Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.1% → 99.9%

Time: 5.6s

Alternatives: 9

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

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

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(x / Float64(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[(x / N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.5%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -57000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;x \leq 112:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) - \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -57000.0)
     (+ 1.0 (/ (+ x -1.0) y))
     (if (<= x -9.6e-32)
       t_0
       (if (<= x -4e-81)
         (* x (- (/ x y) x))
         (if (<= x 112.0) t_0 (- (+ 1.0 (/ x y)) (/ 1.0 y))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -57000.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= -9.6e-32) {
		tmp = t_0;
	} else if (x <= -4e-81) {
		tmp = x * ((x / y) - x);
	} else if (x <= 112.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (x / y)) - (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) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-57000.0d0)) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else if (x <= (-9.6d-32)) then
        tmp = t_0
    else if (x <= (-4d-81)) then
        tmp = x * ((x / y) - x)
    else if (x <= 112.0d0) then
        tmp = t_0
    else
        tmp = (1.0d0 + (x / y)) - (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -57000.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= -9.6e-32) {
		tmp = t_0;
	} else if (x <= -4e-81) {
		tmp = x * ((x / y) - x);
	} else if (x <= 112.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (x / y)) - (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -57000.0:
		tmp = 1.0 + ((x + -1.0) / y)
	elif x <= -9.6e-32:
		tmp = t_0
	elif x <= -4e-81:
		tmp = x * ((x / y) - x)
	elif x <= 112.0:
		tmp = t_0
	else:
		tmp = (1.0 + (x / y)) - (1.0 / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -57000.0)
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	elseif (x <= -9.6e-32)
		tmp = t_0;
	elseif (x <= -4e-81)
		tmp = Float64(x * Float64(Float64(x / y) - x));
	elseif (x <= 112.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(x / y)) - Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -57000.0)
		tmp = 1.0 + ((x + -1.0) / y);
	elseif (x <= -9.6e-32)
		tmp = t_0;
	elseif (x <= -4e-81)
		tmp = x * ((x / y) - x);
	elseif (x <= 112.0)
		tmp = t_0;
	else
		tmp = (1.0 + (x / y)) - (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -57000.0], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.6e-32], t$95$0, If[LessEqual[x, -4e-81], N[(x * N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 112.0], t$95$0, N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] - N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -57000

   1. Initial program 77.2%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 99.0%

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

      1. associate--l+ 99.0%

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

      2. +-commutative 99.0%

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

      3. sub-div 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

   6. Applied egg-rr 99.0%

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

   if -57000 < x < -9.6000000000000005e-32 or -3.9999999999999998e-81 < x < 112

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 83.4%

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

   if -9.6000000000000005e-32 < x < -3.9999999999999998e-81

   1. Initial program 99.7%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y} - 1, {x}^{2}, x\right)} \]

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. unpow2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 66.8%

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

      1. unpow2 66.8%

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

      2. *-commutative 66.8%

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

      3. sub-neg 66.8%

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

      4. metadata-eval 66.8%

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

      5. +-commutative 66.8%

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

      6. distribute-rgt-in 66.8%

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

      7. neg-mul-1 66.8%

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

      8. distribute-lft-neg-in 66.8%

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

      9. associate-*l/ 66.8%

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

      10. *-commutative 66.8%

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

      11. *-rgt-identity 66.8%

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

      12. associate-*l/ 67.0%

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

      13. distribute-rgt-out 67.0%

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

   9. Simplified 67.0%

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

   if 112 < x

   1. Initial program 74.2%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 97.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -57000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;x \leq 112:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) - \frac{1}{y}\\ \end{array} \]

