Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Percentage Accurate: 88.2% → 98.7%

Time: 6.3s

Alternatives: 9

Speedup: 0.5×

• 20.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / 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 * (z - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / 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 * (z - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z} + t\_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-21}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (/ x z)))))
   (if (<= y -7e-5) (+ (/ x z) t_0) (if (<= y 9e-21) (+ y (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - (x / z));
	double tmp;
	if (y <= -7e-5) {
		tmp = (x / z) + t_0;
	} else if (y <= 9e-21) {
		tmp = y + (x / z);
	} else {
		tmp = t_0;
	}
	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 * (1.0d0 - (x / z))
    if (y <= (-7d-5)) then
        tmp = (x / z) + t_0
    else if (y <= 9d-21) then
        tmp = y + (x / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - (x / z));
	double tmp;
	if (y <= -7e-5) {
		tmp = (x / z) + t_0;
	} else if (y <= 9e-21) {
		tmp = y + (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - (x / z))
	tmp = 0
	if y <= -7e-5:
		tmp = (x / z) + t_0
	elif y <= 9e-21:
		tmp = y + (x / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - Float64(x / z)))
	tmp = 0.0
	if (y <= -7e-5)
		tmp = Float64(Float64(x / z) + t_0);
	elseif (y <= 9e-21)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - (x / z));
	tmp = 0.0;
	if (y <= -7e-5)
		tmp = (x / z) + t_0;
	elseif (y <= 9e-21)
		tmp = y + (x / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e-5], N[(N[(x / z), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[y, 9e-21], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.9999999999999994e-5

   1. Initial program 74.2%

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

   3. Taylor expanded in y around 0 99.9%

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

   if -6.9999999999999994e-5 < y < 8.99999999999999936e-21

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 94.2%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 8.99999999999999936e-21 < y

   1. Initial program 79.1%

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

   3. Taylor expanded in y around inf 79.1%

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

      1. associate-/l* 99.9%

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

      2. div-sub 100.0%

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

      3. *-inverses 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-21}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+66} \lor \neg \left(y \leq 9 \cdot 10^{-21}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.65e+66) (not (<= y 9e-21)))
   (* y (- 1.0 (/ x z)))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.65e+66) || !(y <= 9e-21)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = (x + (y * (z - x))) / 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 ((y <= (-1.65d+66)) .or. (.not. (y <= 9d-21))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = (x + (y * (z - x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.65e+66) || !(y <= 9e-21)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.65e+66) or not (y <= 9e-21):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.65e+66) || !(y <= 9e-21))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.65e+66) || ~((y <= 9e-21)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.65e+66], N[Not[LessEqual[y, 9e-21]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6500000000000001e66 or 8.99999999999999936e-21 < y

   1. Initial program 73.1%

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

   3. Taylor expanded in y around inf 73.1%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. *-inverses 99.9%

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

   5. Simplified 99.9%

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

   if -1.6500000000000001e66 < y < 8.99999999999999936e-21

   1. Initial program 99.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+66} \lor \neg \left(y \leq 9 \cdot 10^{-21}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 9 \cdot 10^{-21}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 9e-21))) (* y (- 1.0 (/ x z))) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 9e-21)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / 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 ((y <= (-1.0d0)) .or. (.not. (y <= 9d-21))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 9e-21)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 9e-21):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 9e-21))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 9e-21)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 9e-21]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 8.99999999999999936e-21 < y

   1. Initial program 76.6%

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

   3. Taylor expanded in y around inf 75.6%

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

      1. associate-/l* 98.9%

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

      2. div-sub 98.9%

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

      3. *-inverses 98.9%

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

   5. Simplified 98.9%

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

   if -1 < y < 8.99999999999999936e-21

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 94.3%

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

   4. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 9 \cdot 10^{-21}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-41} \lor \neg \left(z \leq 1.8 \cdot 10^{-81}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.85e-41) (not (<= z 1.8e-81)))
   (+ y (/ x z))
   (* x (/ (- 1.0 y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.85e-41) || !(z <= 1.8e-81)) {
		tmp = y + (x / z);
	} else {
		tmp = x * ((1.0 - 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 <= (-1.85d-41)) .or. (.not. (z <= 1.8d-81))) then
        tmp = y + (x / z)
    else
        tmp = x * ((1.0d0 - y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.85e-41) || !(z <= 1.8e-81)) {
		tmp = y + (x / z);
	} else {
		tmp = x * ((1.0 - y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.85e-41) or not (z <= 1.8e-81):
		tmp = y + (x / z)
	else:
		tmp = x * ((1.0 - y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.85e-41) || !(z <= 1.8e-81))
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.85e-41) || ~((z <= 1.8e-81)))
		tmp = y + (x / z);
	else
		tmp = x * ((1.0 - y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.85e-41], N[Not[LessEqual[z, 1.8e-81]], $MachinePrecision]], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8500000000000001e-41 or 1.7999999999999999e-81 < z

