Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, C

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := \frac{x}{1 - y}\\ \mathbf{if}\;y \leq -8500000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 900000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (/ x (- 1.0 y))))
   (if (<= y -8500000.0)
     t_0
     (if (<= y -1.7e-61)
       t_1
       (if (<= y -6.2e-86) (/ y (+ y -1.0)) (if (<= y 900000.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (y <= -8500000.0) {
		tmp = t_0;
	} else if (y <= -1.7e-61) {
		tmp = t_1;
	} else if (y <= -6.2e-86) {
		tmp = y / (y + -1.0);
	} else if (y <= 900000.0) {
		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 / y)
    t_1 = x / (1.0d0 - y)
    if (y <= (-8500000.0d0)) then
        tmp = t_0
    else if (y <= (-1.7d-61)) then
        tmp = t_1
    else if (y <= (-6.2d-86)) then
        tmp = y / (y + (-1.0d0))
    else if (y <= 900000.0d0) 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 / y);
	double t_1 = x / (1.0 - y);
	double tmp;
	if (y <= -8500000.0) {
		tmp = t_0;
	} else if (y <= -1.7e-61) {
		tmp = t_1;
	} else if (y <= -6.2e-86) {
		tmp = y / (y + -1.0);
	} else if (y <= 900000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	t_1 = x / (1.0 - y)
	tmp = 0
	if y <= -8500000.0:
		tmp = t_0
	elif y <= -1.7e-61:
		tmp = t_1
	elif y <= -6.2e-86:
		tmp = y / (y + -1.0)
	elif y <= 900000.0:
		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(1.0 - y))
	tmp = 0.0
	if (y <= -8500000.0)
		tmp = t_0;
	elseif (y <= -1.7e-61)
		tmp = t_1;
	elseif (y <= -6.2e-86)
		tmp = Float64(y / Float64(y + -1.0));
	elseif (y <= 900000.0)
		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 / (1.0 - y);
	tmp = 0.0;
	if (y <= -8500000.0)
		tmp = t_0;
	elseif (y <= -1.7e-61)
		tmp = t_1;
	elseif (y <= -6.2e-86)
		tmp = y / (y + -1.0);
	elseif (y <= 900000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8500000.0], t$95$0, If[LessEqual[y, -1.7e-61], t$95$1, If[LessEqual[y, -6.2e-86], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 900000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5e6 or 9e5 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. unsub-neg 99.9%

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

      6. mul-1-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

      9. distribute-lft-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-*r/ 99.9%

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

      13. mul-1-neg 99.9%

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

      14. unsub-neg 99.9%

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

      15. sub-neg 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.3%

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

   if -8.5e6 < y < -1.6999999999999999e-61 or -6.19999999999999977e-86 < y < 9e5

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 83.6%

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

   if -1.6999999999999999e-61 < y < -6.19999999999999977e-86

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.1%

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

      1. metadata-eval 80.1%

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

      2. times-frac 80.1%

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

      3. *-lft-identity 80.1%

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

      4. neg-mul-1 80.1%

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

      5. neg-sub0 80.1%

         \[\leadsto \frac{y}{\color{blue}{0 - \left(1 - y\right)}} \]

      6. associate--r- 80.1%

         \[\leadsto \frac{y}{\color{blue}{\left(0 - 1\right) + y}} \]

      7. metadata-eval 80.1%

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

   4. Simplified 80.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8500000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 900000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 3: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -8000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{1}{1 - y}\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 13500000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -8000000.0)
     t_0
     (if (<= y -1.7e-61)
       (* x (/ 1.0 (- 1.0 y)))
       (if (<= y -2.05e-84)
         (/ y (+ y -1.0))
         (if (<= y 13500000.0) (/ x (- 1.0 y)) t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -8000000.0) {
		tmp = t_0;
	} else if (y <= -1.7e-61) {
		tmp = x * (1.0 / (1.0 - y));
	} else if (y <= -2.05e-84) {
		tmp = y / (y + -1.0);
	} else if (y <= 13500000.0) {
		tmp = x / (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-8000000.0d0)) then
        tmp = t_0
    else if (y <= (-1.7d-61)) then
        tmp = x * (1.0d0 / (1.0d0 - y))
    else if (y <= (-2.05d-84)) then
        tmp = y / (y + (-1.0d0))
    else if (y <= 13500000.0d0) then
        tmp = x / (1.0d0 - y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -8000000.0) {
		tmp = t_0;
	} else if (y <= -1.7e-61) {
		tmp = x * (1.0 / (1.0 - y));
	} else if (y <= -2.05e-84) {
		tmp = y / (y + -1.0);
	} else if (y <= 13500000.0) {
		tmp = x / (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -8000000.0:
		tmp = t_0
	elif y <= -1.7e-61:
		tmp = x * (1.0 / (1.0 - y))
	elif y <= -2.05e-84:
		tmp = y / (y + -1.0)
	elif y <= 13500000.0:
		tmp = x / (1.0 - y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -8000000.0)
		tmp = t_0;
	elseif (y <= -1.7e-61)
		tmp = Float64(x * Float64(1.0 / Float64(1.0 - y)));
	elseif (y <= -2.05e-84)
		tmp = Float64(y / Float64(y + -1.0));
	elseif (y <= 13500000.0)
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -8000000.0)
		tmp = t_0;
	elseif (y <= -1.7e-61)
		tmp = x * (1.0 / (1.0 - y));
	elseif (y <= -2.05e-84)
		tmp = y / (y + -1.0);
	elseif (y <= 13500000.0)
		tmp = x / (1.0 - y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8000000.0], t$95$0, If[LessEqual[y, -1.7e-61], N[(x * N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.05e-84], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 13500000.0], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8e6 or 1.35e7 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. unsub-neg 99.9%