Alternative 3: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -950000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 220:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))) (t_1 (/ x (+ x 1.0))))
   (if (<= x -950000.0)
     t_0
     (if (<= x -6e-31)
       t_1
       (if (<= x -9e-83) (* x (/ x y)) (if (<= x 220.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -950000.0) {
		tmp = t_0;
	} else if (x <= -6e-31) {
		tmp = t_1;
	} else if (x <= -9e-83) {
		tmp = x * (x / y);
	} else if (x <= 220.0) {
		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 + (-1.0d0)) / y)
    t_1 = x / (x + 1.0d0)
    if (x <= (-950000.0d0)) then
        tmp = t_0
    else if (x <= (-6d-31)) then
        tmp = t_1
    else if (x <= (-9d-83)) then
        tmp = x * (x / y)
    else if (x <= 220.0d0) 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 + -1.0) / y);
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -950000.0) {
		tmp = t_0;
	} else if (x <= -6e-31) {
		tmp = t_1;
	} else if (x <= -9e-83) {
		tmp = x * (x / y);
	} else if (x <= 220.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	t_1 = x / (x + 1.0)
	tmp = 0
	if x <= -950000.0:
		tmp = t_0
	elif x <= -6e-31:
		tmp = t_1
	elif x <= -9e-83:
		tmp = x * (x / y)
	elif x <= 220.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -950000.0)
		tmp = t_0;
	elseif (x <= -6e-31)
		tmp = t_1;
	elseif (x <= -9e-83)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 220.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x + -1.0) / y);
	t_1 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -950000.0)
		tmp = t_0;
	elseif (x <= -6e-31)
		tmp = t_1;
	elseif (x <= -9e-83)
		tmp = x * (x / y);
	elseif (x <= 220.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -950000.0], t$95$0, If[LessEqual[x, -6e-31], t$95$1, If[LessEqual[x, -9e-83], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 220.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.5e5 or 220 < x

   1. Initial program 75.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 98.5%

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

      1. associate--l+ 98.5%

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

      2. +-commutative 98.5%

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

      3. sub-div 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

   6. Applied egg-rr 98.5%

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

   if -9.5e5 < x < -5.99999999999999962e-31 or -8.99999999999999995e-83 < x < 220

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 83.4%

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

   if -5.99999999999999962e-31 < x < -8.99999999999999995e-83

   1. Initial program 99.7%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y} - 1, {x}^{2}, x\right)} \]

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. unpow2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 66.8%

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

      1. sub-neg 66.8%

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

      2. metadata-eval 66.8%

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

      3. unpow2 66.8%

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

   9. Simplified 66.8%

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

   10. Taylor expanded in y around 0 66.4%

       \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \]
   11. Step-by-step derivation

       1. unpow2 66.4%

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

       2. associate-*r/ 66.6%

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

   12. Simplified 66.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -950000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 220:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]

Alternative 4: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -280000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;x \leq 220:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))) (t_1 (/ x (+ x 1.0))))
   (if (<= x -280000.0)
     t_0
     (if (<= x -2.8e-31)
       t_1
       (if (<= x -4.6e-81) (* x (- (/ x y) x)) (if (<= x 220.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -280000.0) {
		tmp = t_0;
	} else if (x <= -2.8e-31) {
		tmp = t_1;
	} else if (x <= -4.6e-81) {
		tmp = x * ((x / y) - x);
	} else if (x <= 220.0) {
		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 + (-1.0d0)) / y)
    t_1 = x / (x + 1.0d0)
    if (x <= (-280000.0d0)) then
        tmp = t_0
    else if (x <= (-2.8d-31)) then
        tmp = t_1
    else if (x <= (-4.6d-81)) then
        tmp = x * ((x / y) - x)
    else if (x <= 220.0d0) 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 + -1.0) / y);
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -280000.0) {
		tmp = t_0;
	} else if (x <= -2.8e-31) {
		tmp = t_1;
	} else if (x <= -4.6e-81) {
		tmp = x * ((x / y) - x);
	} else if (x <= 220.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	t_1 = x / (x + 1.0)
	tmp = 0
	if x <= -280000.0:
		tmp = t_0
	elif x <= -2.8e-31:
		tmp = t_1
	elif x <= -4.6e-81:
		tmp = x * ((x / y) - x)
	elif x <= 220.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -280000.0)
		tmp = t_0;
	elseif (x <= -2.8e-31)
		tmp = t_1;
	elseif (x <= -4.6e-81)
		tmp = Float64(x * Float64(Float64(x / y) - x));
	elseif (x <= 220.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x + -1.0) / y);
	t_1 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -280000.0)
		tmp = t_0;
	elseif (x <= -2.8e-31)
		tmp = t_1;
	elseif (x <= -4.6e-81)
		tmp = x * ((x / y) - x);
	elseif (x <= 220.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -280000.0], t$95$0, If[LessEqual[x, -2.8e-31], t$95$1, If[LessEqual[x, -4.6e-81], N[(x * N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 220.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8e5 or 220 < x

