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

Percentage Accurate: 88.1% → 99.9%

Time: 9.1s

Alternatives: 10

Speedup: 1.0×

• 20.4% 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: 88.1% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in y around 0 94.8%

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

4. Taylor expanded in x around 0 95.1%

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

   1. +-commutative 95.1%

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

   2. neg-mul-1 95.1%

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

   3. sub-neg 95.1%

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

   4. div-sub 95.1%

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

   5. associate-*r/ 96.6%

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

   6. associate-*l/ 99.9%

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

6. Simplified 99.9%

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

Alternative 2: 99.2% 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 79.6%

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

   3. Taylor expanded in y around inf 79.2%

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

      1. associate-/l* 99.4%

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

      2. div-sub 99.4%

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

      3. sub-neg 99.4%

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

      4. *-inverses 99.4%

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

      5. sub-neg 99.4%

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

   5. Simplified 99.4%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 93.6%

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

   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 99.5%

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

Alternative 3: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+138} \lor \neg \left(x \leq 3.2 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.8e+138) (not (<= x 3.2e-9)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.8e+138) || !(x <= 3.2e-9)) {
		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 <= (-1.8d+138)) .or. (.not. (x <= 3.2d-9))) 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 <= -1.8e+138) || !(x <= 3.2e-9)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.8e+138) or not (x <= 3.2e-9):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.8e+138) || !(x <= 3.2e-9))
		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 <= -1.8e+138) || ~((x <= 3.2e-9)))
		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, -1.8e+138], N[Not[LessEqual[x, 3.2e-9]], $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 < -1.8000000000000001e138 or 3.20000000000000012e-9 < x

   1. Initial program 90.7%

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

   3. Taylor expanded in x around inf 83.4%

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

      1. associate-/l* 86.9%

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

      2. mul-1-neg 86.9%

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

      3. unsub-neg 86.9%

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

   5. Simplified 86.9%

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

   if -1.8000000000000001e138 < x < 3.20000000000000012e-9

   1. Initial program 90.9%

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

   3. Taylor expanded in y around 0 98.7%

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

   4. Taylor expanded in x around 0 87.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+138} \lor \neg \left(x \leq 3.2 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.0)
   (- y (* y (/ 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 - (y * (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 - (y * (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 - (y * (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 - (y * (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(y * Float64(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 - (y * (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[(y * N[(x / z), $MachinePrecision]), $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 91.4%

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

   3. Taylor expanded in y around 0 99.8%

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

   4. Taylor expanded in x around 0 84.5%

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

      1. +-commutative 84.5%

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

      2. neg-mul-1 84.5%

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

      3. sub-neg 84.5%

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

      4. div-sub 84.4%

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

      5. associate-*r/ 96.4%

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

      6. associate-*l/ 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in y around inf 99.0%

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

      1. neg-mul-1 99.0%

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

   9. Simplified 99.0%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 93.6%

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

   4. Taylor expanded in x around 0 99.5%

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

   if 1 < y

   1. Initial program 68.9%

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

   3. Taylor expanded in y around inf 68.9%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - x\right)}{z}} \]
   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. sub-neg 99.9%

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

      4. *-inverses 99.9%

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

      5. sub-neg 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.5%

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

Alternative 5: 62.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-35} \lor \neg \left(y \leq 6 \cdot 10^{-33}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.4e-35) (not (<= y 6e-33))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.4e-35) || !(y <= 6e-33)) {
		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 <= (-1.4d-35)) .or. (.not. (y <= 6d-33))) 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 <= -1.4e-35) || !(y <= 6e-33)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.4e-35) or not (y <= 6e-33):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.4e-35) || !(y <= 6e-33))
		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 <= -1.4e-35) || ~((y <= 6e-33)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.4e-35], N[Not[LessEqual[y, 6e-33]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e-35 or 6.0000000000000003e-33 < y

