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

Percentage Accurate: 88.1% → 99.9%

Time: 9.9s

Alternatives: 7

Speedup: 1.0×

• 21.0% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. mul-1-neg N/A

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

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

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. mul-1-neg N/A

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

   12. unsub-neg N/A

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

   13. div-sub N/A

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

   14. unsub-neg N/A

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

   15. mul-1-neg N/A

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

   16. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{\left(\frac{x + -1 \cdot \left(x \cdot y\right)}{z}\right)}\right) \]

5. Simplified 99.9%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (/ x z)))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (+ y (/ x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - (x / z));
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 78.5%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{z - x}{z}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{z}{z} - \color{blue}{\frac{x}{z}}\right)\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(1 - \frac{\color{blue}{x}}{z}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

      6. /-lowering-/.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 99.7%

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

   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 z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right), z\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), z\right) \]

   5. Simplified 98.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

      10. /-lowering-/.f64 98.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 98.8%

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

4. Final simplification 99.3%

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

Alternative 3: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (/ x z))))
   (if (<= z -1.2e+76) t_0 (if (<= z 6.5e-10) (* x (/ (- 1.0 y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double tmp;
	if (z <= -1.2e+76) {
		tmp = t_0;
	} else if (z <= 6.5e-10) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x / z)
    if (z <= (-1.2d+76)) then
        tmp = t_0
    else if (z <= 6.5d-10) then
        tmp = x * ((1.0d0 - y) / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y + (x / z);
	double tmp;
	if (z <= -1.2e+76) {
		tmp = t_0;
	} else if (z <= 6.5e-10) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x / z)
	tmp = 0
	if z <= -1.2e+76:
		tmp = t_0
	elif z <= 6.5e-10:
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x / z))
	tmp = 0.0
	if (z <= -1.2e+76)
		tmp = t_0;
	elseif (z <= 6.5e-10)
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y + (x / z);
	tmp = 0.0;
	if (z <= -1.2e+76)
		tmp = t_0;
	elseif (z <= 6.5e-10)
		tmp = x * ((1.0 - y) / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.2e+76], t$95$0, If[LessEqual[z, 6.5e-10], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e76 or 6.5000000000000003e-10 < z

   1. Initial program 79.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right), z\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 75.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), z\right) \]

   5. Simplified 75.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

      10. /-lowering-/.f64 92.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

   7. Applied egg-rr 92.3%

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

   if -1.2e76 < z < 6.5000000000000003e-10

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + -1 \cdot y\right), \color{blue}{\left(\frac{x}{z}\right)}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right), \left(\frac{x}{z}\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 - y\right), \left(\frac{\color{blue}{x}}{z}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), \left(\frac{\color{blue}{x}}{z}\right)\right) \]

      7. /-lowering-/.f64 87.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, y\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

   5. Simplified 87.4%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{z} \cdot \left(1 - y\right)\right)}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 \cdot \left(1 - y\right)}{\color{blue}{z}}\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1 - y}{z}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(1 - y\right), \color{blue}{z}\right)\right) \]

      8. --lowering--.f64 83.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, y\right), z\right)\right) \]

   7. Applied egg-rr 83.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+76}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ y z))))
   (if (<= y -2.3e-17) t_0 (if (<= y 1.36e-7) (/ x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y / z);
	double tmp;
	if (y <= -2.3e-17) {
		tmp = t_0;
	} else if (y <= 1.36e-7) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * (y / z)
    if (y <= (-2.3d-17)) then
        tmp = t_0
    else if (y <= 1.36d-7) then
        tmp = x / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y / z);
	double tmp;
	if (y <= -2.3e-17) {
		tmp = t_0;
	} else if (y <= 1.36e-7) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y / z)
	tmp = 0
	if y <= -2.3e-17:
		tmp = t_0
	elif y <= 1.36e-7:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y / z))
	tmp = 0.0
	if (y <= -2.3e-17)
		tmp = t_0;
	elseif (y <= 1.36e-7)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y / z);
	tmp = 0.0;
	if (y <= -2.3e-17)
		tmp = t_0;
	elseif (y <= 1.36e-7)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-17], t$95$0, If[LessEqual[y, 1.36e-7], N[(x / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.30000000000000009e-17 or 1.36e-7 < y

