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

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 10

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: 83.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -32:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -32.0)
   1.0
   (if (<= y 1.25e-5) (- x y) (if (<= y 2.1e+139) (/ x (- 1.0 y)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -32.0) {
		tmp = 1.0;
	} else if (y <= 1.25e-5) {
		tmp = x - y;
	} else if (y <= 2.1e+139) {
		tmp = x / (1.0 - y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-32.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.25d-5) then
        tmp = x - y
    else if (y <= 2.1d+139) then
        tmp = x / (1.0d0 - y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -32.0) {
		tmp = 1.0;
	} else if (y <= 1.25e-5) {
		tmp = x - y;
	} else if (y <= 2.1e+139) {
		tmp = x / (1.0 - y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -32.0:
		tmp = 1.0
	elif y <= 1.25e-5:
		tmp = x - y
	elif y <= 2.1e+139:
		tmp = x / (1.0 - y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -32.0)
		tmp = 1.0;
	elseif (y <= 1.25e-5)
		tmp = Float64(x - y);
	elseif (y <= 2.1e+139)
		tmp = Float64(x / Float64(1.0 - y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -32.0)
		tmp = 1.0;
	elseif (y <= 1.25e-5)
		tmp = x - y;
	elseif (y <= 2.1e+139)
		tmp = x / (1.0 - y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -32.0], 1.0, If[LessEqual[y, 1.25e-5], N[(x - y), $MachinePrecision], If[LessEqual[y, 2.1e+139], N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -32 or 2.0999999999999999e139 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 74.4%

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

   if -32 < y < 1.25000000000000006e-5

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around 0 98.1%

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

   if 1.25000000000000006e-5 < y < 2.0999999999999999e139

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 58.9%

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

      1. sub-neg 58.9%

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

      2. metadata-eval 58.9%

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

      3. associate-*r/ 58.9%

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

      4. mul-1-neg 58.9%

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

      5. +-commutative 58.9%

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

   6. Simplified 58.9%

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

      1. expm1-log1p-u 30.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{-1 + y}\right)\right)} \]

      2. expm1-udef 30.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{-1 + y}\right)} - 1} \]

      3. distribute-frac-neg 30.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-\frac{x}{-1 + y}}\right)} - 1 \]

      4. frac-2neg 30.0%

         \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{-x}{-\left(-1 + y\right)}}\right)} - 1 \]

      5. distribute-neg-in 30.0%

         \[\leadsto e^{\mathsf{log1p}\left(-\frac{-x}{\color{blue}{\left(--1\right) + \left(-y\right)}}\right)} - 1 \]

      6. metadata-eval 30.0%

         \[\leadsto e^{\mathsf{log1p}\left(-\frac{-x}{\color{blue}{1} + \left(-y\right)}\right)} - 1 \]

      7. sub-neg 30.0%

         \[\leadsto e^{\mathsf{log1p}\left(-\frac{-x}{\color{blue}{1 - y}}\right)} - 1 \]

      8. distribute-frac-neg 30.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-\left(-x\right)}{1 - y}}\right)} - 1 \]

      9. remove-double-neg 30.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{1 - y}\right)} - 1 \]

   8. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{1 - y}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 30.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{1 - y}\right)\right)} \]

      2. expm1-log1p 58.9%

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

   10. Simplified 58.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -32:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;x - y\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{1 - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.2) (not (<= y 1.0))) (+ 1.0 (/ (- 1.0 x) y)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.2) || !(y <= 1.0)) {
		tmp = 1.0 + ((1.0 - x) / y);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.2d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 + ((1.0d0 - x) / y)
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.2) || !(y <= 1.0)) {
		tmp = 1.0 + ((1.0 - x) / y);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.2) or not (y <= 1.0):
		tmp = 1.0 + ((1.0 - x) / y)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.2) || !(y <= 1.0))
		tmp = Float64(1.0 + Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(x - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.2) || ~((y <= 1.0)))
		tmp = 1.0 + ((1.0 - x) / y);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.2], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999996 or 1 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. mul-1-neg 98.5%

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

      3. unsub-neg 98.5%

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

      4. div-sub 98.5%

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

   6. Simplified 98.5%

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

   if -1.19999999999999996 < y < 1

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around 0 98.1%

      \[\leadsto \color{blue}{-1} \cdot \left(y - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 + \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 4: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. Step-by-step derivation

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. mul-1-neg 98.5%

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

      3. unsub-neg 98.5%

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

      4. div-sub 98.5%

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

   6. Simplified 98.5%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. unsub-neg 98.4%

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

      3. mul-1-neg 98.4%

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

      4. unsub-neg 98.4%

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

   6. Simplified 98.4%

      \[\leadsto \color{blue}{x - y \cdot \left(1 - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

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

Alternative 5: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.014 \lor \neg \left(y \leq 4 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.014) (not (<= y 4e-6))) (/ y (+ y -1.0)) (- x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.014) || !(y <= 4e-6)) {
		tmp = y / (y + -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) :: tmp
    if ((y <= (-0.014d0)) .or. (.not. (y <= 4d-6))) then
        tmp = y / (y + (-1.0d0))
    else
        tmp = x - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -0.014) || !(y <= 4e-6)) {
		tmp = y / (y + -1.0);
	} else {
		tmp = x - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.014) or not (y <= 4e-6):
		tmp = y / (y + -1.0)
	else:
		tmp = x - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.014) || !(y <= 4e-6))
		tmp = Float64(y / Float64(y + -1.0));
	else
		tmp = Float64(x - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.014) || ~((y <= 4e-6)))
		tmp = y / (y + -1.0);
	else
		tmp = x - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.014], N[Not[LessEqual[y, 4e-6]], $MachinePrecision]], N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(x - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0140000000000000003 or 3.99999999999999982e-6 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 69.3%