   1. Initial program 83.7%

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

   3. Taylor expanded in y around 0 99.4%

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

   4. Taylor expanded in x around 0 89.1%

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

   if -1.8500000000000001e-41 < z < 1.7999999999999999e-81

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 93.3%

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

      1. associate-/l* 88.2%

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

      2. mul-1-neg 88.2%

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

      3. unsub-neg 88.2%

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

   5. Simplified 88.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-41} \lor \neg \left(z \leq 1.8 \cdot 10^{-81}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+22}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 8.9 \cdot 10^{+257}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5e+22)
   (+ y (/ x z))
   (if (<= y 8.9e+257) (* y (/ x (- z))) (- y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e+22) {
		tmp = y + (x / z);
	} else if (y <= 8.9e+257) {
		tmp = y * (x / -z);
	} else {
		tmp = y - (x / 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 (y <= 4.5d+22) then
        tmp = y + (x / z)
    else if (y <= 8.9d+257) then
        tmp = y * (x / -z)
    else
        tmp = y - (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e+22) {
		tmp = y + (x / z);
	} else if (y <= 8.9e+257) {
		tmp = y * (x / -z);
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.5e+22:
		tmp = y + (x / z)
	elif y <= 8.9e+257:
		tmp = y * (x / -z)
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.5e+22)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 8.9e+257)
		tmp = Float64(y * Float64(x / Float64(-z)));
	else
		tmp = Float64(y - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.5e+22)
		tmp = y + (x / z);
	elseif (y <= 8.9e+257)
		tmp = y * (x / -z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.5e+22], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.9e+257], N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.4999999999999998e22

   1. Initial program 92.3%

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

   3. Taylor expanded in y around 0 95.9%

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

   4. Taylor expanded in x around 0 88.5%

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

   if 4.4999999999999998e22 < y < 8.8999999999999998e257

   1. Initial program 83.2%

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

   3. Taylor expanded in y around inf 83.2%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. *-inverses 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 60.2%

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

      1. mul-1-neg 60.2%

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

      2. distribute-frac-neg2 60.2%

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

      3. *-commutative 60.2%

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

      4. associate-*r/ 63.8%

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

   8. Simplified 63.8%

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

   if 8.8999999999999998e257 < y

   1. Initial program 59.2%

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

   3. Taylor expanded in y around 0 88.9%

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

   4. Taylor expanded in x around 0 67.7%

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

      1. add-sqr-sqrt 22.1%

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

      2. sqrt-unprod 67.7%

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

      3. sqr-neg 67.7%

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

      4. sqrt-unprod 45.6%

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

      5. add-sqr-sqrt 78.8%

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

      6. distribute-frac-neg2 78.8%

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

      7. sub-neg 78.8%

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

   6. Applied egg-rr 78.8%

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

Alternative 6: 60.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-24}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e-24) y (if (<= y 1.25e-97) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e-24) {
		tmp = y;
	} else if (y <= 1.25e-97) {
		tmp = x / z;
	} else {
		tmp = 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 (y <= (-3.1d-24)) then
        tmp = y
    else if (y <= 1.25d-97) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e-24) {
		tmp = y;
	} else if (y <= 1.25e-97) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.1e-24:
		tmp = y
	elif y <= 1.25e-97:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e-24)
		tmp = y;
	elseif (y <= 1.25e-97)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.1e-24)
		tmp = y;
	elseif (y <= 1.25e-97)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.1e-24], y, If[LessEqual[y, 1.25e-97], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1e-24 or 1.2499999999999999e-97 < y

   1. Initial program 81.5%

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

   3. Taylor expanded in x around 0 53.7%

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

   if -3.1e-24 < y < 1.2499999999999999e-97

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 78.2%

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

Alternative 7: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-21}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9e-21) (+ y (/ x z)) (- y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e-21) {
		tmp = y + (x / z);
	} else {
		tmp = y - (x / 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 (y <= 9d-21) then
        tmp = y + (x / z)
    else
        tmp = y - (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e-21) {
		tmp = y + (x / z);
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9e-21:
		tmp = y + (x / z)
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9e-21)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9e-21)
		tmp = y + (x / z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9e-21], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.99999999999999936e-21

   1. Initial program 92.7%

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

   3. Taylor expanded in y around 0 95.8%

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

   4. Taylor expanded in x around 0 88.3%

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

   if 8.99999999999999936e-21 < y

   1. Initial program 79.1%

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

   3. Taylor expanded in y around 0 85.7%

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

   4. Taylor expanded in x around 0 45.4%

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

      1. add-sqr-sqrt 25.8%

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

      2. sqrt-unprod 57.0%

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

      3. sqr-neg 57.0%

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

      4. sqrt-unprod 28.5%

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

      5. add-sqr-sqrt 60.9%

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

      6. distribute-frac-neg2 60.9%

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

      7. sub-neg 60.9%

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

   6. Applied egg-rr 60.9%

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

Alternative 8: 78.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return y + (x / 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 = y + (x / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

3. Taylor expanded in y around 0 93.3%

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

4. Taylor expanded in x around 0 77.8%

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

Alternative 9: 41.0% accurate, 9.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = y
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return y
end

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 89.4%

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

3. Taylor expanded in x around 0 40.5%

   \[\leadsto \color{blue}{y} \]
4. Add Preprocessing

Developer Target 1: 94.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y + (x / z)) - (y / (z / 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 + (x / z)) - (y / (z / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ y (/ x z)) (/ y (/ z x))))

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