Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 99.6% → 99.1%

Time: 26.4s

Precision: binary64

Cost: 840

• could not determine a ground truth

  1. x = 1.5004564603928934e-240
  2. y = -1.593288660425601e+149

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\frac{x + 1}{\frac{x}{y}}}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-23}:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y + x \cdot y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e-13)
   (/ x (/ (+ x 1.0) (/ x y)))
   (if (<= x 6.2e-23) (+ x (* x (- (/ x y) x))) (* x (/ x (+ y (* 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 <= -1.4e-13) {
		tmp = x / ((x + 1.0) / (x / y));
	} else if (x <= 6.2e-23) {
		tmp = x + (x * ((x / y) - x));
	} else {
		tmp = x * (x / (y + (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 <= (-1.4d-13)) then
        tmp = x / ((x + 1.0d0) / (x / y))
    else if (x <= 6.2d-23) then
        tmp = x + (x * ((x / y) - x))
    else
        tmp = x * (x / (y + (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 <= -1.4e-13) {
		tmp = x / ((x + 1.0) / (x / y));
	} else if (x <= 6.2e-23) {
		tmp = x + (x * ((x / y) - x));
	} else {
		tmp = x * (x / (y + (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 <= -1.4e-13:
		tmp = x / ((x + 1.0) / (x / y))
	elif x <= 6.2e-23:
		tmp = x + (x * ((x / y) - x))
	else:
		tmp = x * (x / (y + (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 <= -1.4e-13)
		tmp = Float64(x / Float64(Float64(x + 1.0) / Float64(x / y)));
	elseif (x <= 6.2e-23)
		tmp = Float64(x + Float64(x * Float64(Float64(x / y) - x)));
	else
		tmp = Float64(x * Float64(x / Float64(y + 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 <= -1.4e-13)
		tmp = x / ((x + 1.0) / (x / y));
	elseif (x <= 6.2e-23)
		tmp = x + (x * ((x / y) - x));
	else
		tmp = x * (x / (y + (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, -1.4e-13], N[(x / N[(N[(x + 1.0), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e-23], N[(x + N[(x * N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x / N[(y + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	99.6%
Target	99.8%
Herbie	99.1%

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

Derivation

1. Split input into 3 regimes

2. if x < -1.4000000000000001e-13

   1. Initial program 83.6%

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

   2. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]
      Step-by-step derivation

      [Start] 83.6%	\[ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
      associate-/l* [=>] 99.6%	\[ \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Taylor expanded in y around 0 99.3%

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

   4. Simplified 99.6%

      \[\leadsto \frac{x}{\color{blue}{\frac{1 + x}{\frac{x}{y}}}} \]
      Step-by-step derivation

      [Start] 99.3%	\[ \frac{x}{\frac{y \cdot \left(1 + x\right)}{x}} \]
      +-commutative [<=] 99.3%	\[ \frac{x}{\frac{y \cdot \color{blue}{\left(x + 1\right)}}{x}} \]
      *-commutative [=>] 99.3%	\[ \frac{x}{\frac{\color{blue}{\left(x + 1\right) \cdot y}}{x}} \]
      +-commutative [=>] 99.3%	\[ \frac{x}{\frac{\color{blue}{\left(1 + x\right)} \cdot y}{x}} \]
      associate-/l* [=>] 99.6%	\[ \frac{x}{\color{blue}{\frac{1 + x}{\frac{x}{y}}}} \]

   if -1.4000000000000001e-13 < x < 6.1999999999999998e-23

   1. Initial program 99.8%

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

   2. Applied egg-rr 88.6%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot \left({\left(\frac{x}{y}\right)}^{2} + -1\right)}{\frac{x}{y} + -1}}}{x + 1} \]
      Step-by-step derivation

