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

Percentage Accurate: 88.8% → 99.9%

Time: 4.7s

Alternatives: 7

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;y \leq 150000000:\\ \;\;\;\;\frac{x}{y + 1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.9e+41)
   (/ x 1.0)
   (if (<= y 150000000.0) (* (/ x (+ y 1.0)) y) (- x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.9e+41) {
		tmp = x / 1.0;
	} else if (y <= 150000000.0) {
		tmp = (x / (y + 1.0)) * y;
	} else {
		tmp = x - (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 <= (-1.9d+41)) then
        tmp = x / 1.0d0
    else if (y <= 150000000.0d0) then
        tmp = (x / (y + 1.0d0)) * y
    else
        tmp = x - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.9e+41) {
		tmp = x / 1.0;
	} else if (y <= 150000000.0) {
		tmp = (x / (y + 1.0)) * y;
	} else {
		tmp = x - (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.9e+41:
		tmp = x / 1.0
	elif y <= 150000000.0:
		tmp = (x / (y + 1.0)) * y
	else:
		tmp = x - (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.9e+41)
		tmp = Float64(x / 1.0);
	elseif (y <= 150000000.0)
		tmp = Float64(Float64(x / Float64(y + 1.0)) * y);
	else
		tmp = Float64(x - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.9e+41)
		tmp = x / 1.0;
	elseif (y <= 150000000.0)
		tmp = (x / (y + 1.0)) * y;
	else
		tmp = x - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.9e+41], N[(x / 1.0), $MachinePrecision], If[LessEqual[y, 150000000.0], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9000000000000001e41

   1. Initial program 67.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 100.0%

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

   if -1.9000000000000001e41 < y < 1.5e8

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 1.5e8 < y

   1. Initial program 76.8%

      \[\frac{x \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;y \leq 150000000:\\ \;\;\;\;\frac{x}{y + 1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot x}{y + 1}\\ \mathbf{if}\;t\_0 \leq 10^{+297}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* y x) (+ y 1.0)))) (if (<= t_0 1e+297) t_0 (/ x 1.0))))

C

double code(double x, double y) {
	double t_0 = (y * x) / (y + 1.0);
	double tmp;
	if (t_0 <= 1e+297) {
		tmp = t_0;
	} else {
		tmp = x / 1.0;
	}
	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 = (y * x) / (y + 1.0d0)
    if (t_0 <= 1d+297) then
        tmp = t_0
    else
        tmp = x / 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * x) / (y + 1.0);
	double tmp;
	if (t_0 <= 1e+297) {
		tmp = t_0;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) / (y + 1.0)
	tmp = 0
	if t_0 <= 1e+297:
		tmp = t_0
	else:
		tmp = x / 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) / Float64(y + 1.0))
	tmp = 0.0
	if (t_0 <= 1e+297)
		tmp = t_0;
	else
		tmp = Float64(x / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) / (y + 1.0);
	tmp = 0.0;
	if (t_0 <= 1e+297)
		tmp = t_0;
	else
		tmp = x / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+297], t$95$0, N[(x / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (+.f64 y #s(literal 1 binary64))) < 1e297

   1. Initial program 92.5%

      \[\frac{x \cdot y}{y + 1} \]
   2. Add Preprocessing

   if 1e297 < (/.f64 (*.f64 x y) (+.f64 y #s(literal 1 binary64)))

   1. Initial program 6.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 100.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{y + 1} \leq 10^{+297}:\\ \;\;\;\;\frac{y \cdot x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (y * 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 = x - (x / y)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = (y * 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 = x - (x / y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (y * x) * (1.0 - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 74.1%

      \[\frac{x \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 98.5%

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

Alternative 4: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\left(y \cdot x\right) \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (/ x 1.0) (if (<= y 0.75) (* (* y x) (- 1.0 y)) (/ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x / 1.0;
	} else if (y <= 0.75) {
		tmp = (y * x) * (1.0 - y);
	} else {
		tmp = 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)) then
        tmp = x / 1.0d0
    else if (y <= 0.75d0) then
        tmp = (y * x) * (1.0d0 - y)
    else
        tmp = x / 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x / 1.0;
	} else if (y <= 0.75) {
		tmp = (y * x) * (1.0 - y);
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x / 1.0
	elif y <= 0.75:
		tmp = (y * x) * (1.0 - y)
	else:
		tmp = x / 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x / 1.0);
	elseif (y <= 0.75)
		tmp = Float64(Float64(y * x) * Float64(1.0 - y));
	else
		tmp = Float64(x / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x / 1.0;
	elseif (y <= 0.75)
		tmp = (y * x) * (1.0 - y);
	else
		tmp = x / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x / 1.0), $MachinePrecision], If[LessEqual[y, 0.75], N[(N[(y * x), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], N[(x / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.75 < y

   1. Initial program 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 97.9%

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

   if -1 < y < 0.75

   1. Initial program 100.0%

      \[\frac{x \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\left(y \cdot x\right) \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.8× speedup

Math

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

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(x / Float64(Float64(y + 1.0) / y))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 97.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[\frac{x \cdot y}{y + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 50.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

   \[\frac{x \cdot y}{y + 1} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

   \[\leadsto \color{blue}{y \cdot x} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ x (* y y)) (- (/ x y) x))))
   (if (< y -3693.8482788297247)
     t_0
     (if (< y 6799310503.41891) (/ (* x y) (+ y 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x / (y * y)) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = (x * y) / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (y * y)) - ((x / y) - x)
    if (y < (-3693.8482788297247d0)) then
        tmp = t_0
    else if (y < 6799310503.41891d0) then
        tmp = (x * y) / (y + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / (y * y)) - ((x / y) - x);
	double tmp;
	if (y < -3693.8482788297247) {
		tmp = t_0;
	} else if (y < 6799310503.41891) {
		tmp = (x * y) / (y + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / (y * y)) - ((x / y) - x)
	tmp = 0
	if y < -3693.8482788297247:
		tmp = t_0
	elif y < 6799310503.41891:
		tmp = (x * y) / (y + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / Float64(y * y)) - Float64(Float64(x / y) - x))
	tmp = 0.0
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = Float64(Float64(x * y) / Float64(y + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / (y * y)) - ((x / y) - x);
	tmp = 0.0;
	if (y < -3693.8482788297247)
		tmp = t_0;
	elseif (y < 6799310503.41891)
		tmp = (x * y) / (y + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -3693.8482788297247], t$95$0, If[Less[y, 6799310503.41891], N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024242 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -36938482788297247/10000000000000) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 679931050341891/100000) (/ (* x y) (+ y 1)) (- (/ x (* y y)) (- (/ x y) x)))))

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