   1. Initial program 75.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 98.5%

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

      1. associate--l+ 98.5%

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

      2. +-commutative 98.5%

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

      3. sub-div 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

   6. Applied egg-rr 98.5%

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

   if -2.8e5 < x < -2.7999999999999999e-31 or -4.59999999999999982e-81 < x < 220

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 83.4%

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

   if -2.7999999999999999e-31 < x < -4.59999999999999982e-81

   1. Initial program 99.7%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y} - 1, {x}^{2}, x\right)} \]

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. unpow2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 66.8%

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

      1. unpow2 66.8%

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

      2. *-commutative 66.8%

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

      3. sub-neg 66.8%

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

      4. metadata-eval 66.8%

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

      5. +-commutative 66.8%

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

      6. distribute-rgt-in 66.8%

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

      7. neg-mul-1 66.8%

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

      8. distribute-lft-neg-in 66.8%

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

      9. associate-*l/ 66.8%

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

      10. *-commutative 66.8%

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

      11. *-rgt-identity 66.8%

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

      12. associate-*l/ 67.0%

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

      13. distribute-rgt-out 67.0%

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

   9. Simplified 67.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -280000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;x \leq 220:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]

Alternative 5: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.05 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+63}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -7.4e+34)
     (/ x y)
     (if (<= x -1.4e-31)
       t_0
       (if (<= x -4.05e-81) (* x (/ x y)) (if (<= x 2.05e+63) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -7.4e+34) {
		tmp = x / y;
	} else if (x <= -1.4e-31) {
		tmp = t_0;
	} else if (x <= -4.05e-81) {
		tmp = x * (x / y);
	} else if (x <= 2.05e+63) {
		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 + 1.0d0)
    if (x <= (-7.4d+34)) then
        tmp = x / y
    else if (x <= (-1.4d-31)) then
        tmp = t_0
    else if (x <= (-4.05d-81)) then
        tmp = x * (x / y)
    else if (x <= 2.05d+63) 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 + 1.0);
	double tmp;
	if (x <= -7.4e+34) {
		tmp = x / y;
	} else if (x <= -1.4e-31) {
		tmp = t_0;
	} else if (x <= -4.05e-81) {
		tmp = x * (x / y);
	} else if (x <= 2.05e+63) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -7.4e+34:
		tmp = x / y
	elif x <= -1.4e-31:
		tmp = t_0
	elif x <= -4.05e-81:
		tmp = x * (x / y)
	elif x <= 2.05e+63:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -7.4e+34)
		tmp = Float64(x / y);
	elseif (x <= -1.4e-31)
		tmp = t_0;
	elseif (x <= -4.05e-81)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 2.05e+63)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -7.4e+34)
		tmp = x / y;
	elseif (x <= -1.4e-31)
		tmp = t_0;
	elseif (x <= -4.05e-81)
		tmp = x * (x / y);
	elseif (x <= 2.05e+63)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.4e+34], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.4e-31], t$95$0, If[LessEqual[x, -4.05e-81], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05e+63], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.40000000000000017e34 or 2.04999999999999996e63 < x

   1. Initial program 71.7%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 82.2%

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

   if -7.40000000000000017e34 < x < -1.3999999999999999e-31 or -4.0500000000000001e-81 < x < 2.04999999999999996e63

