Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 11

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: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ t_1 := \frac{x}{y + 1}\\ \mathbf{if}\;t\_0 \leq -1000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))) (t_1 (/ x (+ y 1.0))))
   (if (<= t_0 -1000000000000.0)
     t_1
     (if (<= t_0 4e-14)
       (fma y (- 1.0 x) x)
       (if (<= t_0 2.0) (/ 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 <= -1000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 4e-14) {
		tmp = fma(y, (1.0 - x), x);
	} else if (t_0 <= 2.0) {
		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 <= -1000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 4e-14)
		tmp = fma(y, Float64(1.0 - x), x);
	elseif (t_0 <= 2.0)
		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, -1000000000000.0], t$95$1, If[LessEqual[t$95$0, 4e-14], N[(y * N[(1.0 - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 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))) < -1e12 or 2 < (/.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. lower-+.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -1e12 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 4e-14

   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 4e-14 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 2

   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. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

Alternative 3: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ t_1 := \frac{x}{y + 1}\\ \mathbf{if}\;t\_0 \leq -1000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 + \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))) (t_1 (/ x (+ y 1.0))))
   (if (<= t_0 -1000000000000.0)
     t_1
     (if (<= t_0 0.01)
       (fma y (- 1.0 x) x)
       (if (<= t_0 2.0) (+ 1.0 (/ -1.0 y)) 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 <= -1000000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.01) {
		tmp = fma(y, (1.0 - x), x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 + (-1.0 / y);
	} 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 <= -1000000000000.0)
		tmp = t_1;
	elseif (t_0 <= 0.01)
		tmp = fma(y, Float64(1.0 - x), x);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 + Float64(-1.0 / y));
	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, -1000000000000.0], t$95$1, If[LessEqual[t$95$0, 0.01], N[(y * N[(1.0 - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < -1e12 or 2 < (/.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. lower-+.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -1e12 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.0100000000000000002

   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 0.0100000000000000002 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 2

   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. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 97.4%

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

Alternative 4: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + 1}\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+25}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y 1.0))))
   (if (<= t_0 0.01) (fma y 1.0 x) (if (<= t_0 5e+25) 1.0 (fma y 1.0 x)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + 1.0);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = fma(y, 1.0, x);
	} else if (t_0 <= 5e+25) {
		tmp = 1.0;
	} else {
		tmp = fma(y, 1.0, x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = fma(y, 1.0, x);
	elseif (t_0 <= 5e+25)
		tmp = 1.0;
	else
		tmp = fma(y, 1.0, 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[LessEqual[t$95$0, 0.01], N[(y * 1.0 + x), $MachinePrecision], If[LessEqual[t$95$0, 5e+25], 1.0, N[(y * 1.0 + x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.0100000000000000002 or 5.00000000000000024e25 < (/.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 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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.4%

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

   if 0.0100000000000000002 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 5.00000000000000024e25

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 92.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (fma (- (* x y) y) (+ y -1.0) x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma(((x * y) - y), (y + -1.0), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = fma(Float64(Float64(x * y) - y), Float64(y + -1.0), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

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

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 y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-in N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 98.7%

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

4. Final simplification 99.0%

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

Alternative 6: 98.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (fma y (- 1.0 x) x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma(y, (1.0 - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

Wolfram

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

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 y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. associate-*r* N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -1 < y < 1

   1. Initial program 99.9%

      \[\frac{x + y}{y + 1} \]
   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 7: 50.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + y}{y + 1} \leq 0.01:\\ \;\;\;\;y \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (+ x y) (+ y 1.0)) 0.01) (* y 1.0) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (y + 1.0)) <= 0.01) {
		tmp = y * 1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x + y) / (y + 1.0d0)) <= 0.01d0) then
        tmp = y * 1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x + y) / (y + 1.0)) <= 0.01) {
		tmp = y * 1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) / (y + 1.0)) <= 0.01:
		tmp = y * 1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) / Float64(y + 1.0)) <= 0.01)
		tmp = Float64(y * 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) / (y + 1.0)) <= 0.01)
		tmp = y * 1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], 0.01], N[(y * 1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 0.0100000000000000002

   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. lower-+.f64 29.4

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

   5. Applied rewrites 29.4%

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

   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 29.4%

      \[\leadsto y - \color{blue}{y \cdot y} \]
   9. Step-by-step derivation

   10. Applied rewrites 29.4%

       \[\leadsto \left(\left(-y\right) + 1\right) \cdot y \]

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 28.5%

       \[\leadsto 1 \cdot y \]

   if 0.0100000000000000002 < (/.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 y around inf

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

   5. Applied rewrites 71.8%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{y + 1} \leq 0.01:\\ \;\;\;\;y \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= y -1.0) t_0 (if (<= y 0.68) (fma y (- 1.0 x) x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 0.68) {
		tmp = fma(y, (1.0 - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.68)
		tmp = fma(y, Float64(1.0 - x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 0.68], N[(y * N[(1.0 - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.680000000000000049 < y

   1. Initial program 100.0%

      \[\frac{x + y}{y + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 51.3

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

   4. Applied rewrites 51.3%

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

   5. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. associate-*r/ N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

          \[\leadsto 1 + -1 \cdot \frac{\color{blue}{0 - -1 \cdot \left(1 + -1 \cdot x\right)}}{y} \]

      13. metadata-eval N/A

          \[\leadsto 1 + -1 \cdot \frac{\color{blue}{-1 \cdot 0} - -1 \cdot \left(1 + -1 \cdot x\right)}{y} \]

      14. mul0-lft N/A

          \[\leadsto 1 + -1 \cdot \frac{-1 \cdot \color{blue}{\left(0 \cdot {x}^{2}\right)} - -1 \cdot \left(1 + -1 \cdot x\right)}{y} \]

      15. metadata-eval N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-+.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.7%

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

   if -1 < y < 0.680000000000000049

   1. Initial program 99.9%

      \[\frac{x + y}{y + 1} \]
   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 9: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{-1}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -1.0 y))))
   (if (<= y -1.0) t_0 (if (<= y 1.05) (fma y (- 1.0 x) x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-1.0 / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.05) {
		tmp = fma(y, (1.0 - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-1.0 / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.05)
		tmp = fma(y, Float64(1.0 - x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.05], N[(y * N[(1.0 - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.05000000000000004 < y

   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. lower-+.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 79.1%

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

   if -1 < y < 1.05000000000000004

   1. Initial program 99.9%

      \[\frac{x + y}{y + 1} \]
   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 10: 86.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.0) (fma y (- 1.0 x) x) 1.0)))

C

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

Julia

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

Wolfram

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

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 y around inf

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

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{1} \]

   if -1 < y < 1

   1. Initial program 99.9%

      \[\frac{x + y}{y + 1} \]
   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. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 11: 40.0% accurate, 18.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 42.2%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024238 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))