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

Percentage Accurate: 88.0% → 99.9%

Time: 7.4s

Alternatives: 8

Speedup: 1.0×

• 20.6% 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: 88.0% 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

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

FPCore

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

C

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

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 + (-1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.0%

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

3. Taylor expanded in x around -inf 95.9%

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

   1. mul-1-neg 95.9%

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

   2. unsub-neg 95.9%

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

   3. associate-/l* 97.8%

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

   4. associate-/r/ 99.9%

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

   5. sub-neg 99.9%

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

   6. metadata-eval 99.9%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 76.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+188} \lor \neg \left(x \leq -5.8 \cdot 10^{+139}\right) \land x \leq 5 \cdot 10^{+252}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.65e+188) (and (not (<= x -5.8e+139)) (<= x 5e+252)))
   (+ y (/ x z))
   (/ (- x) (/ z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.65e+188) || (!(x <= -5.8e+139) && (x <= 5e+252))) {
		tmp = y + (x / z);
	} else {
		tmp = -x / (z / 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 ((x <= (-1.65d+188)) .or. (.not. (x <= (-5.8d+139))) .and. (x <= 5d+252)) then
        tmp = y + (x / z)
    else
        tmp = -x / (z / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.65e+188) || (!(x <= -5.8e+139) && (x <= 5e+252))) {
		tmp = y + (x / z);
	} else {
		tmp = -x / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.65e+188) or (not (x <= -5.8e+139) and (x <= 5e+252)):
		tmp = y + (x / z)
	else:
		tmp = -x / (z / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.65e+188) || (!(x <= -5.8e+139) && (x <= 5e+252)))
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(Float64(-x) / Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.65e+188) || (~((x <= -5.8e+139)) && (x <= 5e+252)))
		tmp = y + (x / z);
	else
		tmp = -x / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.65e+188], And[N[Not[LessEqual[x, -5.8e+139]], $MachinePrecision], LessEqual[x, 5e+252]]], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[((-x) / N[(z / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.64999999999999991e188 or -5.7999999999999998e139 < x < 4.9999999999999997e252

   1. Initial program 90.0%

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

   3. Taylor expanded in x around -inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. unsub-neg 97.1%

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

      3. associate-/l* 97.5%

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

      4. associate-/r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 81.7%

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

      1. mul-1-neg 81.7%

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

      2. distribute-neg-frac 81.7%

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

   8. Simplified 81.7%

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

   9. Taylor expanded in y around 0 81.7%

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

       1. +-commutative 81.7%

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

   11. Simplified 81.7%

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

   if -1.64999999999999991e188 < x < -5.7999999999999998e139 or 4.9999999999999997e252 < x

   1. Initial program 78.4%

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

   3. Taylor expanded in y around inf 74.1%

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

      1. associate-/l* 95.7%

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

      2. associate-/r/ 95.5%

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

   5. Simplified 95.5%

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

   6. Taylor expanded in z around 0 74.0%

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

      1. mul-1-neg 74.0%

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

      2. associate-/l* 85.6%

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

   8. Simplified 85.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+188} \lor \neg \left(x \leq -5.8 \cdot 10^{+139}\right) \land x \leq 5 \cdot 10^{+252}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+188}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{+140}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+246}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (/ x z))))
   (if (<= x -1.25e+188)
     t_0
     (if (<= x -1.06e+140)
       (/ (- x) (/ z y))
       (if (<= x 5e+246) t_0 (* x (/ (- y) z)))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double tmp;
	if (x <= -1.25e+188) {
		tmp = t_0;
	} else if (x <= -1.06e+140) {
		tmp = -x / (z / y);
	} else if (x <= 5e+246) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y + (x / z)
    if (x <= (-1.25d+188)) then
        tmp = t_0
    else if (x <= (-1.06d+140)) then
        tmp = -x / (z / y)
    else if (x <= 5d+246) then
        tmp = t_0
    else
        tmp = x * (-y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double tmp;
	if (x <= -1.25e+188) {
		tmp = t_0;
	} else if (x <= -1.06e+140) {
		tmp = -x / (z / y);
	} else if (x <= 5e+246) {
		tmp = t_0;
	} else {
		tmp = x * (-y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x / z)
	tmp = 0
	if x <= -1.25e+188:
		tmp = t_0
	elif x <= -1.06e+140:
		tmp = -x / (z / y)
	elif x <= 5e+246:
		tmp = t_0
	else:
		tmp = x * (-y / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x / z))
	tmp = 0.0
	if (x <= -1.25e+188)
		tmp = t_0;
	elseif (x <= -1.06e+140)
		tmp = Float64(Float64(-x) / Float64(z / y));
	elseif (x <= 5e+246)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(Float64(-y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y + (x / z);
	tmp = 0.0;
	if (x <= -1.25e+188)
		tmp = t_0;
	elseif (x <= -1.06e+140)
		tmp = -x / (z / y);
	elseif (x <= 5e+246)
		tmp = t_0;
	else
		tmp = x * (-y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e+188], t$95$0, If[LessEqual[x, -1.06e+140], N[((-x) / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+246], t$95$0, N[(x * N[((-y) / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25e188 or -1.0600000000000001e140 < x < 4.99999999999999976e246

