Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.1% → 99.9%

Time: 5.1s

Alternatives: 10

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: 88.1% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. associate-/l* 99.9%

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

   2. associate-/r/ 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := y + \frac{y}{x}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{1}{t_1}\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{-1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (+ y (/ y x))))
   (if (<= x -7.8e+15)
     t_0
     (if (<= x -8.5e-110)
       (* x (/ 1.0 t_1))
       (if (<= x 1.36e-124)
         x
         (if (<= x 2.7e-7) (/ x t_1) (+ t_0 (/ -1.0 y))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = y + (y / x);
	double tmp;
	if (x <= -7.8e+15) {
		tmp = t_0;
	} else if (x <= -8.5e-110) {
		tmp = x * (1.0 / t_1);
	} else if (x <= 1.36e-124) {
		tmp = x;
	} else if (x <= 2.7e-7) {
		tmp = x / t_1;
	} else {
		tmp = t_0 + (-1.0 / y);
	}
	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 = 1.0d0 + (x / y)
    t_1 = y + (y / x)
    if (x <= (-7.8d+15)) then
        tmp = t_0
    else if (x <= (-8.5d-110)) then
        tmp = x * (1.0d0 / t_1)
    else if (x <= 1.36d-124) then
        tmp = x
    else if (x <= 2.7d-7) then
        tmp = x / t_1
    else
        tmp = t_0 + ((-1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = y + (y / x);
	double tmp;
	if (x <= -7.8e+15) {
		tmp = t_0;
	} else if (x <= -8.5e-110) {
		tmp = x * (1.0 / t_1);
	} else if (x <= 1.36e-124) {
		tmp = x;
	} else if (x <= 2.7e-7) {
		tmp = x / t_1;
	} else {
		tmp = t_0 + (-1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	t_1 = y + (y / x)
	tmp = 0
	if x <= -7.8e+15:
		tmp = t_0
	elif x <= -8.5e-110:
		tmp = x * (1.0 / t_1)
	elif x <= 1.36e-124:
		tmp = x
	elif x <= 2.7e-7:
		tmp = x / t_1
	else:
		tmp = t_0 + (-1.0 / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	t_1 = Float64(y + Float64(y / x))
	tmp = 0.0
	if (x <= -7.8e+15)
		tmp = t_0;
	elseif (x <= -8.5e-110)
		tmp = Float64(x * Float64(1.0 / t_1));
	elseif (x <= 1.36e-124)
		tmp = x;
	elseif (x <= 2.7e-7)
		tmp = Float64(x / t_1);
	else
		tmp = Float64(t_0 + Float64(-1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	t_1 = y + (y / x);
	tmp = 0.0;
	if (x <= -7.8e+15)
		tmp = t_0;
	elseif (x <= -8.5e-110)
		tmp = x * (1.0 / t_1);
	elseif (x <= 1.36e-124)
		tmp = x;
	elseif (x <= 2.7e-7)
		tmp = x / t_1;
	else
		tmp = t_0 + (-1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.8e+15], t$95$0, If[LessEqual[x, -8.5e-110], N[(x * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.36e-124], x, If[LessEqual[x, 2.7e-7], N[(x / t$95$1), $MachinePrecision], N[(t$95$0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -7.8e15

