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

Percentage Accurate: 88.8% → 99.6%

Time: 6.6s

Alternatives: 9

Speedup: 0.8×

• 20.9% 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: 88.8% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+56}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;y \leq 1250000000000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.7e+56)
   (* y (/ (- z x) z))
   (if (<= y 1250000000000.0)
     (/ (+ x (* y (- z x))) z)
     (* y (- 1.0 (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.7e+56) {
		tmp = y * ((z - x) / z);
	} else if (y <= 1250000000000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y * (1.0 - (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 <= (-3.7d+56)) then
        tmp = y * ((z - x) / z)
    else if (y <= 1250000000000.0d0) then
        tmp = (x + (y * (z - x))) / z
    else
        tmp = y * (1.0d0 - (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.7e+56) {
		tmp = y * ((z - x) / z);
	} else if (y <= 1250000000000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y * (1.0 - (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.7e+56:
		tmp = y * ((z - x) / z)
	elif y <= 1250000000000.0:
		tmp = (x + (y * (z - x))) / z
	else:
		tmp = y * (1.0 - (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.7e+56)
		tmp = Float64(y * Float64(Float64(z - x) / z));
	elseif (y <= 1250000000000.0)
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	else
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.7e+56)
		tmp = y * ((z - x) / z);
	elseif (y <= 1250000000000.0)
		tmp = (x + (y * (z - x))) / z;
	else
		tmp = y * (1.0 - (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.7e+56], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1250000000000.0], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.69999999999999997e56

   1. Initial program 71.2%

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

   3. Taylor expanded in y around inf 71.2%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   if -3.69999999999999997e56 < y < 1.25e12

   1. Initial program 99.9%

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

   if 1.25e12 < y

   1. Initial program 62.7%

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

   3. Taylor expanded in y around inf 62.6%

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

      1. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 62.6%

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

      1. associate-/l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. distribute-neg-frac2 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. remove-double-neg 99.8%

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

      8. div-sub 99.8%

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

      9. *-inverses 99.8%

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

   8. Simplified 99.8%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		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 <= 1.0d0))) 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 <= 1.0)) {
		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 <= 1.0):
		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 <= 1.0))
		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 <= 1.0)))
		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, 1.0]], $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 1 < y

   1. Initial program 70.8%

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

   3. Taylor expanded in y around inf 68.6%

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

      1. associate-/l* 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in y around 0 68.6%

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

      1. associate-/l* 97.7%

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

      2. remove-double-neg 97.7%

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

      3. distribute-neg-frac2 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. distribute-neg-frac2 97.7%

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

      6. distribute-frac-neg 97.7%

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

      7. remove-double-neg 97.7%

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

      8. div-sub 97.7%

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

      9. *-inverses 97.7%

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

   8. Simplified 97.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 96.3%

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

   4. Taylor expanded in x around 0 98.3%

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

4. Final simplification 98.0%

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

Alternative 3: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+47} \lor \neg \left(x \leq 1.45 \cdot 10^{+83}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.02e+47) (not (<= x 1.45e+83)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.02e+47) || !(x <= 1.45e+83)) {
		tmp = x * ((1.0 - y) / 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 ((x <= (-1.02d+47)) .or. (.not. (x <= 1.45d+83))) then
        tmp = x * ((1.0d0 - y) / 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 ((x <= -1.02e+47) || !(x <= 1.45e+83)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.02e+47) or not (x <= 1.45e+83):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.02e+47) || !(x <= 1.45e+83))
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.02e+47) || ~((x <= 1.45e+83)))
		tmp = x * ((1.0 - y) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.02e+47], N[Not[LessEqual[x, 1.45e+83]], $MachinePrecision]], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0199999999999999e47 or 1.45e83 < x

