Data.Colour.SRGB:invTransferFunction from colour-2.3.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. x = -5.645753822456889e-114
  2. y = -1.539930342494309e-113

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. Final simplification 100.0%

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

Alternative 2: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+55}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   1.0
   (if (<= y 0.175)
     x
     (if (<= y 1.85e+55) 1.0 (if (<= y 5.2e+104) (/ x y) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 0.175) {
		tmp = x;
	} else if (y <= 1.85e+55) {
		tmp = 1.0;
	} else if (y <= 5.2e+104) {
		tmp = x / y;
	} 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 <= 0.175d0) then
        tmp = x
    else if (y <= 1.85d+55) then
        tmp = 1.0d0
    else if (y <= 5.2d+104) then
        tmp = x / y
    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 <= 0.175) {
		tmp = x;
	} else if (y <= 1.85e+55) {
		tmp = 1.0;
	} else if (y <= 5.2e+104) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 0.175:
		tmp = x
	elif y <= 1.85e+55:
		tmp = 1.0
	elif y <= 5.2e+104:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 0.175)
		tmp = x;
	elseif (y <= 1.85e+55)
		tmp = 1.0;
	elseif (y <= 5.2e+104)
		tmp = Float64(x / y);
	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 <= 0.175)
		tmp = x;
	elseif (y <= 1.85e+55)
		tmp = 1.0;
	elseif (y <= 5.2e+104)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 0.175], x, If[LessEqual[y, 1.85e+55], 1.0, If[LessEqual[y, 5.2e+104], N[(x / y), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 0.17499999999999999 < y < 1.8500000000000001e55 or 5.20000000000000001e104 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 76.8%

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

   if -1 < y < 0.17499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 68.4%

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

   if 1.8500000000000001e55 < y < 5.20000000000000001e104

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 74.2%

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

      1. +-commutative 74.2%

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

   5. Simplified 74.2%

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

   6. Taylor expanded in y around inf 74.2%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.175:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+55}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.26\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.26))) (+ 1.0 (/ (+ x -1.0) y)) (+ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.26)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x + y;
	}
	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)) .or. (.not. (y <= 1.26d0))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.26)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.26):
		tmp = 1.0 + ((x + -1.0) / y)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.26))
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.26)))
		tmp = 1.0 + ((x + -1.0) / y);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.26]], $MachinePrecision]], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1.26000000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.3%

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

      1. +-commutative 98.3%

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

      2. associate--l+ 98.3%

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

      3. +-commutative 98.3%

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

      4. associate--r- 98.3%

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

      5. div-sub 98.3%

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

   5. Simplified 98.3%

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

   if -1 < y < 1.26000000000000001

   1. Initial program 100.0%

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

      1. flip3-+ 100.0%

         \[\leadsto \frac{x + y}{\color{blue}{\frac{{y}^{3} + {1}^{3}}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)}}} \]

      2. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{x + y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)} \]

      3. metadata-eval 99.9%

         \[\leadsto \frac{x + y}{{y}^{3} + \color{blue}{1}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \]

      4. +-commutative 99.9%

         \[\leadsto \frac{x + y}{\color{blue}{1 + {y}^{3}}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(\color{blue}{1} - y \cdot 1\right)\right) \]

      6. *-rgt-identity 99.9%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - \color{blue}{y}\right)\right) \]

      7. associate-+r- 99.9%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \color{blue}{\left(\left(y \cdot y + 1\right) - y\right)} \]

      8. fma-def 99.9%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(\color{blue}{\mathsf{fma}\left(y, y, 1\right)} - y\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x + y}{1 + {y}^{3}} \cdot \left(\mathsf{fma}\left(y, y, 1\right) - y\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{x + y}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{\color{blue}{y + x}}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y + x}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}}} \]

   7. Taylor expanded in y around 0 98.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.26\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.0039\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.0039))) (+ 1.0 (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.0039)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x;
	}
	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)) .or. (.not. (y <= 0.0039d0))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.0039)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.0039):
		tmp = 1.0 + (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.0039))
		tmp = Float64(1.0 + Float64(x / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.0039)))
		tmp = 1.0 + (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.0039]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.0038999999999999998 < y