   1. Initial program 76.2%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -7.8e15 < x < -8.50000000000000029e-110

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. remove-double-neg 99.5%

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

      5. unsub-neg 99.5%

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

      6. div-sub 99.4%

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

      7. distribute-frac-neg 99.4%

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

      8. *-inverses 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 71.1%

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

      1. distribute-lft-in 71.1%

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

      2. *-rgt-identity 71.1%

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

      3. associate-*r/ 71.2%

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

      4. *-rgt-identity 71.2%

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

   6. Simplified 71.2%

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

      1. clear-num 71.1%

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

      2. associate-/r/ 71.3%

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

   8. Applied egg-rr 71.3%

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

   if -8.50000000000000029e-110 < x < 1.3599999999999999e-124

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 89.5%

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

   if 1.3599999999999999e-124 < x < 2.70000000000000009e-7

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

      4. remove-double-neg 99.8%

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

      5. unsub-neg 99.8%

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

      6. div-sub 99.8%

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

      7. distribute-frac-neg 99.8%

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

      8. *-inverses 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 67.2%

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

      1. distribute-lft-in 67.2%

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

      2. *-rgt-identity 67.2%

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

      3. associate-*r/ 67.4%

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

      4. *-rgt-identity 67.4%

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

   6. Simplified 67.4%

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

   if 2.70000000000000009e-7 < x

   1. Initial program 82.2%

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

   2. Taylor expanded in x around inf 97.3%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{y}\right) - \frac{1}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+15}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{1}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \end{array} \]

Alternative 3: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 98000000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ x y))))
   (if (<= x -1.0)
     (/ x y)
     (if (<= x -8.2e-108)
       t_0
       (if (<= x 1.36e-124)
         x
         (if (<= x 5.1e-49)
           t_0
           (if (<= x 98000000.0) (/ x (+ x 1.0)) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = x * (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -8.2e-108) {
		tmp = t_0;
	} else if (x <= 1.36e-124) {
		tmp = x;
	} else if (x <= 5.1e-49) {
		tmp = t_0;
	} else if (x <= 98000000.0) {
		tmp = x / (x + 1.0);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * (x / y)
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= (-8.2d-108)) then
        tmp = t_0
    else if (x <= 1.36d-124) then
        tmp = x
    else if (x <= 5.1d-49) then
        tmp = t_0
    else if (x <= 98000000.0d0) then
        tmp = x / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -8.2e-108) {
		tmp = t_0;
	} else if (x <= 1.36e-124) {
		tmp = x;
	} else if (x <= 5.1e-49) {
		tmp = t_0;
	} else if (x <= 98000000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x / y)
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= -8.2e-108:
		tmp = t_0
	elif x <= 1.36e-124:
		tmp = x
	elif x <= 5.1e-49:
		tmp = t_0
	elif x <= 98000000.0:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x / y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= -8.2e-108)
		tmp = t_0;
	elseif (x <= 1.36e-124)
		tmp = x;
	elseif (x <= 5.1e-49)
		tmp = t_0;
	elseif (x <= 98000000.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x / y);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= -8.2e-108)
		tmp = t_0;
	elseif (x <= 1.36e-124)
		tmp = x;
	elseif (x <= 5.1e-49)
		tmp = t_0;
	elseif (x <= 98000000.0)
		tmp = x / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, -8.2e-108], t$95$0, If[LessEqual[x, 1.36e-124], x, If[LessEqual[x, 5.1e-49], t$95$0, If[LessEqual[x, 98000000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1 or 9.8e7 < x

   1. Initial program 79.0%

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

   2. Taylor expanded in x around inf 79.1%

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

   if -1 < x < -8.20000000000000074e-108 or 1.3599999999999999e-124 < x < 5.10000000000000026e-49

