Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.4 → 0.2

Time: 7.2s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2954784032016266:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 8766795.636704825:\\ \;\;\;\;\frac{x + x \cdot \frac{x}{y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -2954784032016266.0)
   (+ 1.0 (/ (+ x -1.0) y))
   (if (<= x 8766795.636704825)
     (/ (+ x (* x (/ x y))) (+ x 1.0))
     (+ 1.0 (/ x y)))))

C

double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

↓

double code(double x, double y) {
	double tmp;
	if (x <= -2954784032016266.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 8766795.636704825) {
		tmp = (x + (x * (x / y))) / (x + 1.0);
	} else {
		tmp = 1.0 + (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2954784032016266.0d0)) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else if (x <= 8766795.636704825d0) then
        tmp = (x + (x * (x / y))) / (x + 1.0d0)
    else
        tmp = 1.0d0 + (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

↓

public static double code(double x, double y) {
	double tmp;
	if (x <= -2954784032016266.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 8766795.636704825) {
		tmp = (x + (x * (x / y))) / (x + 1.0);
	} else {
		tmp = 1.0 + (x / y);
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if x <= -2954784032016266.0:
		tmp = 1.0 + ((x + -1.0) / y)
	elif x <= 8766795.636704825:
		tmp = (x + (x * (x / y))) / (x + 1.0)
	else:
		tmp = 1.0 + (x / y)
	return tmp

Julia

function code(x, y)
	return Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
end

↓

function code(x, y)
	tmp = 0.0
	if (x <= -2954784032016266.0)
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	elseif (x <= 8766795.636704825)
		tmp = Float64(Float64(x + Float64(x * Float64(x / y))) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 + Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = (x * ((x / y) + 1.0)) / (x + 1.0);
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2954784032016266.0)
		tmp = 1.0 + ((x + -1.0) / y);
	elseif (x <= 8766795.636704825)
		tmp = (x + (x * (x / y))) / (x + 1.0);
	else
		tmp = 1.0 + (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := If[LessEqual[x, -2954784032016266.0], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8766795.636704825], N[(N[(x + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Target

Original	9.4
Target	0.1
Herbie	0.2

\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1} \]

Derivation

1. Split input into 3 regimes

2. if x < -2954784032016266

   1. Initial program 23.3

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

   2. Taylor expanded in x around inf 0.0

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

   3. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\frac{x - 1}{y} + 1} \]

   if -2954784032016266 < x < 8766795.6367048249

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \frac{x}{y}, x\right)}{x + 1}} \]
      Proof
      (/.f64 (fma.f64 x (/.f64 x y) x) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 x (/.f64 x y) (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))) (+.f64 x 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (/.f64 x y)) (*.f64 x 1))) (+.f64 x 1)): 1 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 x (+.f64 (/.f64 x y) 1))) (+.f64 x 1)): 1 points increase in error, 1 points decrease in error

   3. Applied egg-rr 0.1

      \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y} + x}}{x + 1} \]

   if 8766795.6367048249 < x

   1. Initial program 22.1

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

   2. Taylor expanded in x around inf 0.4

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

   3. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{x - 1}{y} + 1} \]

   4. Applied egg-rr 0.6

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

   5. Taylor expanded in x around inf 0.6

      \[\leadsto \color{blue}{\frac{x}{y}} + 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2954784032016266:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 8766795.636704825:\\ \;\;\;\;\frac{x + x \cdot \frac{x}{y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -2954784032016266:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 43834224986894050:\\ \;\;\;\;\frac{x + \frac{x}{\frac{y}{x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 2
Error	1.7
Cost	840

\[\begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -391764256296.16785:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0018699366583554077:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.0
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+134}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.302593520506833 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -391764256296.16785:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 7.234194828708834 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	704

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

Alternative 5
Error	10.5
Cost	584

\[\begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -391764256296.16785:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.234194828708834 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	9.7
Cost	584

\[\begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -391764256296.16785:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8766795.636704825:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	9.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -391764256296.16785:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 8766795.636704825:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 8
Error	28.2
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -243963859682639.4:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.0018699366583554077:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	53.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))