   1. Initial program 99.3%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 79.5%

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

   if -1.3999999999999999e-31 < x < -4.0500000000000001e-81

   1. Initial program 99.7%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y} - 1, {x}^{2}, x\right)} \]

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. unpow2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 66.8%

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

      1. sub-neg 66.8%

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

      2. metadata-eval 66.8%

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

      3. unpow2 66.8%

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

   9. Simplified 66.8%

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

   10. Taylor expanded in y around 0 66.4%

       \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \]
   11. Step-by-step derivation

       1. unpow2 66.4%

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

       2. associate-*r/ 66.6%

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

   12. Simplified 66.6%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -4.05 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-32}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 100:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x -4.7e-32)
     x
     (if (<= x -4.6e-81) (* x (/ x y)) (if (<= x 100.0) x (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -4.7e-32) {
		tmp = x;
	} else if (x <= -4.6e-81) {
		tmp = x * (x / y);
	} else if (x <= 100.0) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= (-4.7d-32)) then
        tmp = x
    else if (x <= (-4.6d-81)) then
        tmp = x * (x / y)
    else if (x <= 100.0d0) then
        tmp = x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -4.7e-32) {
		tmp = x;
	} else if (x <= -4.6e-81) {
		tmp = x * (x / y);
	} else if (x <= 100.0) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= -4.7e-32:
		tmp = x
	elif x <= -4.6e-81:
		tmp = x * (x / y)
	elif x <= 100.0:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= -4.7e-32)
		tmp = x;
	elseif (x <= -4.6e-81)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 100.0)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= -4.7e-32)
		tmp = x;
	elseif (x <= -4.6e-81)
		tmp = x * (x / y);
	elseif (x <= 100.0)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, -4.7e-32], x, If[LessEqual[x, -4.6e-81], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 100.0], x, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 100 < x

   1. Initial program 75.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 74.0%

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

   if -1 < x < -4.70000000000000019e-32 or -4.59999999999999982e-81 < x < 100

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 82.6%

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

   if -4.70000000000000019e-32 < x < -4.59999999999999982e-81

   1. Initial program 99.7%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y} - 1, {x}^{2}, x\right)} \]

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. unpow2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 66.8%

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

      1. sub-neg 66.8%

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

      2. metadata-eval 66.8%

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

      3. unpow2 66.8%

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

   9. Simplified 66.8%

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

   10. Taylor expanded in y around 0 66.4%

       \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \]
   11. Step-by-step derivation

       1. unpow2 66.4%

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

       2. associate-*r/ 66.6%

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

   12. Simplified 66.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-32}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 100:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 95:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ x y) (if (<= x 95.0) x (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 95.0) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 95.0d0) then
        tmp = x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 95.0) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 95.0:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 95.0)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 95.0)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 95.0], x, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 95 < x

   1. Initial program 75.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 74.0%

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

   if -1 < x < 95

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 77.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 95:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 8: 48.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2400000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2400000.0) 1.0 (if (<= x 2e-7) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2400000.0) {
		tmp = 1.0;
	} else if (x <= 2e-7) {
		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 <= (-2400000.0d0)) then
        tmp = 1.0d0
    else if (x <= 2d-7) 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 <= -2400000.0) {
		tmp = 1.0;
	} else if (x <= 2e-7) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2400000.0:
		tmp = 1.0
	elif x <= 2e-7:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2400000.0)
		tmp = 1.0;
	elseif (x <= 2e-7)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2400000.0)
		tmp = 1.0;
	elseif (x <= 2e-7)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2400000.0], 1.0, If[LessEqual[x, 2e-7], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e6 or 1.9999999999999999e-7 < x

   1. Initial program 75.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.2%

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

   5. Taylor expanded in y around inf 23.7%

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

   if -2.4e6 < x < 1.9999999999999999e-7

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 78.3%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2400000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 13.9% 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 88.5%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 47.8%

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

5. Taylor expanded in y around inf 13.2%

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

6. Final simplification 13.2%

   \[\leadsto 1 \]

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 2023176 
(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)))