   1. Initial program 100.0%

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

      1. flip3-+ 37.1%

         \[\leadsto \frac{x + y}{\color{blue}{\frac{{y}^{3} + {1}^{3}}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)}}} \]

      2. associate-/r/ 37.2%

         \[\leadsto \color{blue}{\frac{x + y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)} \]

      3. metadata-eval 37.2%

         \[\leadsto \frac{x + y}{{y}^{3} + \color{blue}{1}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \]

      4. +-commutative 37.2%

         \[\leadsto \frac{x + y}{\color{blue}{1 + {y}^{3}}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \]

      5. metadata-eval 37.2%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(\color{blue}{1} - y \cdot 1\right)\right) \]

      6. *-rgt-identity 37.2%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - \color{blue}{y}\right)\right) \]

      7. associate-+r- 37.2%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \color{blue}{\left(\left(y \cdot y + 1\right) - y\right)} \]

      8. fma-def 37.2%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(\color{blue}{\mathsf{fma}\left(y, y, 1\right)} - y\right) \]

   4. Applied egg-rr 37.2%

      \[\leadsto \color{blue}{\frac{x + y}{1 + {y}^{3}} \cdot \left(\mathsf{fma}\left(y, y, 1\right) - y\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ 37.1%

         \[\leadsto \color{blue}{\frac{x + y}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}}} \]

      2. +-commutative 37.1%

         \[\leadsto \frac{\color{blue}{y + x}}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}} \]

   6. Simplified 37.1%

      \[\leadsto \color{blue}{\frac{y + x}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}}} \]

   7. Taylor expanded in y around inf 97.2%

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

   8. Taylor expanded in y around 0 97.2%

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

      1. +-commutative 97.2%

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

   10. Simplified 97.2%

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

   if -1 < y < 0.0038999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 68.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.0039\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3900 \lor \neg \left(y \leq 14000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3900.0) (not (<= y 14000.0))) (+ 1.0 (/ x y)) (/ x (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3900.0) || !(y <= 14000.0)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x / (y + 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 <= (-3900.0d0)) .or. (.not. (y <= 14000.0d0))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = x / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3900.0) || !(y <= 14000.0)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3900.0) or not (y <= 14000.0):
		tmp = 1.0 + (x / y)
	else:
		tmp = x / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3900.0) || !(y <= 14000.0))
		tmp = Float64(1.0 + Float64(x / y));
	else
		tmp = Float64(x / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3900.0) || ~((y <= 14000.0)))
		tmp = 1.0 + (x / y);
	else
		tmp = x / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3900.0], N[Not[LessEqual[y, 14000.0]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3900 or 14000 < y

   1. Initial program 100.0%

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

      1. flip3-+ 35.2%

         \[\leadsto \frac{x + y}{\color{blue}{\frac{{y}^{3} + {1}^{3}}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)}}} \]

      2. associate-/r/ 35.2%

         \[\leadsto \color{blue}{\frac{x + y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)} \]

      3. metadata-eval 35.2%

         \[\leadsto \frac{x + y}{{y}^{3} + \color{blue}{1}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \]

      4. +-commutative 35.2%

         \[\leadsto \frac{x + y}{\color{blue}{1 + {y}^{3}}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \]

      5. metadata-eval 35.2%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(\color{blue}{1} - y \cdot 1\right)\right) \]

      6. *-rgt-identity 35.2%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - \color{blue}{y}\right)\right) \]

      7. associate-+r- 35.2%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \color{blue}{\left(\left(y \cdot y + 1\right) - y\right)} \]

      8. fma-def 35.2%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(\color{blue}{\mathsf{fma}\left(y, y, 1\right)} - y\right) \]

   4. Applied egg-rr 35.2%

      \[\leadsto \color{blue}{\frac{x + y}{1 + {y}^{3}} \cdot \left(\mathsf{fma}\left(y, y, 1\right) - y\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ 35.2%

         \[\leadsto \color{blue}{\frac{x + y}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}}} \]

      2. +-commutative 35.2%

         \[\leadsto \frac{\color{blue}{y + x}}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}} \]

   6. Simplified 35.2%

      \[\leadsto \color{blue}{\frac{y + x}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}}} \]

