Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{y + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{y + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{y + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x + y}{y + 1} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ t_1 := \frac{x}{y - -1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\frac{y}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))) (t_1 (/ x (- y -1.0))))
   (if (<= t_0 -4e+16)
     t_1
     (if (<= t_0 0.02) (fma 1.0 y x) (if (<= t_0 4.0) (/ y (- y -1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + 1.0);
	double t_1 = x / (y - -1.0);
	double tmp;
	if (t_0 <= -4e+16) {
		tmp = t_1;
	} else if (t_0 <= 0.02) {
		tmp = fma(1.0, y, x);
	} else if (t_0 <= 4.0) {
		tmp = y / (y - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + 1.0))
	t_1 = Float64(x / Float64(y - -1.0))
	tmp = 0.0
	if (t_0 <= -4e+16)
		tmp = t_1;
	elseif (t_0 <= 0.02)
		tmp = fma(1.0, y, x);
	elseif (t_0 <= 4.0)
		tmp = Float64(y / Float64(y - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+16], t$95$1, If[LessEqual[t$95$0, 0.02], N[(1.0 * y + x), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(y / N[(y - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < -4e16 or 4 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

      5. distribute-lft-neg-out N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -4e16 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot x}, y, x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(1 - \color{blue}{1} \cdot x, y, x\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(1 - \color{blue}{x}, y, x\right) \]

      10. lower--.f64 99.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{1 - x}, y, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, y, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(1, y, x\right) \]

   if 0.0200000000000000004 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 4

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

      5. distribute-lft-neg-out N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{y}{y - -1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 61.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+16} \lor \neg \left(t\_0 \leq 0.9999999999977595\right):\\ \;\;\;\;\frac{x}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))))
   (if (or (<= t_0 -4e+16) (not (<= t_0 0.9999999999977595)))
     (/ x (- y -1.0))
     (fma 1.0 y x))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + 1.0);
	double tmp;
	if ((t_0 <= -4e+16) || !(t_0 <= 0.9999999999977595)) {
		tmp = x / (y - -1.0);
	} else {
		tmp = fma(1.0, y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + 1.0))
	tmp = 0.0
	if ((t_0 <= -4e+16) || !(t_0 <= 0.9999999999977595))
		tmp = Float64(x / Float64(y - -1.0));
	else
		tmp = fma(1.0, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e+16], N[Not[LessEqual[t$95$0, 0.9999999999977595]], $MachinePrecision]], N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < -4e16 or 0.999999999997759459 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

      5. distribute-lft-neg-out N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if -4e16 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.999999999997759459

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot x}, y, x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(1 - \color{blue}{1} \cdot x, y, x\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(1 - \color{blue}{x}, y, x\right) \]

      10. lower--.f64 98.7

          \[\leadsto \mathsf{fma}\left(\color{blue}{1 - x}, y, x\right) \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, y, x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.7%

      \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{y + 1} \leq -4 \cdot 10^{+16} \lor \neg \left(\frac{x + y}{y + 1} \leq 0.9999999999977595\right):\\ \;\;\;\;\frac{x}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (/ x y) (fma (- 1.0 x) y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x / y;
	} else {
		tmp = fma((1.0 - x), y, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(x / y);
	else
		tmp = fma(Float64(1.0 - x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

      5. distribute-lft-neg-out N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 29.5%

      \[\leadsto \frac{x}{\color{blue}{y}} \]

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot x}, y, x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(1 - \color{blue}{1} \cdot x, y, x\right) \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(1 - \color{blue}{x}, y, x\right) \]

      10. lower--.f64 99.6

          \[\leadsto \mathsf{fma}\left(\color{blue}{1 - x}, y, x\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-134} \lor \neg \left(x \leq 1.15 \cdot 10^{-239}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.55e-134) (not (<= x 1.15e-239))) (* 1.0 x) (* 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.55e-134) || !(x <= 1.15e-239)) {
		tmp = 1.0 * x;
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.55d-134)) .or. (.not. (x <= 1.15d-239))) then
        tmp = 1.0d0 * x
    else
        tmp = 1.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.55e-134) || !(x <= 1.15e-239)) {
		tmp = 1.0 * x;
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.55e-134) or not (x <= 1.15e-239):
		tmp = 1.0 * x
	else:
		tmp = 1.0 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.55e-134) || !(x <= 1.15e-239))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(1.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.55e-134) || ~((x <= 1.15e-239)))
		tmp = 1.0 * x;
	else
		tmp = 1.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.55e-134], N[Not[LessEqual[x, 1.15e-239]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(1.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000003e-134 or 1.1499999999999999e-239 < x

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

      5. distribute-lft-neg-out N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 69.2

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

   5. Applied rewrites 69.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.4%

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

   9. Taylor expanded in y around inf

      \[\leadsto \left(-1 \cdot y\right) \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 2.9%

       \[\leadsto \left(-y\right) \cdot x \]

   12. Taylor expanded in y around 0

       \[\leadsto 1 \cdot x \]
   13. Step-by-step derivation

   14. Applied rewrites 53.8%

       \[\leadsto 1 \cdot x \]

   if -1.55000000000000003e-134 < x < 1.1499999999999999e-239

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

      5. distribute-lft-neg-out N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + -1 \cdot y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.6%

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

   9. Taylor expanded in y around inf

      \[\leadsto \left(-1 \cdot y\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 3.4%

       \[\leadsto \left(-y\right) \cdot y \]

   12. Taylor expanded in y around 0

       \[\leadsto 1 \cdot y \]
   13. Step-by-step derivation

   14. Applied rewrites 45.5%

       \[\leadsto 1 \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-134} \lor \neg \left(x \leq 1.15 \cdot 10^{-239}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, y, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma 1.0 y x))

C

double code(double x, double y) {
	return fma(1.0, y, x);
}

Julia

function code(x, y)
	return fma(1.0, y, x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x + y}{y + 1} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. lower-fma.f64 N/A

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

   7. fp-cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot x}, y, x\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(1 - \color{blue}{1} \cdot x, y, x\right) \]

   9. *-lft-identity N/A

      \[\leadsto \mathsf{fma}\left(1 - \color{blue}{x}, y, x\right) \]

   10. lower--.f64 56.6

       \[\leadsto \mathsf{fma}\left(\color{blue}{1 - x}, y, x\right) \]

5. Applied rewrites 56.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - x, y, x\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 56.7%

   \[\leadsto \mathsf{fma}\left(1, y, x\right) \]
9. Add Preprocessing

Alternative 7: 38.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x + y}{y + 1} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. rgt-mult-inverse N/A

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

   4. cancel-sign-sub N/A

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

   5. distribute-lft-neg-out N/A

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

   6. rgt-mult-inverse N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 59.3

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

5. Applied rewrites 59.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 46.4%

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

9. Taylor expanded in y around inf

   \[\leadsto \left(-1 \cdot y\right) \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 3.2%

    \[\leadsto \left(-y\right) \cdot x \]

12. Taylor expanded in y around 0

    \[\leadsto 1 \cdot x \]
13. Step-by-step derivation

14. Applied rewrites 46.7%

    \[\leadsto 1 \cdot x \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))