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

      6. mul-1-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

      9. distribute-lft-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-*r/ 99.9%

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

      13. mul-1-neg 99.9%

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

      14. unsub-neg 99.9%

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

      15. sub-neg 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.3%

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

   if -8e6 < y < -1.6999999999999999e-61

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 94.0%

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

      1. clear-num 93.8%

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

      2. associate-/r/ 94.1%

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

   4. Applied egg-rr 94.1%

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

   if -1.6999999999999999e-61 < y < -2.05000000000000003e-84

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.1%

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

      1. metadata-eval 80.1%

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

      2. times-frac 80.1%

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

      3. *-lft-identity 80.1%

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

      4. neg-mul-1 80.1%

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

      5. neg-sub0 80.1%

         \[\leadsto \frac{y}{\color{blue}{0 - \left(1 - y\right)}} \]

      6. associate--r- 80.1%

         \[\leadsto \frac{y}{\color{blue}{\left(0 - 1\right) + y}} \]

      7. metadata-eval 80.1%

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

   4. Simplified 80.1%

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

   if -2.05000000000000003e-84 < y < 1.35e7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 82.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8000000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{1}{1 - y}\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 13500000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 4: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -110000:\\ \;\;\;\;1 - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{1}{1 - y}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 28000000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -110000.0)
   (- 1.0 (/ (+ x -1.0) y))
   (if (<= y -2.1e-61)
     (* x (/ 1.0 (- 1.0 y)))
     (if (<= y -1.7e-84)
       (/ y (+ y -1.0))
       (if (<= y 28000000.0) (/ x (- 1.0 y)) (- 1.0 (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -110000.0) {
		tmp = 1.0 - ((x + -1.0) / y);
	} else if (y <= -2.1e-61) {
		tmp = x * (1.0 / (1.0 - y));
	} else if (y <= -1.7e-84) {
		tmp = y / (y + -1.0);
	} else if (y <= 28000000.0) {
		tmp = x / (1.0 - y);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-110000.0d0)) then
        tmp = 1.0d0 - ((x + (-1.0d0)) / y)
    else if (y <= (-2.1d-61)) then
        tmp = x * (1.0d0 / (1.0d0 - y))
    else if (y <= (-1.7d-84)) then
        tmp = y / (y + (-1.0d0))
    else if (y <= 28000000.0d0) then
        tmp = x / (1.0d0 - y)
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -110000.0) {
		tmp = 1.0 - ((x + -1.0) / y);
	} else if (y <= -2.1e-61) {
		tmp = x * (1.0 / (1.0 - y));
	} else if (y <= -1.7e-84) {
		tmp = y / (y + -1.0);
	} else if (y <= 28000000.0) {
		tmp = x / (1.0 - y);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -110000.0:
		tmp = 1.0 - ((x + -1.0) / y)
	elif y <= -2.1e-61:
		tmp = x * (1.0 / (1.0 - y))
	elif y <= -1.7e-84:
		tmp = y / (y + -1.0)
	elif y <= 28000000.0:
		tmp = x / (1.0 - y)
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -110000.0)
		tmp = Float64(1.0 - Float64(Float64(x + -1.0) / y));
	elseif (y <= -2.1e-61)
		tmp = Float64(x * Float64(1.0 / Float64(1.0 - y)));
	elseif (y <= -1.7e-84)
		tmp = Float64(y / Float64(y + -1.0));
	elseif (y <= 28000000.0)
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -110000.0)
		tmp = 1.0 - ((x + -1.0) / y);
	elseif (y <= -2.1e-61)
		tmp = x * (1.0 / (1.0 - y));
	elseif (y <= -1.7e-84)
		tmp = y / (y + -1.0);
	elseif (y <= 28000000.0)
		tmp = x / (1.0 - y);
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -110000.0], N[(1.0 - N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.1e-61], N[(x * N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.7e-84], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 28000000.0], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.1e5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. div-sub 100.0%

