Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

   \[\leadsto \frac{y + x}{y - -1} \]
4. Add Preprocessing

Alternative 2: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{y - -1}\\ t_1 := \frac{x}{y - -1}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\left(\left(-y\right) - x\right) \cdot \left(y - 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- y -1.0))) (t_1 (/ x (- y -1.0))))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 1e-5)
       (* (- (- y) x) (- y 1.0))
       (if (<= t_0 2.0) (/ y (- y -1.0)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (y + x) / (y - -1.0);
	double t_1 = x / (y - -1.0);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (-y - x) * (y - 1.0);
	} else if (t_0 <= 2.0) {
		tmp = y / (y - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y + x) / (y - (-1.0d0))
    t_1 = x / (y - (-1.0d0))
    if (t_0 <= (-10.0d0)) then
        tmp = t_1
    else if (t_0 <= 1d-5) then
        tmp = (-y - x) * (y - 1.0d0)
    else if (t_0 <= 2.0d0) then
        tmp = y / (y - (-1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) / (y - -1.0);
	double t_1 = x / (y - -1.0);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (-y - x) * (y - 1.0);
	} else if (t_0 <= 2.0) {
		tmp = y / (y - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) / (y - -1.0)
	t_1 = x / (y - -1.0)
	tmp = 0
	if t_0 <= -10.0:
		tmp = t_1
	elif t_0 <= 1e-5:
		tmp = (-y - x) * (y - 1.0)
	elif t_0 <= 2.0:
		tmp = y / (y - -1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) / Float64(y - -1.0))
	t_1 = Float64(x / Float64(y - -1.0))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = Float64(Float64(Float64(-y) - x) * Float64(y - 1.0));
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) / (y - -1.0);
	t_1 = x / (y - -1.0);
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = (-y - x) * (y - 1.0);
	elseif (t_0 <= 2.0)
		tmp = y / (y - -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 1e-5], N[(N[((-y) - x), $MachinePrecision] * N[(y - 1.0), $MachinePrecision]), $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))) < -10 or 2 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 99.9%

      \[\frac{x + y}{y + 1} \]
   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. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -10 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if 1.00000000000000008e-5 < (/.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. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{y - -1} \leq -10:\\ \;\;\;\;\frac{x}{y - -1}\\ \mathbf{elif}\;\frac{y + x}{y - -1} \leq 10^{-5}:\\ \;\;\;\;\left(\left(-y\right) - x\right) \cdot \left(y - 1\right)\\ \mathbf{elif}\;\frac{y + x}{y - -1} \leq 2:\\ \;\;\;\;\frac{y}{y - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{y - -1}\\ t_1 := \frac{x}{y - -1}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\left(\left(-y\right) - x\right) \cdot \left(y - 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- y -1.0))) (t_1 (/ x (- y -1.0))))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 1e-5) (* (- (- y) x) (- y 1.0)) (if (<= t_0 2.0) 1.0 t_1)))))

C

double code(double x, double y) {
	double t_0 = (y + x) / (y - -1.0);
	double t_1 = x / (y - -1.0);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (-y - x) * (y - 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y + x) / (y - (-1.0d0))
    t_1 = x / (y - (-1.0d0))
    if (t_0 <= (-10.0d0)) then
        tmp = t_1
    else if (t_0 <= 1d-5) then
        tmp = (-y - x) * (y - 1.0d0)
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) / (y - -1.0);
	double t_1 = x / (y - -1.0);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (-y - x) * (y - 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) / (y - -1.0)
	t_1 = x / (y - -1.0)
	tmp = 0
	if t_0 <= -10.0:
		tmp = t_1
	elif t_0 <= 1e-5:
		tmp = (-y - x) * (y - 1.0)
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) / Float64(y - -1.0))
	t_1 = Float64(x / Float64(y - -1.0))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = Float64(Float64(Float64(-y) - x) * Float64(y - 1.0));
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) / (y - -1.0);
	t_1 = x / (y - -1.0);
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = (-y - x) * (y - 1.0);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 1e-5], N[(N[((-y) - x), $MachinePrecision] * N[(y - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < -10 or 2 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 99.9%

      \[\frac{x + y}{y + 1} \]
   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. lower-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -10 < (/.f64 (+.f64 x y) (+.f64 y #s(literal 1 binary64))) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if 1.00000000000000008e-5 < (/.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 y around inf