   1. Initial program 91.4%

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

   3. Taylor expanded in x around inf 89.2%

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

      1. associate-/l* 90.1%

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

      2. mul-1-neg 90.1%

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

      3. unsub-neg 90.1%

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

   5. Simplified 90.1%

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

   if -1.0199999999999999e47 < x < 1.45e83

   1. Initial program 83.2%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 88.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+47} \lor \neg \left(x \leq 1.45 \cdot 10^{+83}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 75.6%

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

   3. Taylor expanded in y around inf 74.4%

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

      1. associate-/l* 98.7%

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

   5. Simplified 98.7%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 96.3%

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

   4. Taylor expanded in x around 0 98.3%

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

   if 1 < y

   1. Initial program 65.2%

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

   3. Taylor expanded in y around inf 62.0%

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

      1. associate-/l* 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in y around 0 62.0%

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

      1. associate-/l* 96.6%

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

      2. remove-double-neg 96.6%

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

      3. distribute-neg-frac2 96.6%

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

      4. distribute-neg-frac 96.6%

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

      5. distribute-neg-frac2 96.6%

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

      6. distribute-frac-neg 96.6%

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

      7. remove-double-neg 96.6%

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

      8. div-sub 96.6%

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

      9. *-inverses 96.6%

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

   8. Simplified 96.6%

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

Alternative 5: 61.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-26} \lor \neg \left(y \leq 1.06 \cdot 10^{-80}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.5e-26) (not (<= y 1.06e-80))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.5e-26) || !(y <= 1.06e-80)) {
		tmp = z * (y / z);
	} else {
		tmp = 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 <= (-2.5d-26)) .or. (.not. (y <= 1.06d-80))) then
        tmp = z * (y / z)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.5e-26) || !(y <= 1.06e-80)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.5e-26) or not (y <= 1.06e-80):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.5e-26) || !(y <= 1.06e-80))
		tmp = Float64(z * Float64(y / z));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.5e-26) || ~((y <= 1.06e-80)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.5e-26], N[Not[LessEqual[y, 1.06e-80]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5000000000000001e-26 or 1.0600000000000001e-80 < y

   1. Initial program 75.7%

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

   3. Taylor expanded in z around inf 49.5%

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

   4. Taylor expanded in x around 0 36.7%

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

      1. *-commutative 36.7%

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

      2. associate-/l* 60.0%

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

   6. Applied egg-rr 60.0%

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

   if -2.5000000000000001e-26 < y < 1.0600000000000001e-80

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.8%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-26} \lor \neg \left(y \leq 1.06 \cdot 10^{-80}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-27}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3e-27) y (if (<= y 7.2e-81) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e-27) {
		tmp = y;
	} else if (y <= 7.2e-81) {
		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 <= (-3d-27)) then
        tmp = y
    else if (y <= 7.2d-81) 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 <= -3e-27) {
		tmp = y;
	} else if (y <= 7.2e-81) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3e-27:
		tmp = y
	elif y <= 7.2e-81:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3e-27)
		tmp = y;
	elseif (y <= 7.2e-81)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3e-27)
		tmp = y;
	elseif (y <= 7.2e-81)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3e-27], y, If[LessEqual[y, 7.2e-81], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -3.0000000000000001e-27 or 7.1999999999999997e-81 < y

   1. Initial program 75.7%

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

   3. Taylor expanded in x around 0 57.8%

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

   if -3.0000000000000001e-27 < y < 7.1999999999999997e-81

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.8%

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

Alternative 7: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (x / z) + (y * (1.0 - (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 / z) + (y * (1.0d0 - (x / z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

3. Taylor expanded in y around 0 97.2%

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

4. Final simplification 97.2%

   \[\leadsto \frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right) \]
5. Add Preprocessing

Alternative 8: 78.3% 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 86.0%

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

3. Taylor expanded in y around 0 97.2%

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

4. Taylor expanded in x around 0 83.1%

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

Alternative 9: 40.7% 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 86.0%

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

3. Taylor expanded in x around 0 44.0%

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

Developer target: 94.5% 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 2024084 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :alt
  (- (+ y (/ x z)) (/ y (/ z x)))

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