   1. Initial program 79.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y \cdot z\right)}, z\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 33.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), z\right) \]

   5. Simplified 33.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

         \[\leadsto \left(\frac{1}{z} \cdot \left(y \cdot 1\right)\right) \cdot z \]

      5. *-inverses N/A

         \[\leadsto \left(\frac{1}{z} \cdot \left(y \cdot \frac{z}{z}\right)\right) \cdot z \]

      6. associate-/l* N/A

         \[\leadsto \left(\frac{1}{z} \cdot \frac{y \cdot z}{z}\right) \cdot z \]

      7. div-inv N/A

         \[\leadsto \left(\frac{1}{z} \cdot \left(\left(y \cdot z\right) \cdot \frac{1}{z}\right)\right) \cdot z \]

      8. associate-*l* N/A

         \[\leadsto \left(\left(\frac{1}{z} \cdot \left(y \cdot z\right)\right) \cdot \frac{1}{z}\right) \cdot z \]

      9. associate-/r/ N/A

         \[\leadsto \left(\frac{1}{\frac{z}{y \cdot z}} \cdot \frac{1}{z}\right) \cdot z \]

      10. clear-num N/A

          \[\leadsto \left(\frac{y \cdot z}{z} \cdot \frac{1}{z}\right) \cdot z \]

      11. associate-*l/ N/A

          \[\leadsto \frac{\left(y \cdot z\right) \cdot \frac{1}{z}}{z} \cdot z \]

      12. associate-*l* N/A

          \[\leadsto \frac{y \cdot \left(z \cdot \frac{1}{z}\right)}{z} \cdot z \]

      13. div-inv N/A

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

      14. *-inverses N/A

          \[\leadsto \frac{y \cdot 1}{z} \cdot z \]

      15. *-rgt-identity N/A

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

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{z}\right), \color{blue}{z}\right) \]

      17. /-lowering-/.f64 51.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), z\right) \]

   7. Applied egg-rr 51.1%

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

   if -2.30000000000000009e-17 < y < 1.36e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 73.0%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 73.0%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-17}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+83}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.22e+83) y (if (<= z 1.6e+43) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.22e+83) {
		tmp = y;
	} else if (z <= 1.6e+43) {
		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 <= (-1.22d+83)) then
        tmp = y
    else if (z <= 1.6d+43) 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 <= -1.22e+83) {
		tmp = y;
	} else if (z <= 1.6e+43) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.22e+83:
		tmp = y
	elif z <= 1.6e+43:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.22e+83)
		tmp = y;
	elseif (z <= 1.6e+43)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.22e+83)
		tmp = y;
	elseif (z <= 1.6e+43)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.22e+83], y, If[LessEqual[z, 1.6e+43], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if z < -1.22e83 or 1.60000000000000007e43 < z

   1. Initial program 75.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.4%

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

   if -1.22e83 < z < 1.60000000000000007e43

   1. Initial program 98.2%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 53.7%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 53.7%

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

Alternative 6: 77.0% 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 89.3%

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

3. Taylor expanded in z around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right), z\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 69.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), z\right) \]

5. Simplified 69.9%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. distribute-rgt-in N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. div-inv N/A

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

   7. *-inverses N/A

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

   8. *-rgt-identity N/A

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

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{z}\right), \color{blue}{y}\right) \]

   10. /-lowering-/.f64 77.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right) \]

7. Applied egg-rr 77.3%

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

8. Final simplification 77.3%

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

Alternative 7: 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 89.3%

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

3. Taylor expanded in x around 0

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

5. Simplified 38.3%

   \[\leadsto \color{blue}{y} \]
6. 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 2024192 
(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))