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

      5. unsub-neg 100.0%

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

      6. mul-1-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-*r/ 100.0%

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

      13. mul-1-neg 100.0%

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

      14. unsub-neg 100.0%

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

      15. sub-neg 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   if -1.1e5 < y < -2.0999999999999999e-61

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 94.0%

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

      1. clear-num 93.8%

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

      2. associate-/r/ 94.1%

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

   4. Applied egg-rr 94.1%

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

   if -2.0999999999999999e-61 < y < -1.7000000000000001e-84

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.1%

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

      1. metadata-eval 80.1%

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

      2. times-frac 80.1%

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

      3. *-lft-identity 80.1%

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

      4. neg-mul-1 80.1%

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

      5. neg-sub0 80.1%

         \[\leadsto \frac{y}{\color{blue}{0 - \left(1 - y\right)}} \]

      6. associate--r- 80.1%

         \[\leadsto \frac{y}{\color{blue}{\left(0 - 1\right) + y}} \]

      7. metadata-eval 80.1%

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

   4. Simplified 80.1%

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

   if -1.7000000000000001e-84 < y < 2.8e7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 82.2%

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

   if 2.8e7 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. div-sub 99.6%

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

      5. unsub-neg 99.6%

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

      6. mul-1-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-lft-in 99.6%

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

      10. metadata-eval 99.6%

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

      11. sub-neg 99.6%

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

      12. associate-*r/ 99.6%

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

      13. mul-1-neg 99.6%

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

      14. unsub-neg 99.6%

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

      15. sub-neg 99.6%

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

      16. metadata-eval 99.6%

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

      17. +-commutative 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -110000:\\ \;\;\;\;1 - \frac{x + -1}{y}\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{1}{1 - y}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{elif}\;y \leq 28000000:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 5: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0156 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.0156) (not (<= y 1.0))) (- 1.0 (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.0156) || !(y <= 1.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-0.0156d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -0.0156) || !(y <= 1.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.0156) or not (y <= 1.0):
		tmp = 1.0 - (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.0156) || !(y <= 1.0))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.0156) || ~((y <= 1.0)))
		tmp = 1.0 - (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.0156], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -0.015599999999999999 or 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.3%

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

      1. +-commutative 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unsub-neg 99.3%

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

      4. div-sub 99.3%

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

      5. unsub-neg 99.3%

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

      6. mul-1-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. metadata-eval 99.3%

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

      9. distribute-lft-in 99.3%

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

      10. metadata-eval 99.3%

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

      11. sub-neg 99.3%

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

      12. associate-*r/ 99.3%

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

      13. mul-1-neg 99.3%

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

      14. unsub-neg 99.3%

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

      15. sub-neg 99.3%

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

      16. metadata-eval 99.3%

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

      17. +-commutative 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around inf 98.8%

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

   if -0.015599999999999999 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0156 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3750000 \lor \neg \left(y \leq 62000000\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3750000.0) (not (<= y 62000000.0)))
   (- 1.0 (/ x y))
   (/ x (- 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3750000.0) || !(y <= 62000000.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (1.0 - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3750000.0d0)) .or. (.not. (y <= 62000000.0d0))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / (1.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3750000.0) || !(y <= 62000000.0)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3750000.0) or not (y <= 62000000.0):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (1.0 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3750000.0) || !(y <= 62000000.0))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / Float64(1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3750000.0) || ~((y <= 62000000.0)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3750000.0], N[Not[LessEqual[y, 62000000.0]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.75e6 or 6.2e7 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. unsub-neg 99.9%

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

      6. mul-1-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. metadata-eval 99.9%

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

      9. distribute-lft-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-*r/ 99.9%

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

      13. mul-1-neg 99.9%

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

      14. unsub-neg 99.9%

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

      15. sub-neg 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.3%

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

   if -3.75e6 < y < 6.2e7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 80.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3750000 \lor \neg \left(y \leq 62000000\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - y}\\ \end{array} \]

Alternative 7: 73.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.0d0)) then
        tmp = 1.0d0
    else if (y <= 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 (y <= -1.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 71.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.0%

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

4. Final simplification 75.3%

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

Alternative 8: 38.1% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around inf 34.5%

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

3. Final simplification 34.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))