Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3

Percentage Accurate: 84.7% → 96.2%

Time: 8.8s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (- y z)) y))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return (x * (y - z)) / y

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (- y z)) y))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return (x * (y - z)) / y

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

   \[\frac{x \cdot \left(y - z\right)}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 96.9

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

4. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\frac{x}{\frac{y}{y - z}}} \]
5. Add Preprocessing

Alternative 2: 72.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(y - z\right) \cdot x}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-80}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(y - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (- y z) x) y)))
   (if (<= t_0 -5e-222)
     (/ (* (- z) x) y)
     (if (<= t_0 1e-80) (* 1.0 x) (* (/ x y) (- y z))))))

C

double code(double x, double y, double z) {
	double t_0 = ((y - z) * x) / y;
	double tmp;
	if (t_0 <= -5e-222) {
		tmp = (-z * x) / y;
	} else if (t_0 <= 1e-80) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / y) * (y - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y - z) * x) / y
    if (t_0 <= (-5d-222)) then
        tmp = (-z * x) / y
    else if (t_0 <= 1d-80) then
        tmp = 1.0d0 * x
    else
        tmp = (x / y) * (y - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y - z) * x) / y;
	double tmp;
	if (t_0 <= -5e-222) {
		tmp = (-z * x) / y;
	} else if (t_0 <= 1e-80) {
		tmp = 1.0 * x;
	} else {
		tmp = (x / y) * (y - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y - z) * x) / y
	tmp = 0
	if t_0 <= -5e-222:
		tmp = (-z * x) / y
	elif t_0 <= 1e-80:
		tmp = 1.0 * x
	else:
		tmp = (x / y) * (y - z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y - z) * x) / y)
	tmp = 0.0
	if (t_0 <= -5e-222)
		tmp = Float64(Float64(Float64(-z) * x) / y);
	elseif (t_0 <= 1e-80)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(x / y) * Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y - z) * x) / y;
	tmp = 0.0;
	if (t_0 <= -5e-222)
		tmp = (-z * x) / y;
	elseif (t_0 <= 1e-80)
		tmp = 1.0 * x;
	else
		tmp = (x / y) * (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-222], N[(N[((-z) * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 1e-80], N[(1.0 * x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -5.00000000000000008e-222

   1. Initial program 84.5%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 53.8

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

   5. Applied rewrites 53.8%

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

   if -5.00000000000000008e-222 < (/.f64 (*.f64 x (-.f64 y z)) y) < 9.99999999999999961e-81

   1. Initial program 90.4%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 80.0%

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

   if 9.99999999999999961e-81 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 84.4%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot x}{y} \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{y}\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot x}{y} \leq 10^{-80}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-107}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.8e-107)
   (* 1.0 x)
   (if (<= y 6.5e+23) (/ (* (- z) x) y) (* 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e-107) {
		tmp = 1.0 * x;
	} else if (y <= 6.5e+23) {
		tmp = (-z * x) / y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.8d-107)) then
        tmp = 1.0d0 * x
    else if (y <= 6.5d+23) then
        tmp = (-z * x) / y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e-107) {
		tmp = 1.0 * x;
	} else if (y <= 6.5e+23) {
		tmp = (-z * x) / y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.8e-107:
		tmp = 1.0 * x
	elif y <= 6.5e+23:
		tmp = (-z * x) / y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.8e-107)
		tmp = Float64(1.0 * x);
	elseif (y <= 6.5e+23)
		tmp = Float64(Float64(Float64(-z) * x) / y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.8e-107)
		tmp = 1.0 * x;
	elseif (y <= 6.5e+23)
		tmp = (-z * x) / y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.8e-107], N[(1.0 * x), $MachinePrecision], If[LessEqual[y, 6.5e+23], N[(N[((-z) * x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8000000000000002e-107 or 6.4999999999999996e23 < y

   1. Initial program 78.9%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 72.7%

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

   if -3.8000000000000002e-107 < y < 6.4999999999999996e23

   1. Initial program 95.6%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 82.1

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

   5. Applied rewrites 82.1%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-107}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+75}:\\ \;\;\;\;\frac{-z}{y} \cdot x\\ \mathbf{elif}\;z \leq 10^{+37}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{y} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5e+75)
   (* (/ (- z) y) x)
   (if (<= z 1e+37) (* 1.0 x) (* (/ (- x) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+75) {
		tmp = (-z / y) * x;
	} else if (z <= 1e+37) {
		tmp = 1.0 * x;
	} else {
		tmp = (-x / y) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5d+75)) then
        tmp = (-z / y) * x
    else if (z <= 1d+37) then
        tmp = 1.0d0 * x
    else
        tmp = (-x / y) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+75) {
		tmp = (-z / y) * x;
	} else if (z <= 1e+37) {
		tmp = 1.0 * x;
	} else {
		tmp = (-x / y) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5e+75:
		tmp = (-z / y) * x
	elif z <= 1e+37:
		tmp = 1.0 * x
	else:
		tmp = (-x / y) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5e+75)
		tmp = Float64(Float64(Float64(-z) / y) * x);
	elseif (z <= 1e+37)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(Float64(-x) / y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5e+75)
		tmp = (-z / y) * x;
	elseif (z <= 1e+37)
		tmp = 1.0 * x;
	else
		tmp = (-x / y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5e+75], N[(N[((-z) / y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1e+37], N[(1.0 * x), $MachinePrecision], N[(N[((-x) / y), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.0000000000000002e75

