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

Percentage Accurate: 88.2% → 99.8%

Time: 7.7s

Alternatives: 8

Speedup: 0.9×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + (y * (z - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + (y * (z - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - \frac{x}{z} \cdot y\\ \mathbf{if}\;y \leq -1900000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 15500000000000:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y + x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (* (/ x z) y))))
   (if (<= y -1900000000000.0)
     t_0
     (if (<= y 15500000000000.0) (/ (+ (* (- z x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y - ((x / z) * y);
	double tmp;
	if (y <= -1900000000000.0) {
		tmp = t_0;
	} else if (y <= 15500000000000.0) {
		tmp = (((z - x) * 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 - ((x / z) * y)
    if (y <= (-1900000000000.0d0)) then
        tmp = t_0
    else if (y <= 15500000000000.0d0) then
        tmp = (((z - x) * 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 - ((x / z) * y);
	double tmp;
	if (y <= -1900000000000.0) {
		tmp = t_0;
	} else if (y <= 15500000000000.0) {
		tmp = (((z - x) * y) + x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y - ((x / z) * y)
	tmp = 0
	if y <= -1900000000000.0:
		tmp = t_0
	elif y <= 15500000000000.0:
		tmp = (((z - x) * y) + x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y - Float64(Float64(x / z) * y))
	tmp = 0.0
	if (y <= -1900000000000.0)
		tmp = t_0;
	elseif (y <= 15500000000000.0)
		tmp = Float64(Float64(Float64(Float64(z - x) * y) + x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y - ((x / z) * y);
	tmp = 0.0;
	if (y <= -1900000000000.0)
		tmp = t_0;
	elseif (y <= 15500000000000.0)
		tmp = (((z - x) * y) + x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y - N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1900000000000.0], t$95$0, If[LessEqual[y, 15500000000000.0], N[(N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9e12 or 1.55e13 < y

   1. Initial program 77.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. associate-*l/ N/A

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

      11. lower--.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if -1.9e12 < y < 1.55e13

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1900000000000:\\ \;\;\;\;y - \frac{x}{z} \cdot y\\ \mathbf{elif}\;y \leq 15500000000000:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y + x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - \frac{x}{z} \cdot y\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (* (/ x z) y))))
   (if (<= y -6.5e+31) t_0 (if (<= y 8e-11) (+ (/ x z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y - ((x / z) * y);
	double tmp;
	if (y <= -6.5e+31) {
		tmp = t_0;
	} else if (y <= 8e-11) {
		tmp = (x / z) + y;
	} 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) * y)
    if (y <= (-6.5d+31)) then
        tmp = t_0
    else if (y <= 8d-11) then
        tmp = (x / z) + y
    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) * y);
	double tmp;
	if (y <= -6.5e+31) {
		tmp = t_0;
	} else if (y <= 8e-11) {
		tmp = (x / z) + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y - ((x / z) * y)
	tmp = 0
	if y <= -6.5e+31:
		tmp = t_0
	elif y <= 8e-11:
		tmp = (x / z) + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y - Float64(Float64(x / z) * y))
	tmp = 0.0
	if (y <= -6.5e+31)
		tmp = t_0;
	elseif (y <= 8e-11)
		tmp = Float64(Float64(x / z) + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y - ((x / z) * y);
	tmp = 0.0;
	if (y <= -6.5e+31)
		tmp = t_0;
	elseif (y <= 8e-11)
		tmp = (x / z) + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y - N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+31], t$95$0, If[LessEqual[y, 8e-11], N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5000000000000004e31 or 7.99999999999999952e-11 < y

   1. Initial program 78.4%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. associate-*l/ N/A

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

      11. lower--.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if -6.5000000000000004e31 < y < 7.99999999999999952e-11