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

   if -0.0140000000000000003 < y < 3.99999999999999982e-6

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around 0 98.5%

      \[\leadsto \color{blue}{-1} \cdot \left(y - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.014 \lor \neg \left(y \leq 4 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{y}{y + -1}\\ \mathbf{else}:\\ \;\;\;\;x - y\\ \end{array} \]

Alternative 6: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. mul-1-neg 98.5%

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

      3. unsub-neg 98.5%

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

      4. div-sub 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in x around inf 97.9%

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

      1. neg-mul-1 97.9%

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

      2. distribute-neg-frac 97.9%

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

   9. Simplified 97.9%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around 0 98.1%

      \[\leadsto \color{blue}{-1} \cdot \left(y - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

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

Alternative 7: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00015:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-47}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.00015) 1.0 (if (<= y -8.2e-47) (- y) (if (<= y 1.0) x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.00015) {
		tmp = 1.0;
	} else if (y <= -8.2e-47) {
		tmp = -y;
	} 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 <= (-0.00015d0)) then
        tmp = 1.0d0
    else if (y <= (-8.2d-47)) then
        tmp = -y
    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 <= -0.00015) {
		tmp = 1.0;
	} else if (y <= -8.2e-47) {
		tmp = -y;
	} else if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.00015:
		tmp = 1.0
	elif y <= -8.2e-47:
		tmp = -y
	elif y <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.00015)
		tmp = 1.0;
	elseif (y <= -8.2e-47)
		tmp = Float64(-y);
	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 <= -0.00015)
		tmp = 1.0;
	elseif (y <= -8.2e-47)
		tmp = -y;
	elseif (y <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.00015], 1.0, If[LessEqual[y, -8.2e-47], (-y), If[LessEqual[y, 1.0], x, 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.49999999999999987e-4 or 1 < y

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 66.9%

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

   if -1.49999999999999987e-4 < y < -8.20000000000000003e-47

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-/r/ 99.6%

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

      4. associate-/l* 99.8%

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

      5. neg-mul-1 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-out 99.8%

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

      10. remove-double-neg 99.8%

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

      11. sub-neg 99.8%

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

      12. associate-/l/ 99.8%

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

      13. neg-mul-1 99.8%

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

      14. sub0-neg 99.8%

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

      15. associate--r- 99.8%

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

      16. metadata-eval 99.8%

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

      17. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 69.9%

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

   5. Taylor expanded in y around 0 68.0%

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

      1. neg-mul-1 68.0%

         \[\leadsto \color{blue}{-y} \]

   7. Simplified 68.0%

      \[\leadsto \color{blue}{-y} \]

   if -8.20000000000000003e-47 < y < 1

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 79.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00015:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-47}:\\ \;\;\;\;-y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -32:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -32.0) 1.0 (if (<= y 1.0) (- x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -32.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-32.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.0d0) then
        tmp = x - y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -32.0) {
		tmp = 1.0;
	} else if (y <= 1.0) {
		tmp = x - y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -32.0:
		tmp = 1.0
	elif y <= 1.0:
		tmp = x - y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -32.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = Float64(x - y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -32.0)
		tmp = 1.0;
	elseif (y <= 1.0)
		tmp = x - y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -32.0], 1.0, If[LessEqual[y, 1.0], N[(x - y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -32 or 1 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 67.9%

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

   if -32 < y < 1

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around 0 98.1%

      \[\leadsto \color{blue}{-1} \cdot \left(y - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -32:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 74.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y -9.2) 1.0 (if (<= y 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.2) {
		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 <= (-9.2d0)) 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 <= -9.2) {
		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 <= -9.2:
		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 <= -9.2)
		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 <= -9.2)
		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, -9.2], 1.0, If[LessEqual[y, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -9.1999999999999993 or 1 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 100.0%

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

      5. neg-mul-1 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-out 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

      12. associate-/l/ 100.0%

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

      13. neg-mul-1 100.0%

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

      14. sub0-neg 100.0%

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

      15. associate--r- 100.0%

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

      16. metadata-eval 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 67.9%

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

   if -9.1999999999999993 < y < 1

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-out 99.9%

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

      10. remove-double-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. associate-/l/ 99.9%

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

      13. neg-mul-1 99.9%

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

      14. sub0-neg 99.9%

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

      15. associate--r- 99.9%

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

      16. metadata-eval 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 74.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 38.5% 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. Step-by-step derivation

   1. *-lft-identity 100.0%

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

   2. metadata-eval 100.0%

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

   3. associate-/r/ 99.8%

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

   4. associate-/l* 100.0%

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

   5. neg-mul-1 100.0%

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

   6. distribute-frac-neg 100.0%

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

   7. sub-neg 100.0%

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

   8. +-commutative 100.0%

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

   9. distribute-neg-out 100.0%

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

   10. remove-double-neg 100.0%

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

   11. sub-neg 100.0%

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

   12. associate-/l/ 100.0%

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

   13. neg-mul-1 100.0%

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

   14. sub0-neg 100.0%

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

   15. associate--r- 100.0%

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

   16. metadata-eval 100.0%

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

   17. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around inf 33.6%

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

5. Final simplification 33.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023311 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, C"
  :precision binary64
  (/ (- x y) (- 1.0 y)))