   1. Initial program 90.0%

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

   3. Taylor expanded in x around -inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. unsub-neg 97.1%

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

      3. associate-/l* 97.5%

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

      4. associate-/r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 81.7%

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

      1. mul-1-neg 81.7%

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

      2. distribute-neg-frac 81.7%

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

   8. Simplified 81.7%

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

   9. Taylor expanded in y around 0 81.7%

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

       1. +-commutative 81.7%

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

   11. Simplified 81.7%

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

   if -1.25e188 < x < -1.0600000000000001e140

   1. Initial program 81.3%

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

   3. Taylor expanded in y around inf 81.3%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around 0 81.6%

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

      1. mul-1-neg 81.6%

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

      2. associate-/l* 90.6%

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

   8. Simplified 90.6%

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

   if 4.99999999999999976e246 < x

   1. Initial program 76.0%

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

   3. Taylor expanded in y around inf 68.1%

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

      1. associate-/l* 92.0%

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

      2. associate-/r/ 91.9%

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

   5. Simplified 91.9%

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

   6. Taylor expanded in z around 0 67.8%

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

      1. mul-1-neg 67.8%

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

      2. associate-*r/ 81.5%

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

      3. distribute-rgt-neg-in 81.5%

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

      4. distribute-neg-frac 81.5%

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

   8. Simplified 81.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+188}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{+140}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+246}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.7 \cdot 10^{-8}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 2.7e-8))) (* y (- 1.0 (/ x z))) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 2.7e-8)) {
		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 <= 2.7d-8))) 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 <= 2.7e-8)) {
		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 <= 2.7e-8):
		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 <= 2.7e-8))
		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 <= 2.7e-8)))
		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, 2.7e-8]], $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 2.70000000000000002e-8 < y

   1. Initial program 78.2%

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

   3. Taylor expanded in x around -inf 91.8%

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

      1. mul-1-neg 91.8%

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

      2. unsub-neg 91.8%

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

      3. associate-/l* 95.6%

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

      4. associate-/r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around inf 99.1%

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

   if -1 < y < 2.70000000000000002e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around 0 99.8%

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

       1. +-commutative 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.7 \cdot 10^{-8}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-8}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.0)
   (/ y (/ z (- z x)))
   (if (<= y 2.7e-8) (+ 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 / (z - x));
	} else if (y <= 2.7e-8) {
		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 / (z - x))
    else if (y <= 2.7d-8) 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 / (z - x));
	} else if (y <= 2.7e-8) {
		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 / (z - x))
	elif y <= 2.7e-8:
		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(z / Float64(z - x)));
	elseif (y <= 2.7e-8)
		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 / (z - x));
	elseif (y <= 2.7e-8)
		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[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e-8], 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 77.3%

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

   3. Taylor expanded in y around inf 76.4%

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

      1. associate-/l* 98.9%

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

   5. Simplified 98.9%

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

   if -1 < y < 2.70000000000000002e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around 0 99.8%

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

       1. +-commutative 99.8%

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

   11. Simplified 99.8%

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

   if 2.70000000000000002e-8 < y

   1. Initial program 79.5%

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

   3. Taylor expanded in x around -inf 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. associate-/l* 96.3%

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

      4. associate-/r/ 100.0%

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

      5. sub-neg 100.0%

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

      6. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around inf 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-8}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-28}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.7e-28) y (if (<= z 7.8e-63) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.7e-28) {
		tmp = y;
	} else if (z <= 7.8e-63) {
		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 <= (-2.7d-28)) then
        tmp = y
    else if (z <= 7.8d-63) 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 <= -2.7e-28) {
		tmp = y;
	} else if (z <= 7.8e-63) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.7e-28:
		tmp = y
	elif z <= 7.8e-63:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.7e-28)
		tmp = y;
	elseif (z <= 7.8e-63)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.7e-28)
		tmp = y;
	elseif (z <= 7.8e-63)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.7e-28], y, If[LessEqual[z, 7.8e-63], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if z < -2.6999999999999999e-28 or 7.80000000000000044e-63 < z

   1. Initial program 81.6%

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

   3. Taylor expanded in x around 0 61.8%

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

   if -2.6999999999999999e-28 < z < 7.80000000000000044e-63

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 59.4%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-28}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 89.0%

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

3. Taylor expanded in x around -inf 95.9%

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

   1. mul-1-neg 95.9%

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

   2. unsub-neg 95.9%

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

   3. associate-/l* 97.8%

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

   4. associate-/r/ 99.9%

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

   5. sub-neg 99.9%

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

   6. metadata-eval 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in y around 0 77.3%

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

   1. mul-1-neg 77.3%

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

   2. distribute-neg-frac 77.3%

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

8. Simplified 77.3%

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

9. Taylor expanded in y around 0 77.3%

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

    1. +-commutative 77.3%

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

11. Simplified 77.3%

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

12. Final simplification 77.3%

    \[\leadsto y + \frac{x}{z} \]
13. Add Preprocessing

Alternative 8: 40.5% 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 89.0%

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

3. Taylor expanded in x around 0 40.7%

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

4. Final simplification 40.7%

   \[\leadsto y \]
5. Add Preprocessing

Developer target: 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 2024019 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

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