      [Start] 99.8%	\[ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
      flip-+ [=>] 88.7%	\[ \frac{x \cdot \color{blue}{\frac{\frac{x}{y} \cdot \frac{x}{y} - 1 \cdot 1}{\frac{x}{y} - 1}}}{x + 1} \]
      associate-*r/ [=>] 88.6%	\[ \frac{\color{blue}{\frac{x \cdot \left(\frac{x}{y} \cdot \frac{x}{y} - 1 \cdot 1\right)}{\frac{x}{y} - 1}}}{x + 1} \]
      metadata-eval [=>] 88.6%	\[ \frac{\frac{x \cdot \left(\frac{x}{y} \cdot \frac{x}{y} - \color{blue}{1}\right)}{\frac{x}{y} - 1}}{x + 1} \]
      sub-neg [=>] 88.6%	\[ \frac{\frac{x \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y} + \left(-1\right)\right)}}{\frac{x}{y} - 1}}{x + 1} \]
      pow2 [=>] 88.6%	\[ \frac{\frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} + \left(-1\right)\right)}{\frac{x}{y} - 1}}{x + 1} \]
      metadata-eval [=>] 88.6%	\[ \frac{\frac{x \cdot \left({\left(\frac{x}{y}\right)}^{2} + \color{blue}{-1}\right)}{\frac{x}{y} - 1}}{x + 1} \]
      sub-neg [=>] 88.6%	\[ \frac{\frac{x \cdot \left({\left(\frac{x}{y}\right)}^{2} + -1\right)}{\color{blue}{\frac{x}{y} + \left(-1\right)}}}{x + 1} \]
      metadata-eval [=>] 88.6%	\[ \frac{\frac{x \cdot \left({\left(\frac{x}{y}\right)}^{2} + -1\right)}{\frac{x}{y} + \color{blue}{-1}}}{x + 1} \]

   3. Simplified 88.5%

      \[\leadsto \frac{\color{blue}{\frac{-1 + {\left(\frac{x}{y}\right)}^{2}}{\frac{-1 + \frac{x}{y}}{x}}}}{x + 1} \]
      Step-by-step derivation

      [Start] 88.6%	\[ \frac{\frac{x \cdot \left({\left(\frac{x}{y}\right)}^{2} + -1\right)}{\frac{x}{y} + -1}}{x + 1} \]
      *-commutative [=>] 88.6%	\[ \frac{\frac{\color{blue}{\left({\left(\frac{x}{y}\right)}^{2} + -1\right) \cdot x}}{\frac{x}{y} + -1}}{x + 1} \]
      associate-/l* [=>] 88.5%	\[ \frac{\color{blue}{\frac{{\left(\frac{x}{y}\right)}^{2} + -1}{\frac{\frac{x}{y} + -1}{x}}}}{x + 1} \]
      +-commutative [=>] 88.5%	\[ \frac{\frac{\color{blue}{-1 + {\left(\frac{x}{y}\right)}^{2}}}{\frac{\frac{x}{y} + -1}{x}}}{x + 1} \]
      +-commutative [=>] 88.5%	\[ \frac{\frac{-1 + {\left(\frac{x}{y}\right)}^{2}}{\frac{\color{blue}{-1 + \frac{x}{y}}}{x}}}{x + 1} \]

   4. Taylor expanded in x around 0 87.8%

      \[\leadsto \color{blue}{\left(\frac{1}{y} - 1\right) \cdot {x}^{2} + x} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{x + x \cdot \left(\frac{x}{y} - x\right)} \]
      Step-by-step derivation

