)) - Float64(Float64(y * x) / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + (y * (z - x))) / z;
end

↓

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - (x / z));
	tmp = 0.0;
	if (y <= -5.192274197996505e+48)
		tmp = t_0;
	elseif (y <= 3.8138946845704933e+56)
		tmp = (y + (x / z)) - ((y * x) / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.192274197996505e+48], t$95$0, If[LessEqual[y, 3.8138946845704933e+56], N[(N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.1
Target	0.1
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if y < -5.19227419799650512e48 or 3.813894684570493e56 < y

   1. Initial program 26.8

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

   2. Simplified 26.8

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

   3. Taylor expanded in y around inf 26.8

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

   4. Simplified 0.1

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

   if -5.19227419799650512e48 < y < 3.813894684570493e56

   1. Initial program 0.7

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

   2. Simplified 0.7

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

   3. Taylor expanded in y around 0 0.3

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.192274197996505 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 3.8138946845704933 \cdot 10^{+56}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))