Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.4% → 99.8%

Time: 7.1s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1}\\ t_1 := \frac{x}{\frac{y}{x} + y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 200000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- (/ x y) -1.0) x) (- x -1.0))) (t_1 (/ x (+ (/ y x) y))))
   (if (<= t_0 -1e+162) t_1 (if (<= t_0 200000000.0) t_0 t_1))))

C

double code(double x, double y) {
	double t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	double t_1 = x / ((y / x) + y);
	double tmp;
	if (t_0 <= -1e+162) {
		tmp = t_1;
	} else if (t_0 <= 200000000.0) {
		tmp = t_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 = (((x / y) - (-1.0d0)) * x) / (x - (-1.0d0))
    t_1 = x / ((y / x) + y)
    if (t_0 <= (-1d+162)) then
        tmp = t_1
    else if (t_0 <= 200000000.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	double t_1 = x / ((y / x) + y);
	double tmp;
	if (t_0 <= -1e+162) {
		tmp = t_1;
	} else if (t_0 <= 200000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) - -1.0) * x) / (x - -1.0)
	t_1 = x / ((y / x) + y)
	tmp = 0
	if t_0 <= -1e+162:
		tmp = t_1
	elif t_0 <= 200000000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) - -1.0) * x) / Float64(x - -1.0))
	t_1 = Float64(x / Float64(Float64(y / x) + y))
	tmp = 0.0
	if (t_0 <= -1e+162)
		tmp = t_1;
	elseif (t_0 <= 200000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	t_1 = x / ((y / x) + y);
	tmp = 0.0;
	if (t_0 <= -1e+162)
		tmp = t_1;
	elseif (t_0 <= 200000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+162], t$95$1, If[LessEqual[t$95$0, 200000000.0], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999994e161 or 2e8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 99.9%

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

   if -9.9999999999999994e161 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e8

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -1 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{\frac{y}{x} + y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 200000000:\\ \;\;\;\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{x} + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1}\\ t_1 := \frac{x - 1}{y}\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- (/ x y) -1.0) x) (- x -1.0))) (t_1 (/ (- x 1.0) y)))
   (if (<= t_0 -1000.0)
     t_1
     (if (<= t_0 0.5)
       (* (fma (- x 1.0) x 1.0) x)
       (if (<= t_0 2.0) (- 1.0 (/ 1.0 x)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	double t_1 = (x - 1.0) / y;
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.5) {
		tmp = fma((x - 1.0), x, 1.0) * x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (1.0 / x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) - -1.0) * x) / Float64(x - -1.0))
	t_1 = Float64(Float64(x - 1.0) / y)
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 0.5)
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(1.0 / x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], t$95$1, If[LessEqual[t$95$0, 0.5], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e3 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 79.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 87.0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 81.1%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 86.0%

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

   if 0.5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 93.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -1000:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 2:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1}\\ t_1 := \frac{x - 1}{y}\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- (/ x y) -1.0) x) (- x -1.0))) (t_1 (/ (- x 1.0) y)))
   (if (<= t_0 -1000.0)
     t_1
     (if (<= t_0 0.5)
       (fma (- x) x x)
       (if (<= t_0 2.0) (- 1.0 (/ 1.0 x)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	double t_1 = (x - 1.0) / y;
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.5) {
		tmp = fma(-x, x, x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (1.0 / x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) - -1.0) * x) / Float64(x - -1.0))
	t_1 = Float64(Float64(x - 1.0) / y)
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 0.5)
		tmp = fma(Float64(-x), x, x);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(1.0 / x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], t$95$1, If[LessEqual[t$95$0, 0.5], N[((-x) * x + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e3 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 79.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 87.0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 81.1%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.5

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.9%

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

   if 0.5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 93.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -1000:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 2:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1}\\ t_1 := \frac{x}{\frac{y}{x} + y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 200000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- (/ x y) -1.0) x) (- x -1.0))) (t_1 (/ x (+ (/ y x) y))))
   (if (<= t_0 -1e+162)
     t_1
     (if (<= t_0 200000000.0) (/ (fma (/ x y) x x) (- x -1.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	double t_1 = x / ((y / x) + y);
	double tmp;
	if (t_0 <= -1e+162) {
		tmp = t_1;
	} else if (t_0 <= 200000000.0) {
		tmp = fma((x / y), x, x) / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) - -1.0) * x) / Float64(x - -1.0))
	t_1 = Float64(x / Float64(Float64(y / x) + y))
	tmp = 0.0
	if (t_0 <= -1e+162)
		tmp = t_1;
	elseif (t_0 <= 200000000.0)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+162], t$95$1, If[LessEqual[t$95$0, 200000000.0], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -9.9999999999999994e161 or 2e8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 99.9%

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

   if -9.9999999999999994e161 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e8