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. div-sub 99.5%

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

      7. distribute-frac-neg 99.5%

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

      8. *-inverses 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 71.8%

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

   5. Taylor expanded in x around 0 69.1%

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

      1. associate-/r/ 69.2%

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

   7. Applied egg-rr 69.2%

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

   if -8.20000000000000074e-108 < x < 1.3599999999999999e-124

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 89.5%

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

   if 5.10000000000000026e-49 < x < 9.8e7

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 58.7%

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

      1. +-commutative 58.7%

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

   4. Simplified 58.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 98000000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 82.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := \frac{x}{y + \frac{y}{x}}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (/ x (+ y (/ y x)))))
   (if (<= x -7.8e+15)
     t_0
     (if (<= x -3.6e-106)
       t_1
       (if (<= x 1.36e-124) x (if (<= x 2.7e-7) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = x / (y + (y / x));
	double tmp;
	if (x <= -7.8e+15) {
		tmp = t_0;
	} else if (x <= -3.6e-106) {
		tmp = t_1;
	} else if (x <= 1.36e-124) {
		tmp = x;
	} else if (x <= 2.7e-7) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    t_1 = x / (y + (y / x))
    if (x <= (-7.8d+15)) then
        tmp = t_0
    else if (x <= (-3.6d-106)) then
        tmp = t_1
    else if (x <= 1.36d-124) then
        tmp = x
    else if (x <= 2.7d-7) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = x / (y + (y / x));
	double tmp;
	if (x <= -7.8e+15) {
		tmp = t_0;
	} else if (x <= -3.6e-106) {
		tmp = t_1;
	} else if (x <= 1.36e-124) {
		tmp = x;
	} else if (x <= 2.7e-7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	t_1 = x / (y + (y / x))
	tmp = 0
	if x <= -7.8e+15:
		tmp = t_0
	elif x <= -3.6e-106:
		tmp = t_1
	elif x <= 1.36e-124:
		tmp = x
	elif x <= 2.7e-7:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	t_1 = Float64(x / Float64(y + Float64(y / x)))
	tmp = 0.0
	if (x <= -7.8e+15)
		tmp = t_0;
	elseif (x <= -3.6e-106)
		tmp = t_1;
	elseif (x <= 1.36e-124)
		tmp = x;
	elseif (x <= 2.7e-7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	t_1 = x / (y + (y / x));
	tmp = 0.0;
	if (x <= -7.8e+15)
		tmp = t_0;
	elseif (x <= -3.6e-106)
		tmp = t_1;
	elseif (x <= 1.36e-124)
		tmp = x;
	elseif (x <= 2.7e-7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.8e+15], t$95$0, If[LessEqual[x, -3.6e-106], t$95$1, If[LessEqual[x, 1.36e-124], x, If[LessEqual[x, 2.7e-7], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.8e15 or 2.70000000000000009e-7 < x

   1. Initial program 79.2%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 98.3%

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

   if -7.8e15 < x < -3.60000000000000013e-106 or 1.3599999999999999e-124 < x < 2.70000000000000009e-7

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. div-sub 99.6%

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

      7. distribute-frac-neg 99.6%

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

      8. *-inverses 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 69.6%

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

      1. distribute-lft-in 69.6%

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

      2. *-rgt-identity 69.6%

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

      3. associate-*r/ 69.7%

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

      4. *-rgt-identity 69.7%

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

   6. Simplified 69.7%

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

   if -3.60000000000000013e-106 < x < 1.3599999999999999e-124

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 89.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+15}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 5: 82.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := y + \frac{y}{x}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \frac{1}{t_1}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-125}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (+ y (/ y x))))
   (if (<= x -7.8e+15)
     t_0
     (if (<= x -5.2e-106)
       (* x (/ 1.0 t_1))
       (if (<= x 8e-125) x (if (<= x 2.7e-7) (/ x t_1) t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = y + (y / x);
	double tmp;
	if (x <= -7.8e+15) {
		tmp = t_0;
	} else if (x <= -5.2e-106) {
		tmp = x * (1.0 / t_1);
	} else if (x <= 8e-125) {
		tmp = x;
	} else if (x <= 2.7e-7) {
		tmp = x / t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    t_1 = y + (y / x)
    if (x <= (-7.8d+15)) then
        tmp = t_0
    else if (x <= (-5.2d-106)) then
        tmp = x * (1.0d0 / t_1)
    else if (x <= 8d-125) then
        tmp = x
    else if (x <= 2.7d-7) then
        tmp = x / t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = y + (y / x);
	double tmp;
	if (x <= -7.8e+15) {
		tmp = t_0;
	} else if (x <= -5.2e-106) {
		tmp = x * (1.0 / t_1);
	} else if (x <= 8e-125) {
		tmp = x;
	} else if (x <= 2.7e-7) {
		tmp = x / t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	t_1 = y + (y / x)
	tmp = 0
	if x <= -7.8e+15:
		tmp = t_0
	elif x <= -5.2e-106:
		tmp = x * (1.0 / t_1)
	elif x <= 8e-125:
		tmp = x
	elif x <= 2.7e-7:
		tmp = x / t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	t_1 = Float64(y + Float64(y / x))
	tmp = 0.0
	if (x <= -7.8e+15)
		tmp = t_0;
	elseif (x <= -5.2e-106)
		tmp = Float64(x * Float64(1.0 / t_1));
	elseif (x <= 8e-125)
		tmp = x;
	elseif (x <= 2.7e-7)
		tmp = Float64(x / t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	t_1 = y + (y / x);
	tmp = 0.0;
	if (x <= -7.8e+15)
		tmp = t_0;
	elseif (x <= -5.2e-106)
		tmp = x * (1.0 / t_1);
	elseif (x <= 8e-125)
		tmp = x;
	elseif (x <= 2.7e-7)
		tmp = x / t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.8e+15], t$95$0, If[LessEqual[x, -5.2e-106], N[(x * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e-125], x, If[LessEqual[x, 2.7e-7], N[(x / t$95$1), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.8e15 or 2.70000000000000009e-7 < x