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 99.9%

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

Alternative 3: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+161}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+119}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 y) (/ x z))))
   (if (<= x -6.8e+161) t_0 (if (<= x 2.95e+119) (+ (/ x z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * (x / z);
	double tmp;
	if (x <= -6.8e+161) {
		tmp = t_0;
	} else if (x <= 2.95e+119) {
		tmp = (x / z) + y;
	} 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 = (1.0d0 - y) * (x / z)
    if (x <= (-6.8d+161)) then
        tmp = t_0
    else if (x <= 2.95d+119) then
        tmp = (x / z) + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * (x / z);
	double tmp;
	if (x <= -6.8e+161) {
		tmp = t_0;
	} else if (x <= 2.95e+119) {
		tmp = (x / z) + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 - y) * (x / z)
	tmp = 0
	if x <= -6.8e+161:
		tmp = t_0
	elif x <= 2.95e+119:
		tmp = (x / z) + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 - y) * Float64(x / z))
	tmp = 0.0
	if (x <= -6.8e+161)
		tmp = t_0;
	elseif (x <= 2.95e+119)
		tmp = Float64(Float64(x / z) + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - y) * (x / z);
	tmp = 0.0;
	if (x <= -6.8e+161)
		tmp = t_0;
	elseif (x <= 2.95e+119)
		tmp = (x / z) + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.8e+161], t$95$0, If[LessEqual[x, 2.95e+119], N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.79999999999999986e161 or 2.95e119 < x

   1. Initial program 92.2%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out-- N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. unsub-neg N/A

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

      18. mul-1-neg N/A

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

      19. lower-/.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 95.1

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 95.1%

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

   if -6.79999999999999986e161 < x < 2.95e119

   1. Initial program 86.7%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 84.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+161}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+119}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 - y}{z} \cdot x + y\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+81) (+ (* (/ (- 1.0 y) z) x) y) (- y (* (/ x z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+81) {
		tmp = (((1.0 - y) / z) * x) + y;
	} else {
		tmp = y - ((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 (y <= 5d+81) then
        tmp = (((1.0d0 - y) / z) * x) + y
    else
        tmp = y - ((x / z) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+81) {
		tmp = (((1.0 - y) / z) * x) + y;
	} else {
		tmp = y - ((x / z) * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e+81:
		tmp = (((1.0 - y) / z) * x) + y
	else:
		tmp = y - ((x / z) * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e+81)
		tmp = Float64(Float64(Float64(Float64(1.0 - y) / z) * x) + y);
	else
		tmp = Float64(y - Float64(Float64(x / z) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e+81)
		tmp = (((1.0 - y) / z) * x) + y;
	else
		tmp = y - ((x / z) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e+81], N[(N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision], N[(y - N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999998e81

   1. Initial program 90.4%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

   if 4.9999999999999998e81 < y

   1. Initial program 79.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. associate-*l/ N/A

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

      11. lower--.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

Alternative 5: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - y}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+81) (fma (/ (- 1.0 y) z) x y) (- y (* (/ x z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+81) {
		tmp = fma(((1.0 - y) / z), x, y);
	} else {
		tmp = y - ((x / z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e+81)
		tmp = fma(Float64(Float64(1.0 - y) / z), x, y);
	else
		tmp = Float64(y - Float64(Float64(x / z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e+81], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x + y), $MachinePrecision], N[(y - N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999998e81

   1. Initial program 90.4%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 98.0%

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

   if 4.9999999999999998e81 < y

   1. Initial program 79.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. associate-*l/ N/A

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

      11. lower--.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

Alternative 6: 76.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+140}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.5e+140) (+ (/ x z) y) (* (/ (- y) z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e+140) {
		tmp = (x / z) + y;
	} else {
		tmp = (-y / z) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.5d+140) then
        tmp = (x / z) + y
    else
        tmp = (-y / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e+140) {
		tmp = (x / z) + y;
	} else {
		tmp = (-y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.5e+140:
		tmp = (x / z) + y
	else:
		tmp = (-y / z) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.5e+140)
		tmp = Float64(Float64(x / z) + y);
	else
		tmp = Float64(Float64(Float64(-y) / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.5e+140)
		tmp = (x / z) + y;
	else
		tmp = (-y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.5e+140], N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision], N[(N[((-y) / z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.49999999999999989e140

   1. Initial program 90.2%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 85.0%

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

   if 3.49999999999999989e140 < y

   1. Initial program 76.0%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out-- N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. unsub-neg N/A

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

      18. mul-1-neg N/A

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

      19. lower-/.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 64.2%

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

Alternative 7: 77.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 96.2%

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

7. Applied rewrites 96.2%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 78.7%

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

Alternative 8: 40.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 36.5

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

5. Applied rewrites 36.5%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
6. Add Preprocessing

Developer Target 1: 94.0% accurate, 0.6× 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 2024244 
(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))