      [Start] 87.8%	\[ \left(\frac{1}{y} - 1\right) \cdot {x}^{2} + x \]
      +-commutative [=>] 87.8%	\[ \color{blue}{x + \left(\frac{1}{y} - 1\right) \cdot {x}^{2}} \]
      *-commutative [=>] 87.8%	\[ x + \color{blue}{{x}^{2} \cdot \left(\frac{1}{y} - 1\right)} \]
      sub-neg [=>] 87.8%	\[ x + {x}^{2} \cdot \color{blue}{\left(\frac{1}{y} + \left(-1\right)\right)} \]
      metadata-eval [=>] 87.8%	\[ x + {x}^{2} \cdot \left(\frac{1}{y} + \color{blue}{-1}\right) \]
      distribute-rgt-in [=>] 87.8%	\[ x + \color{blue}{\left(\frac{1}{y} \cdot {x}^{2} + -1 \cdot {x}^{2}\right)} \]
      associate-*l/ [=>] 87.9%	\[ x + \left(\color{blue}{\frac{1 \cdot {x}^{2}}{y}} + -1 \cdot {x}^{2}\right) \]
      *-lft-identity [=>] 87.9%	\[ x + \left(\frac{\color{blue}{{x}^{2}}}{y} + -1 \cdot {x}^{2}\right) \]
      unpow2 [=>] 87.9%	\[ x + \left(\frac{\color{blue}{x \cdot x}}{y} + -1 \cdot {x}^{2}\right) \]
      associate-/l* [=>] 99.8%	\[ x + \left(\color{blue}{\frac{x}{\frac{y}{x}}} + -1 \cdot {x}^{2}\right) \]
      associate-/r/ [=>] 99.9%	\[ x + \left(\color{blue}{\frac{x}{y} \cdot x} + -1 \cdot {x}^{2}\right) \]
      neg-mul-1 [<=] 99.9%	\[ x + \left(\frac{x}{y} \cdot x + \color{blue}{\left(-{x}^{2}\right)}\right) \]
      unpow2 [=>] 99.9%	\[ x + \left(\frac{x}{y} \cdot x + \left(-\color{blue}{x \cdot x}\right)\right) \]
      distribute-lft-neg-in [=>] 99.9%	\[ x + \left(\frac{x}{y} \cdot x + \color{blue}{\left(-x\right) \cdot x}\right) \]
      distribute-rgt-in [<=] 99.9%	\[ x + \color{blue}{x \cdot \left(\frac{x}{y} + \left(-x\right)\right)} \]
      unsub-neg [=>] 99.9%	\[ x + x \cdot \color{blue}{\left(\frac{x}{y} - x\right)} \]

   if 6.1999999999999998e-23 < x

   1. Initial program 99.2%

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

   2. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]
      Step-by-step derivation

      [Start] 99.2%	\[ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
      associate-/l* [=>] 99.2%	\[ \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Taylor expanded in y around 0 99.6%

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

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{x}{y + y \cdot x} \cdot x} \]
      Step-by-step derivation

      [Start] 99.6%	\[ \frac{x}{\frac{y \cdot \left(1 + x\right)}{x}} \]
      associate-/r/ [=>] 99.4%	\[ \color{blue}{\frac{x}{y \cdot \left(1 + x\right)} \cdot x} \]
      distribute-lft-in [=>] 99.6%	\[ \frac{x}{\color{blue}{y \cdot 1 + y \cdot x}} \cdot x \]
      *-rgt-identity [=>] 99.6%	\[ \frac{x}{\color{blue}{y} + y \cdot x} \cdot x \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\frac{x + 1}{\frac{x}{y}}}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-23}:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y + x \cdot y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\frac{x + 1}{\frac{x}{y}}}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-23}:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y + x \cdot y}\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-15} \lor \neg \left(x \leq 6.2 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	99.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-13} \lor \neg \left(x \leq 6.2 \cdot 10^{-23}\right):\\ \;\;\;\;x \cdot \frac{x}{y + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-13} \lor \neg \left(x \leq 6.2 \cdot 10^{-23}\right):\\ \;\;\;\;x \cdot \frac{x}{y + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \end{array} \]

Alternative 5
Accuracy	70.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-168} \lor \neg \left(y \leq 4.4 \cdot 10^{-212}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{x}{y} - x\right)\\ \end{array} \]

Alternative 6
Accuracy	96.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{x}{y}\\ \end{array} \]

Alternative 7
Accuracy	99.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-15} \lor \neg \left(x \leq 6.2 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{x}{y}\\ \end{array} \]

Alternative 8
Accuracy	99.7%
Cost	704

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

Alternative 9
Accuracy	70.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-166} \lor \neg \left(y \leq 9.6 \cdot 10^{-212}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 10
Accuracy	52.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 11
Accuracy	52.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 12
Accuracy	3.7%
Cost	64

\[1 \]

Alternative 13
Accuracy	50.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023277 
(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)))