   1. Initial program 79.2%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 98.3%

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

   if -7.8e15 < x < -5.2000000000000001e-106

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.5%

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

      3. +-commutative 99.5%

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

      4. remove-double-neg 99.5%

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

      5. unsub-neg 99.5%

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

      6. div-sub 99.4%

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

      7. distribute-frac-neg 99.4%

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

      8. *-inverses 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 71.1%

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

      1. distribute-lft-in 71.1%

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

      2. *-rgt-identity 71.1%

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

      3. associate-*r/ 71.2%

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

      4. *-rgt-identity 71.2%

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

   6. Simplified 71.2%

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

      1. clear-num 71.1%

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

      2. associate-/r/ 71.3%

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

   8. Applied egg-rr 71.3%

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

   if -5.2000000000000001e-106 < x < 8.0000000000000001e-125

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 89.5%

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

   if 8.0000000000000001e-125 < x < 2.70000000000000009e-7

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

      4. remove-double-neg 99.8%

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

      5. unsub-neg 99.8%

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

      6. div-sub 99.8%

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

      7. distribute-frac-neg 99.8%

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

      8. *-inverses 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 67.2%

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

      1. distribute-lft-in 67.2%

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

      2. *-rgt-identity 67.2%

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

      3. associate-*r/ 67.4%

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

      4. *-rgt-identity 67.4%

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

   6. Simplified 67.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+15}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \frac{1}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-125}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 6: 70.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ x y))))
   (if (<= x -1.0)
     (/ x y)
     (if (<= x -4.1e-106)
       t_0
       (if (<= x 1.36e-124) x (if (<= x 1.0) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x * (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -4.1e-106) {
		tmp = t_0;
	} else if (x <= 1.36e-124) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * (x / y)
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= (-4.1d-106)) then
        tmp = t_0
    else if (x <= 1.36d-124) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -4.1e-106) {
		tmp = t_0;
	} else if (x <= 1.36e-124) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x / y)
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= -4.1e-106:
		tmp = t_0
	elif x <= 1.36e-124:
		tmp = x
	elif x <= 1.0:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x / y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= -4.1e-106)
		tmp = t_0;
	elseif (x <= 1.36e-124)
		tmp = x;
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x / y);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= -4.1e-106)
		tmp = t_0;
	elseif (x <= 1.36e-124)
		tmp = x;
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, -4.1e-106], t$95$0, If[LessEqual[x, 1.36e-124], x, If[LessEqual[x, 1.0], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 1 < x

   1. Initial program 79.4%

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

   2. Taylor expanded in x around inf 77.9%

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

   if -1 < x < -4.0999999999999999e-106 or 1.3599999999999999e-124 < x < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. div-sub 99.6%