   1. Initial program 82.6%

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

   3. Taylor expanded in z around inf 51.3%

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

   4. Taylor expanded in x around 0 43.7%

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

      1. *-commutative 43.7%

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

      2. associate-/l* 57.7%

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

   6. Applied egg-rr 57.7%

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

   if -1.4e-35 < y < 6.0000000000000003e-33

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 73.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-35} \lor \neg \left(y \leq 6 \cdot 10^{-33}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 0.205:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7e-8) y (if (<= z 0.205) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e-8) {
		tmp = y;
	} else if (z <= 0.205) {
		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 (z <= (-7d-8)) then
        tmp = y
    else if (z <= 0.205d0) 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 (z <= -7e-8) {
		tmp = y;
	} else if (z <= 0.205) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7e-8:
		tmp = y
	elif z <= 0.205:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7e-8)
		tmp = y;
	elseif (z <= 0.205)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7e-8)
		tmp = y;
	elseif (z <= 0.205)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7e-8], y, If[LessEqual[z, 0.205], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if z < -7.00000000000000048e-8 or 0.204999999999999988 < z

   1. Initial program 81.7%

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

   3. Taylor expanded in x around 0 70.5%

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

   if -7.00000000000000048e-8 < z < 0.204999999999999988

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 59.7%

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

Alternative 7: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+146}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.45e+146) (* y (/ (- x) z)) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.45e+146) {
		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 <= (-2.45d+146)) 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 <= -2.45e+146) {
		tmp = y * (-x / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.45e+146:
		tmp = y * (-x / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.45e+146)
		tmp = Float64(y * Float64(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 <= -2.45e+146)
		tmp = y * (-x / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.45e+146], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4500000000000001e146

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0 99.9%

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

   4. Taylor expanded in x around 0 82.9%

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

      1. +-commutative 82.9%

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

      2. neg-mul-1 82.9%

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

      3. sub-neg 82.9%

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

      4. div-sub 82.9%

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

      5. associate-*r/ 91.1%

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

      6. associate-*l/ 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around inf 99.9%

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

      1. neg-mul-1 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around inf 73.1%

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

       1. associate-*r/ 73.1%

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

       2. mul-1-neg 73.1%

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

       3. distribute-lft-neg-out 73.1%

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

       4. associate-*l/ 77.1%

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

   12. Simplified 77.1%

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

   if -2.4500000000000001e146 < y

   1. Initial program 91.9%

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

   3. Taylor expanded in y around 0 94.4%

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

   4. Taylor expanded in x around 0 84.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+146}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{+146}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.25e+146) (* x (/ (- y) z)) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.25e+146) {
		tmp = x * (-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 (y <= (-2.25d+146)) then
        tmp = x * (-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 (y <= -2.25e+146) {
		tmp = x * (-y / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.25e+146:
		tmp = x * (-y / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.25e+146)
		tmp = Float64(x * Float64(Float64(-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 (y <= -2.25e+146)
		tmp = x * (-y / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.25e+146], N[(x * N[((-y) / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.25000000000000013e146

   1. Initial program 78.8%

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

   3. Taylor expanded in x around inf 73.1%

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

      1. associate-/l* 64.9%

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

      2. mul-1-neg 64.9%

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

      3. unsub-neg 64.9%

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

   5. Simplified 64.9%

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

   6. Taylor expanded in y around inf 64.9%

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

      1. neg-mul-1 64.9%

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

      2. distribute-neg-frac 64.9%

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

   8. Simplified 64.9%

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

   if -2.25000000000000013e146 < y

   1. Initial program 91.9%

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

   3. Taylor expanded in y around 0 94.4%

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

   4. Taylor expanded in x around 0 84.9%

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

Alternative 9: 77.9% 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 90.8%

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

3. Taylor expanded in y around 0 94.8%

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

4. Taylor expanded in x around 0 80.6%

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

Alternative 10: 40.6% 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 90.8%

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

3. Taylor expanded in x around 0 42.7%

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

Developer Target 1: 94.0% 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 2024145 
(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))