   7. Taylor expanded in y around inf 99.4%

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

   8. Taylor expanded in y around 0 99.4%

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

      1. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   if -3900 < y < 14000

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 68.0%

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

      1. +-commutative 68.0%

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

   5. Simplified 68.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3900 \lor \neg \left(y \leq 14000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x + y;
	}
	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)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = 1.0 + (x / y)
	else:
		tmp = x + y
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = 1.0 + (x / y);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

      1. flip3-+ 36.7%

         \[\leadsto \frac{x + y}{\color{blue}{\frac{{y}^{3} + {1}^{3}}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)}}} \]

      2. associate-/r/ 36.7%

         \[\leadsto \color{blue}{\frac{x + y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)} \]

      3. metadata-eval 36.7%

         \[\leadsto \frac{x + y}{{y}^{3} + \color{blue}{1}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \]

      4. +-commutative 36.7%

         \[\leadsto \frac{x + y}{\color{blue}{1 + {y}^{3}}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \]

      5. metadata-eval 36.7%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(\color{blue}{1} - y \cdot 1\right)\right) \]

      6. *-rgt-identity 36.7%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - \color{blue}{y}\right)\right) \]

      7. associate-+r- 36.7%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \color{blue}{\left(\left(y \cdot y + 1\right) - y\right)} \]

      8. fma-def 36.7%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(\color{blue}{\mathsf{fma}\left(y, y, 1\right)} - y\right) \]

   4. Applied egg-rr 36.7%

      \[\leadsto \color{blue}{\frac{x + y}{1 + {y}^{3}} \cdot \left(\mathsf{fma}\left(y, y, 1\right) - y\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ 36.7%

         \[\leadsto \color{blue}{\frac{x + y}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}}} \]

      2. +-commutative 36.7%

         \[\leadsto \frac{\color{blue}{y + x}}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}} \]

   6. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{y + x}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}}} \]

   7. Taylor expanded in y around inf 97.8%

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

   8. Taylor expanded in y around 0 97.8%

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

      1. +-commutative 97.8%

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

   10. Simplified 97.8%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. flip3-+ 100.0%

         \[\leadsto \frac{x + y}{\color{blue}{\frac{{y}^{3} + {1}^{3}}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)}}} \]

      2. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{x + y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)} \]

      3. metadata-eval 99.9%

         \[\leadsto \frac{x + y}{{y}^{3} + \color{blue}{1}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \]

      4. +-commutative 99.9%

         \[\leadsto \frac{x + y}{\color{blue}{1 + {y}^{3}}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(\color{blue}{1} - y \cdot 1\right)\right) \]

      6. *-rgt-identity 99.9%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - \color{blue}{y}\right)\right) \]

      7. associate-+r- 99.9%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \color{blue}{\left(\left(y \cdot y + 1\right) - y\right)} \]

      8. fma-def 99.9%

         \[\leadsto \frac{x + y}{1 + {y}^{3}} \cdot \left(\color{blue}{\mathsf{fma}\left(y, y, 1\right)} - y\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x + y}{1 + {y}^{3}} \cdot \left(\mathsf{fma}\left(y, y, 1\right) - y\right)} \]
   5. Step-by-step derivation

      1. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{x + y}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{\color{blue}{y + x}}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{y + x}{\frac{1 + {y}^{3}}{\mathsf{fma}\left(y, y, 1\right) - y}}} \]

   7. Taylor expanded in y around 0 98.4%

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

4. Final simplification 98.1%

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

Alternative 7: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.006:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -1.0) 1.0 (if (<= y 0.006) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 0.006) {
		tmp = 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 <= 0.006d0) then
        tmp = 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 <= 0.006) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = 1.0
	elif y <= 0.006:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = 1.0;
	elseif (y <= 0.006)
		tmp = 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 <= 0.006)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], 1.0, If[LessEqual[y, 0.006], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.0060000000000000001 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 72.9%

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

   if -1 < y < 0.0060000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 68.4%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.006:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.1% accurate, 7.0× 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 39.7%

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

4. Final simplification 39.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x y)
  :name "Data.Colour.SRGB:invTransferFunction from colour-2.3.3"
  :precision binary64
  (/ (+ x y) (+ y 1.0)))