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

Percentage Accurate: 88.1% → 100.0%

Time: 5.1s

Alternatives: 7

Speedup: 1.1×

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: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{z}, 1 - y, y\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

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

   2. mul-1-neg N/A

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

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

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

5. Applied rewrites 99.9%

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

Alternative 2: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z - x}{z} \cdot y\\ \mathbf{if}\;y \leq -16.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- z x) z) y)))
   (if (<= y -16.5) t_0 (if (<= y 1.0) (fma (/ x z) 1.0 y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((z - x) / z) * y;
	double tmp;
	if (y <= -16.5) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma((x / z), 1.0, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z - x) / z) * y)
	tmp = 0.0
	if (y <= -16.5)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = fma(Float64(x / z), 1.0, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -16.5], t$95$0, If[LessEqual[y, 1.0], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -16.5 or 1 < y

   1. Initial program 80.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 79.5

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

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 99.0%

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

   if -16.5 < y < 1

   1. Initial program 99.9%

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

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.7%

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 1, y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -16.5:\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 350000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+117}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 350000000000.0)
   (fma (/ x z) 1.0 y)
   (if (<= y 6.5e+117) (* (/ (- y) z) x) (* 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 350000000000.0) {
		tmp = fma((x / z), 1.0, y);
	} else if (y <= 6.5e+117) {
		tmp = (-y / z) * x;
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 350000000000.0)
		tmp = fma(Float64(x / z), 1.0, y);
	elseif (y <= 6.5e+117)
		tmp = Float64(Float64(Float64(-y) / z) * x);
	else
		tmp = Float64(1.0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 350000000000.0], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], If[LessEqual[y, 6.5e+117], N[(N[((-y) / z), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.5e11

   1. Initial program 94.1%

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

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.8%

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

   if 3.5e11 < y < 6.5000000000000004e117

   1. Initial program 90.3%

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

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

      2. associate-*l/ N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 70.8%

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

   if 6.5000000000000004e117 < y

   1. Initial program 73.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto y \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 58.3%

       \[\leadsto y \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 350000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+117}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 350000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+117}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 350000000000.0)
   (fma (/ x z) 1.0 y)
   (if (<= y 6.5e+117) (* (- y) (/ x z)) (* 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 350000000000.0) {
		tmp = fma((x / z), 1.0, y);
	} else if (y <= 6.5e+117) {
		tmp = -y * (x / z);
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 350000000000.0)
		tmp = fma(Float64(x / z), 1.0, y);
	elseif (y <= 6.5e+117)
		tmp = Float64(Float64(-y) * Float64(x / z));
	else
		tmp = Float64(1.0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 350000000000.0], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], If[LessEqual[y, 6.5e+117], N[((-y) * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(1.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.5e11

   1. Initial program 94.1%

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

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

      2. mul-1-neg N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.8%

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

   if 3.5e11 < y < 6.5000000000000004e117

   1. Initial program 90.3%

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

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

      2. associate-*l/ N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. div-sub N/A

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

      14. unsub-neg N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 70.8%

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

   10. Applied rewrites 70.6%

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

   if 6.5000000000000004e117 < y

   1. Initial program 73.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto y \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 58.3%

       \[\leadsto y \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 350000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+117}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;1 \cdot y\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.5e-18) (* 1.0 y) (if (<= y 6.2e-8) (/ x z) (* 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e-18) {
		tmp = 1.0 * y;
	} else if (y <= 6.2e-8) {
		tmp = x / z;
	} else {
		tmp = 1.0 * 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 <= (-1.5d-18)) then
        tmp = 1.0d0 * y
    else if (y <= 6.2d-8) then
        tmp = x / z
    else
        tmp = 1.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e-18) {
		tmp = 1.0 * y;
	} else if (y <= 6.2e-8) {
		tmp = x / z;
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.5e-18:
		tmp = 1.0 * y
	elif y <= 6.2e-8:
		tmp = x / z
	else:
		tmp = 1.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.5e-18)
		tmp = Float64(1.0 * y);
	elseif (y <= 6.2e-8)
		tmp = Float64(x / z);
	else
		tmp = Float64(1.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.5e-18)
		tmp = 1.0 * y;
	elseif (y <= 6.2e-8)
		tmp = x / z;
	else
		tmp = 1.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.5e-18], N[(1.0 * y), $MachinePrecision], If[LessEqual[y, 6.2e-8], N[(x / z), $MachinePrecision], N[(1.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.49999999999999991e-18 or 6.2e-8 < y

   1. Initial program 81.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto y \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 50.6%

       \[\leadsto y \cdot 1 \]

   if -1.49999999999999991e-18 < y < 6.2e-8

   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. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-18}:\\ \;\;\;\;1 \cdot y\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{z}, 1, y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (/ x z) 1.0 y))

C

double code(double x, double y, double z) {
	return fma((x / z), 1.0, y);
}

Julia

function code(x, y, z)
	return fma(Float64(x / z), 1.0, y)
end

Wolfram

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

Derivation

1. Initial program 90.4%

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

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

   2. mul-1-neg N/A

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

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

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

   4. associate-/l* N/A

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

   5. mul-1-neg N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 79.8%

   \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 1, y\right) \]
9. Add Preprocessing

Alternative 7: 41.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 1.0 y))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(1.0 * y), $MachinePrecision]

Derivation

1. Initial program 90.4%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 54.4

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

5. Applied rewrites 54.4%

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

7. Applied rewrites 65.8%

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

8. Taylor expanded in x around 0

   \[\leadsto y \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 37.9%

    \[\leadsto y \cdot 1 \]

11. Final simplification 37.9%

    \[\leadsto 1 \cdot y \]
12. Add Preprocessing

Developer Target 1: 94.5% 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 2024332 
(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))