Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.1 → 0.1

Time: 8.7s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -1.0288939178730458 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4023103654.38204:\\ \;\;\;\;\frac{x + \frac{x}{\frac{y}{x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))))
   (if (<= x -1.0288939178730458e+23)
     t_0
     (if (<= x 4023103654.38204) (/ (+ x (/ x (/ y x))) (+ x 1.0)) t_0))))

C

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

↓

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (x <= -1.0288939178730458e+23) {
		tmp = t_0;
	} else if (x <= 4023103654.38204) {
		tmp = (x + (x / (y / x))) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((x + (-1.0d0)) / y)
    if (x <= (-1.0288939178730458d+23)) then
        tmp = t_0
    else if (x <= 4023103654.38204d0) then
        tmp = (x + (x / (y / x))) / (x + 1.0d0)
    else
        tmp = t_0
    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 t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (x <= -1.0288939178730458e+23) {
		tmp = t_0;
	} else if (x <= 4023103654.38204) {
		tmp = (x + (x / (y / x))) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	tmp = 0
	if x <= -1.0288939178730458e+23:
		tmp = t_0
	elif x <= 4023103654.38204:
		tmp = (x + (x / (y / x))) / (x + 1.0)
	else:
		tmp = t_0
	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)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	tmp = 0.0
	if (x <= -1.0288939178730458e+23)
		tmp = t_0;
	elseif (x <= 4023103654.38204)
		tmp = Float64(Float64(x + Float64(x / Float64(y / x))) / Float64(x + 1.0));
	else
		tmp = t_0;
	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)
	t_0 = 1.0 + ((x + -1.0) / y);
	tmp = 0.0;
	if (x <= -1.0288939178730458e+23)
		tmp = t_0;
	elseif (x <= 4023103654.38204)
		tmp = (x + (x / (y / x))) / (x + 1.0);
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0288939178730458e+23], t$95$0, If[LessEqual[x, 4023103654.38204], N[(N[(x + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	9.1
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if x < -1.0288939178730458e23 or 4023103654.38204 < x

   1. Initial program 22.7

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

   2. Simplified 22.7

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \frac{x}{y}, x\right)}{x + 1}} \]
      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)): 0 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, 0 points decrease in error

   3. Taylor expanded in x around inf 0.1

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

   4. Applied egg-rr 0.1

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

   if -1.0288939178730458e23 < x < 4023103654.38204

   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)): 0 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, 0 points decrease in error

   3. Applied egg-rr 0.1

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

   4. Applied egg-rr 0.1

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y}{x}}} + x}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0288939178730458 \cdot 10^{+23}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 4023103654.38204:\\ \;\;\;\;\frac{x + \frac{x}{\frac{y}{x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.6
Cost	1112

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -2.35 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.0745064058610068 \cdot 10^{+37}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.0288939178730458 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.769201411975239 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.768763379594474 \cdot 10^{-167}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 4023103654.38204:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	20.7
Cost	984

\[\begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.0745064058610068 \cdot 10^{+37}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -11.696171011692593:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.769201411975239 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.112952261697714 \cdot 10^{-149}:\\ \;\;\;\;\frac{x \cdot x}{y}\\ \mathbf{elif}\;x \leq 4023103654.38204:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	21.2
Cost	984

\[\begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.0745064058610068 \cdot 10^{+37}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -11.696171011692593:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.769201411975239 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.768763379594474 \cdot 10^{-167}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 4023103654.38204:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	21.1
Cost	984

\[\begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.0745064058610068 \cdot 10^{+37}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -11.696171011692593:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq -1.769201411975239 \cdot 10^{-135}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.768763379594474 \cdot 10^{-167}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 4023103654.38204:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Error	10.6
Cost	976

\[\begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -11.696171011692593:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.769201411975239 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.768763379594474 \cdot 10^{-167}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 4023103654.38204:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.4
Cost	840

\[\begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -11.696171011692593:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.30214891483612 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + \frac{x}{y}}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.2
Cost	840

\[\begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -11.696171011692593:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.30214891483612 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\frac{x}{y} + \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	0.2
Cost	832

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

Alternative 9
Error	20.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.0745064058610068 \cdot 10^{+37}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -11.696171011692593:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4023103654.38204:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 10
Error	0.1
Cost	704

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

Alternative 11
Error	28.0
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -2.700742729996183 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.30214891483612 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	35.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022291 
(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)))