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

   5. Applied rewrites 97.2%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{y - -1} \leq -10:\\ \;\;\;\;\frac{x}{y - -1}\\ \mathbf{elif}\;\frac{y + x}{y - -1} \leq 10^{-5}:\\ \;\;\;\;\left(\left(-y\right) - x\right) \cdot \left(y - 1\right)\\ \mathbf{elif}\;\frac{y + x}{y - -1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(1.0 - x) / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = fma(Float64(Float64(x - 1.0) * 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[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(N[(N[(x - 1.0), $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. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   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 98.1%

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

Alternative 5: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = 1.0 - ((1.0 - x) / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (-y - x) * (y - 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((1.0d0 - x) / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = (-y - x) * (y - 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	t_0 = 1.0 - ((1.0 - x) / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = (-y - x) * (y - 1.0)
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - ((1.0 - x) / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = (-y - x) * (y - 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(N[((-y) - x), $MachinePrecision] * N[(y - 1.0), $MachinePrecision]), $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. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 97.6

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

   7. Applied rewrites 97.6%

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

Alternative 6: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{-x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\left(\left(-y\right) - x\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ (- x) y))))
   (if (<= y -1.0) t_0 (if (<= y 0.75) (* (- (- y) x) (- y 1.0)) 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 <= 0.75) {
		tmp = (-y - x) * (y - 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (-x / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 0.75d0) then
        tmp = (-y - x) * (y - 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static 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 <= 0.75) {
		tmp = (-y - x) * (y - 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (-x / y)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 0.75:
		tmp = (-y - x) * (y - 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(-x) / y))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.75)
		tmp = Float64(Float64(Float64(-y) - x) * Float64(y - 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (-x / y);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 0.75)
		tmp = (-y - x) * (y - 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = 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, 0.75], N[(N[((-y) - x), $MachinePrecision] * N[(y - 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.75 < 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. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.0%

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

   if -1 < y < 0.75

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 97.6

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

   7. Applied rewrites 97.6%

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

Alternative 7: 86.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = (-y - x) * (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 (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = (-y - x) * (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 (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = (-y - x) * (y - 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 1.0:
		tmp = (-y - x) * (y - 1.0)
	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 = Float64(Float64(Float64(-y) - x) * Float64(y - 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 1.0], N[(N[((-y) - x), $MachinePrecision] * N[(y - 1.0), $MachinePrecision]), $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 77.6%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 97.6

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

   7. Applied rewrites 97.6%

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

Alternative 8: 85.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 1.0) (fma (- 1.0 x) y 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((1.0 - x), y, 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(Float64(1.0 - x), y, 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[(N[(1.0 - x), $MachinePrecision] * y + 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 77.6%

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

   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. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 9: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 135000000:\\ \;\;\;\;\mathsf{fma}\left(1, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 135000000.0) (fma 1.0 y x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 135000000.0) {
		tmp = fma(1.0, y, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 135000000.0)
		tmp = fma(1.0, y, x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 135000000.0], N[(1.0 * y + x), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.35e8 < 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 78.2%

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

   if -1 < y < 1.35e8

   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. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 96.6

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.6%

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

Alternative 10: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 135000000:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) 1.0 (if (<= y 135000000.0) (* 1.0 x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 135000000.0) {
		tmp = 1.0 * x;
	} 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 (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 135000000.0d0) then
        tmp = 1.0d0 * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 135000000.0) {
		tmp = 1.0 * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 135000000.0:
		tmp = 1.0 * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 135000000.0)
		tmp = Float64(1.0 * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 135000000.0)
		tmp = 1.0 * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 135000000.0], N[(1.0 * x), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.35e8 < 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 78.2%

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

   if -1 < y < 1.35e8

   1. Initial program 99.9%

      \[\frac{x + y}{y + 1} \]
   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. lower-+.f64 73.2

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

   5. Applied rewrites 73.2%

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

   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 70.7%

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

   10. Applied rewrites 70.7%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 69.6%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 37.7% 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 37.6%

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

Reproduce

herbie shell --seed 2024277 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))