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

      7. distribute-frac-neg 99.6%

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

      8. *-inverses 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 65.9%

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

   5. Taylor expanded in x around 0 62.9%

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

      1. associate-/r/ 63.0%

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

   7. Applied egg-rr 63.0%

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

   if -4.0999999999999999e-106 < x < 1.3599999999999999e-124

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 89.5%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := x \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-125}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (* x (/ x y))))
   (if (<= x -1.0)
     t_0
     (if (<= x -1.45e-111)
       t_1
       (if (<= x 9.2e-125) x (if (<= x 2.7e-7) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = x * (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -1.45e-111) {
		tmp = t_1;
	} else if (x <= 9.2e-125) {
		tmp = x;
	} else if (x <= 2.7e-7) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    t_1 = x * (x / y)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-1.45d-111)) then
        tmp = t_1
    else if (x <= 9.2d-125) then
        tmp = x
    else if (x <= 2.7d-7) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = x * (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -1.45e-111) {
		tmp = t_1;
	} else if (x <= 9.2e-125) {
		tmp = x;
	} else if (x <= 2.7e-7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	t_1 = x * (x / y)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= -1.45e-111:
		tmp = t_1
	elif x <= 9.2e-125:
		tmp = x
	elif x <= 2.7e-7:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	t_1 = Float64(x * Float64(x / y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -1.45e-111)
		tmp = t_1;
	elseif (x <= 9.2e-125)
		tmp = x;
	elseif (x <= 2.7e-7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	t_1 = x * (x / y);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -1.45e-111)
		tmp = t_1;
	elseif (x <= 9.2e-125)
		tmp = x;
	elseif (x <= 2.7e-7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -1.45e-111], t$95$1, If[LessEqual[x, 9.2e-125], x, If[LessEqual[x, 2.7e-7], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 2.70000000000000009e-7 < x

   1. Initial program 79.6%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 98.3%

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

   if -1 < x < -1.45000000000000001e-111 or 9.1999999999999996e-125 < x < 2.70000000000000009e-7

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. div-sub 99.6%

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

      7. distribute-frac-neg 99.6%

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

      8. *-inverses 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 67.4%

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

   5. Taylor expanded in x around 0 64.4%

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

      1. associate-/r/ 64.5%

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

   7. Applied egg-rr 64.5%

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

   if -1.45000000000000001e-111 < x < 9.1999999999999996e-125

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 89.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-111}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-125}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 8: 73.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 6200\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 6200.0))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 6200.0)) {
		tmp = 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 ((x <= (-1.0d0)) .or. (.not. (x <= 6200.0d0))) then
        tmp = x / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 6200.0)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 6200.0):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 6200.0))
		tmp = Float64(x / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 6200.0)))
		tmp = x / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 6200.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 6200 < x

   1. Initial program 79.3%

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

   2. Taylor expanded in x around inf 78.4%

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

   if -1 < x < 6200

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 67.1%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 6200\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 49.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -105000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -105000000000.0) 1.0 (if (<= x 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -105000000000.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		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 (x <= (-105000000000.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) 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 (x <= -105000000000.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -105000000000.0:
		tmp = 1.0
	elif x <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -105000000000.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -105000000000.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -105000000000.0], 1.0, If[LessEqual[x, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e11 or 1 < x

   1. Initial program 79.3%

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

   2. Taylor expanded in y around inf 24.2%

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

      1. +-commutative 24.2%

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

   4. Simplified 24.2%

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

   5. Taylor expanded in x around inf 23.5%

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

   if -1.05e11 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 67.0%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -105000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 14.8% accurate, 11.0× 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 88.0%

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

2. Taylor expanded in y around inf 42.4%

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

   1. +-commutative 42.4%

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

4. Simplified 42.4%

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

5. Taylor expanded in x around inf 15.1%

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

6. Final simplification 15.1%

   \[\leadsto 1 \]

Developer target: 99.8% 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 2023308 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

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