   1. Initial program 81.5%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if -5.0000000000000002e75 < z < 9.99999999999999954e36

   1. Initial program 83.5%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 75.0%

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

   if 9.99999999999999954e36 < z

   1. Initial program 93.0%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 16.6

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

   5. Applied rewrites 16.6%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 76.2

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

   8. Applied rewrites 76.2%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+75}:\\ \;\;\;\;\frac{-z}{y} \cdot x\\ \mathbf{elif}\;z \leq 10^{+37}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{y} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-108}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.5e-108)
   (* 1.0 x)
   (if (<= y 6.5e+23) (* (/ (- x) y) z) (* 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e-108) {
		tmp = 1.0 * x;
	} else if (y <= 6.5e+23) {
		tmp = (-x / y) * z;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.5d-108)) then
        tmp = 1.0d0 * x
    else if (y <= 6.5d+23) then
        tmp = (-x / y) * z
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e-108) {
		tmp = 1.0 * x;
	} else if (y <= 6.5e+23) {
		tmp = (-x / y) * z;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.5e-108:
		tmp = 1.0 * x
	elif y <= 6.5e+23:
		tmp = (-x / y) * z
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.5e-108)
		tmp = Float64(1.0 * x);
	elseif (y <= 6.5e+23)
		tmp = Float64(Float64(Float64(-x) / y) * z);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.5e-108)
		tmp = 1.0 * x;
	elseif (y <= 6.5e+23)
		tmp = (-x / y) * z;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.5e-108], N[(1.0 * x), $MachinePrecision], If[LessEqual[y, 6.5e+23], N[(N[((-x) / y), $MachinePrecision] * z), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5e-108 or 6.4999999999999996e23 < y

   1. Initial program 78.9%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 72.7%

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

   if -2.5e-108 < y < 6.4999999999999996e23

   1. Initial program 95.6%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 15.1

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

   5. Applied rewrites 15.1%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 76.5

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

   8. Applied rewrites 76.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-108}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z}{y}, -x, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (/ z y) (- x) x))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.6%

   \[\frac{x \cdot \left(y - z\right)}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 96.9

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

4. Applied rewrites 96.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. remove-double-neg N/A

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

   8. lift-neg.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. associate-*l/ N/A

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

   11. lift-/.f64 N/A

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

   12. distribute-rgt-in N/A

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

   13. +-commutative N/A

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

   14. associate-*r* N/A

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

   15. lift-/.f64 N/A

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

   16. frac-2neg N/A

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

   17. metadata-eval N/A

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

   18. div-inv N/A

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

   19. lift-neg.f64 N/A

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

   20. frac-2neg N/A

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

   21. lift-/.f64 N/A

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

   22. associate-*l/ N/A

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

6. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{y}, -x, x\right)} \]
7. Add Preprocessing

Alternative 7: 95.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* (/ (- y z) y) x))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return ((y - z) / y) * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

   \[\frac{x \cdot \left(y - z\right)}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 96.3

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\frac{y - z}{y} \cdot x} \]
5. Add Preprocessing

Alternative 8: 50.2% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* 1.0 x))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return 1.0 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

   \[\frac{x \cdot \left(y - z\right)}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 96.3

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

4. Applied rewrites 96.3%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 50.3%

   \[\leadsto \color{blue}{1} \cdot x \]
8. Add Preprocessing

Developer Target 1: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z -2.060202331921739e+104)
   (- x (/ (* z x) y))
   (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < -2.060202331921739e+104) {
		tmp = x - ((z * x) / y);
	} else if (z < 1.6939766013828526e+213) {
		tmp = x / (y / (y - z));
	} else {
		tmp = (y - z) * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < (-2.060202331921739d+104)) then
        tmp = x - ((z * x) / y)
    else if (z < 1.6939766013828526d+213) then
        tmp = x / (y / (y - z))
    else
        tmp = (y - z) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < -2.060202331921739e+104) {
		tmp = x - ((z * x) / y);
	} else if (z < 1.6939766013828526e+213) {
		tmp = x / (y / (y - z));
	} else {
		tmp = (y - z) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < -2.060202331921739e+104:
		tmp = x - ((z * x) / y)
	elif z < 1.6939766013828526e+213:
		tmp = x / (y / (y - z))
	else:
		tmp = (y - z) * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < -2.060202331921739e+104)
		tmp = Float64(x - Float64(Float64(z * x) / y));
	elseif (z < 1.6939766013828526e+213)
		tmp = Float64(x / Float64(y / Float64(y - z)));
	else
		tmp = Float64(Float64(y - z) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < -2.060202331921739e+104)
		tmp = x - ((z * x) / y);
	elseif (z < 1.6939766013828526e+213)
		tmp = x / (y / (y - z));
	else
		tmp = (y - z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, -2.060202331921739e+104], N[(x - N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[Less[z, 1.6939766013828526e+213], N[(x / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024243 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -206020233192173900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- x (/ (* z x) y)) (if (< z 1693976601382852600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ x (/ y (- y z))) (* (- y z) (/ x y)))))

  (/ (* x (- y z)) y))