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -1 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{\frac{y}{x} + y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 200000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{x} + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1}\\ t_1 := \frac{1}{y} \cdot x\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- (/ x y) -1.0) x) (- x -1.0))) (t_1 (* (/ 1.0 y) x)))
   (if (<= t_0 -1000.0) t_1 (if (<= t_0 2.0) (/ x (- x -1.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	double t_1 = (1.0 / y) * x;
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -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 = (((x / y) - (-1.0d0)) * x) / (x - (-1.0d0))
    t_1 = (1.0d0 / y) * x
    if (t_0 <= (-1000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	double t_1 = (1.0 / y) * x;
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) - -1.0) * x) / (x - -1.0)
	t_1 = (1.0 / y) * x
	tmp = 0
	if t_0 <= -1000.0:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) - -1.0) * x) / Float64(x - -1.0))
	t_1 = Float64(Float64(1.0 / y) * x)
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	t_1 = (1.0 / y) * x;
	tmp = 0.0;
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], t$95$1, If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e3 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 79.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 87.0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.8%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 89.0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -1000:\\ \;\;\;\;\frac{1}{y} \cdot x\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1}\\ t_1 := \frac{x - 1}{y}\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- (/ x y) -1.0) x) (- x -1.0))) (t_1 (/ (- x 1.0) y)))
   (if (<= t_0 -1000.0) t_1 (if (<= t_0 2.0) (/ x (- x -1.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	double t_1 = (x - 1.0) / y;
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -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 = (((x / y) - (-1.0d0)) * x) / (x - (-1.0d0))
    t_1 = (x - 1.0d0) / y
    if (t_0 <= (-1000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	double t_1 = (x - 1.0) / y;
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) - -1.0) * x) / (x - -1.0)
	t_1 = (x - 1.0) / y
	tmp = 0
	if t_0 <= -1000.0:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) - -1.0) * x) / Float64(x - -1.0))
	t_1 = Float64(Float64(x - 1.0) / y)
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	t_1 = (x - 1.0) / y;
	tmp = 0.0;
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], t$95$1, If[LessEqual[t$95$0, 2.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e3 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 79.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 87.0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 81.1%

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

   if -1e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 89.0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -1000:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 2:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (- (/ x y) -1.0) x) (- x -1.0)) -2e+44) (* (- x) x) (* 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (((((x / y) - -1.0) * x) / (x - -1.0)) <= -2e+44) {
		tmp = -x * x;
	} else {
		tmp = 1.0 * x;
	}
	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) - (-1.0d0)) * x) / (x - (-1.0d0))) <= (-2d+44)) then
        tmp = -x * x
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((((x / y) - -1.0) * x) / (x - -1.0)) <= -2e+44) {
		tmp = -x * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((((x / y) - -1.0) * x) / (x - -1.0)) <= -2e+44:
		tmp = -x * x
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x / y) - -1.0) * x) / Float64(x - -1.0)) <= -2e+44)
		tmp = Float64(Float64(-x) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((((x / y) - -1.0) * x) / (x - -1.0)) <= -2e+44)
		tmp = -x * x;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], -2e+44], N[((-x) * x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2.0000000000000002e44

   1. Initial program 76.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 1.1

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

   5. Applied rewrites 1.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto -1 \cdot {x}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.7%

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

   if -2.0000000000000002e44 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 94.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 50.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 47.5%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. clear-num-rev N/A

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

   6. lower-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-/.f64 99.8

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 81.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]
   6. Step-by-step derivation

      1. div-add-rev N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{1 + x}}}{y} \]

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 68.4

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-in N/A

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

      3. rgt-mult-inverse N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 98.4

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

   10. Applied rewrites 98.4%

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

   if -1 < x < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-in N/A

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

      8. neg-mul-1 N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 98.3%

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

Alternative 10: 86.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5200000000000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -5.4e+17) t_0 (if (<= x 5200000000000.0) (/ x (- x -1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -5.4e+17) {
		tmp = t_0;
	} else if (x <= 5200000000000.0) {
		tmp = x / (x - -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 = ((x - 1.0d0) / y) + 1.0d0
    if (x <= (-5.4d+17)) then
        tmp = t_0
    else if (x <= 5200000000000.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -5.4e+17) {
		tmp = t_0;
	} else if (x <= 5200000000000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - 1.0) / y) + 1.0
	tmp = 0
	if x <= -5.4e+17:
		tmp = t_0
	elif x <= 5200000000000.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -5.4e+17)
		tmp = t_0;
	elseif (x <= 5200000000000.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - 1.0) / y) + 1.0;
	tmp = 0.0;
	if (x <= -5.4e+17)
		tmp = t_0;
	elseif (x <= 5200000000000.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -5.4e+17], t$95$0, If[LessEqual[x, 5200000000000.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4e17 or 5.2e12 < x

   1. Initial program 81.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]
   6. Step-by-step derivation

      1. div-add-rev N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{1 + x}}}{y} \]

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 67.1

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

   7. Applied rewrites 67.1%

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

   8. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-in N/A

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

      3. rgt-mult-inverse N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -5.4e17 < x < 5.2e12

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+17}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{elif}\;x \leq 5200000000000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 81.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 74.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 74.1%

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

   if -1 < x < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.7%

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

Alternative 12: 42.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return fma(Float64(-x), x, x)
end

Wolfram

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

Derivation

1. Initial program 91.2%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 51.4

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

5. Applied rewrites 51.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 44.8%

   \[\leadsto \mathsf{fma}\left(-x, \color{blue}{x}, x\right) \]
9. Add Preprocessing

Alternative 13: 38.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 51.7%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 44.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 39.9%

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

Developer Target 1: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024298 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1))))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))